Welcome traders to another tick mill earnings season preview with me, Patrick Munnally, before we jump into today's report, as always, want to adhere to the risk disclaimer. And most pertinent to today's presentation is the fact that the views and opinions expressed by me are solely mine, they're not addictive or representative of those held by tick mill UK or tick mill Europe limited. Okay, let's jump into today's report. We are looking at Netflix. Netflix are set to announce earnings after the close of New York trade today. The earnings per share estimates is 55 cents on revenue of 7.86 billion. Now, I would say there is a whisper number on the street that the EPS could come in as high as 65 cents. Since its last earnings report, Netflix has continued to a decent run of share out performance on the back of investor confidence that's twin initiatives of launching an ad supported tier and cracking down on possible sharing will reignite higher levels of net revenue growth for 2023-2024. In addition, Netflix has returned to compelling content launches in the back half of 2022, after a mixture of net addition pull forward and the pandemic coupled with content production disruption. On the ad supported initiative, the market debate is really around whether the market supports the Netflix management's own narrative of crawling, walking and then running. However, we did get press reports in December that the initial response to this initiative was muted. The opportunity set for Netflix is the expectation that a host of large scale brands advertisers will adopt the offering, but its current framework of large minimum upfront commitment and the above industry pricing and limited metrics for measuring success could cap the advertising dollar opportunity, absent a wider scale of users and greater ability to measure attribution. In addition, concerns that additional subscriber offerings could cause a dial down into lower price plans by users in any potential consumer recession over the next six to 12 months also weigh on the forward outlook. On the pathway sharing initiatives, markets continue to monitor how Netflix management will actually implement and execute on such an initiative in 2023. In terms of tonight's report, there is the potential for Netflix to surprise to the upside on subscriber performance based on survey data, leaving the majority of forward subscriber estimates largely unchanged. In terms of the initiatives I've discussed, working assumption is that they act as a potential catalyst in 2023 in terms of changing that subscriber growth dynamic, but in just the last quarter, the FX landscape has shifted significantly with the US dollar performance having a start reversal and down 7% in the last quarter of last year since Netflix last reported. As such, adjustments are going to be made to the operating income estimates for the following quarters. Looking ahead into 2023, concerns around the impact of a consumer recession as well as heightened levels of competition and demand trends and margin expansion and levels of content spend and a view that Netflix is becoming a bit of a show me story with a relatively night catalyst part in the next six to 12 months on new product initiatives are all concerns for investors. Upside catalysts come in the form of the potential for subscriber growth to surprise to the upside. Price increases with a minimal churn resulting in higher operating estimates would be well received. Better than expected cost savings coupled with pricing power and a slow down on content spend resulting in higher margins versus market estimates and faster than expected execution on both that password sharing and ad supported initiative. An industry competition abating and an impact on subscriber growth and more original content and better consumer attention will all be positive for the stock. Let's jump into some of the statistical trading patterns we see around Netflix earnings. Netflix shares have actually moved lower in the immediate aftermath of earnings nine out of the 12 previous reports. On average, the stock has actually moved down next to 4.4% in the first day of trading after the company reported earnings. Based on the previous 12 earnings releases, Netflix is more likely to trade higher one day after earnings just shy of 1%. The average gain of 0.7%. On average, the stock has moved lower by 0.8% one week after earnings. Moving to the analyst community, Netflix remains a buy, is no longer a strong buy though and we are looking at a broad range here in terms of price forecast for the next 12 months. On the top side, capping out at about $405 on the downside 215 with an average of 319 as the level for the 12 months ahead. Looking at options blow and investor sentiment going into the company's earnings release has 46% expecting an earnings beat. Notable options activity has specific interest in the $325 calls expiring tomorrow Friday. However, options order flow sentiment in general has been bearish with put to call ratio currently 1 to 2, 42% calls, 58% puts. Expected move post earnings based on implied volatility gives about a 9.4% move versus an average of actual earnings move of 10.6% in recent quarters. The options market however has overestimated the Netflix stock earnings move 58% at the time given the last 12 quarters of data. Let's pull up a chart here and see if we can identify any near-term trading opportunities with respect to stock. I'm using the daily timeframe here, obviously we have this five-way decline. Last time out we were looking for a break of this congestion zone here, looking for a move through the 250 level to target 301. That move played out nicely and hit the target post last earnings. Heading into today's earnings report, let's look at some key trade levels and opportunities. So if the bearish sentiment in terms of those those about higher put ratio in the market at the moment plays out and the earnings come in in line or below expectations, I'd be looking for any pullbacks into the initial support here coming in at the 290 level. Watch here for bullish reverse patterns on the daily timeframe to engage on the long side, but I'd actually be tugging and then move up into the yearly pivot here at 354 on the upside. Also that comes in with the objective of sending trend channel resistance as well. If we don't find support in the low 290s, then I'd look for a deeper corrective move into the main trend channel support coming in at that $265 level and then prior resistance also could access support 255. So that would be another area, I'd be paying close attention on the daily timeframe looking for bullish reversal patterns again to engage on the long side, looking for that move up into our low 350s yearly pivot target. Now if as some are anticipating we could see a potential pop in subscriber growth numbers of Netflix tonight, I have actually when I was scanning the options chains noted that there was some interest in the 390 call expiring tomorrow and so if that's if we do get that upside surprise, I'd be looking for any daily close through the 350 yearly pivot to target and move up into our prior technical wave 4 high at that $400 mark and you'll remember from the analyst community that was the upper end of expectations heading into the year ahead. So I'd be looking to take some profits there if that move plays out alternatively on the downside, a really negative report would see us take out that 250 sport, I'd be deploying short positions then looking for a move back down into that 211 we will see that prior support zone before we got this recent move to the upside also now marks the high volume note there at 225 really any close below there will be a dramatically bearish development opening move back down to test the price cycle lows in the low 160s. If you have any questions or comments feel free to email me patrick dot monthly at tickmail partners.com as always trade as plan the trade trade the plan and most importantly manage your risk until next time thanks very much


Big zone Here's Devin Farrell Farrell known for his outside shooting but takes it inside to score the first seven points of the game So let's hold it a little bit Maybe one or two counts longer than they normally would you got to pop that ball And like you said that makes the defense shift and you have those those gaps when you do move it It is two straight times down the floor though that they've scored But an answer after the free throws with a three ball from the left side simian cuddle from Atlanta Tough for the Northeast on offense right now is the three-pointer was just missed by Thomas Whalen up top Devin Farrell He drives and he slithers through the defense gets all the way to the cup to score It's a 20 to 6 first quarter so far for the Southeast team You don't want to get to where it's a 30-point game and it's so far to reach Well on the boys side of the bracket there had been very few lopsided results so far The Northeast was part of one of them losing their first game of pool play 87-55 There's Devin Farrell the escape dribble on a pretty looking jump shot at the level and they're doing this at such a young age It's great for you. This is the first junior NBA world championship 32 teams coming here to Orlando this week That's a deep three Wow simian cuddle. Hello Okay But they're again they're in rhythm. It is back to ten as McCabe misses on a three-pointer final minute of the third quarter Farrell who just scored Spark swings it coddle for three on the money simian cuddles for three of the game Simian cuddle has a great-looking shot under control either against the central team Drive-five power elite from the Kansas City area or team FOE team family over everything from Dallas, Texas That's the next semi-final the first one is done Devin Farrell 24 points fellows in the southeast defeats the Northeast 73 55


There's disagreement over marriage equality in the Democratic Party platform, with some party members resisting the move. We have a date set for arguments in a new federal case to overturn Nevada's marriage ban. And an Iowa judge who upheld marriage equality has support from across the political spectrum, but is it enough to protect him from a new attack by anti-gay groups? At the American Foundation for Equal Rights, I'm Matt Baume, and welcome to Marriage News Watch for August 20th, 2012. We're one step closer to seeing the freedom to marry in the official Democratic Party platform. Last weekend, the platform committee unanimously voted in favor of the plank. Committee co-chair and Newark Mayor Cory Booker told reporters, quote, the 14th Amendment is very clear, equal protection under the law, unquote. But some party members are resisting the move. Utah gubernatorial candidate Peter Cook cited his Mormon faith as influencing his decision to oppose marriage equality. A Reno judge has set a date for arguments in a case to overturn Nevada's marriage ban. Once it was filed in federal court by Lambda Legal, which will present oral arguments on November 26th. Lambda claims that Nevada's ban creates an arbitrary selective bar, which violates the Equal Protection Clause of the 14th Amendment. There's a judicial showdown shaping up in Iowa, where anti-gay activists have launched a campaign to unseat a judge who overturned the state's marriage equality ban in 2009. Iowa Supreme Court Justice David Wiggins joined his colleagues in unanimously upholding the freedom to marry. He now faces a retention vote in November. But despite the anti-gay backlash, Wiggins is supported by a broad coalition of Democrats and Republicans. That's not the only big election battle on the horizon. The organization Freedom to Marry has raised $3 million for state ballot measures two months ahead of a deadline that they established last March. At least a third of that money came from Republican donors, and the funds will go towards races in Maine, Minnesota, and Washington state. That's in addition to a million dollars that was raised by the Human Rights Campaign. Pulling in all four battleground states is very close, so this is an extremely important time to get involved with those campaigns, wherever you are. Those are the headlines. Visit AFER.org for more on the fight to overturn Prop 8. At the American Foundation for Equal Rights, I'm Matt Baume. We'll see you next week.


I think it's really important that people just are aware that this exists, that social media and the internet has facilitated access to children that didn't exist before. Previously I guess all you needed to worry about was just making sure your kids were safe while they were playing in the park, but now there is concern about their safety even within their own house, at their friends' places, at schools. Wherever they take their mobile device now, they are accessible. Carly was groomed online by an internet predator who had over 200 fake profiles he was using to lure young people. He used his own son to be the profile of a young boy and proceeded to establish a relationship with her online over an 18 month period and went to an awful lot of trouble to manipulate and deceive. And basically we were no match for a criminal like this. Unfortunately convinced Carly to meet him alone and lure her to her death in 2007. Twenty years ago, well 1996, my husband and my daughter, she was seven at the time or just about to turn seven, went to New Zealand on a holiday with his, to visit my husband's father and the wider family. We were very close, his father was very special to all of us.  He did offend against our daughter, she told my husband thankfully and of course we went to police which then brought unstuck 40 years of offending against two generations of women. We've had extensive and long relationships with victim advocacy groups. It's critical, it's critical to the success that we have as well and they provide a support service to help people work through and in some cases disclose what's happened to them. There are I guess a less intimidating agency to go to than a law enforcement body so we've fostered and maintained really good relationships with the likes of Brave Hearts and Carly Ryan. There's just so many children and youth out there that are so willing and open to share every aspect of their lives online, so trusting and so therefore become really good targets for criminals looking to take advantage of the innocence and that trust and that's exactly what happened to Carly. When I got on the internet I looked at what happens to these kids and we know they're committing suicide. They're lying in gutters with syringes up their arms.  You know all of this and then you look and you think but it's preventable. A lot of the kids that we do with can't even comprehend that there would be people like this perusing the internet trying to take advantage of young people and when they find out or when we help young people connect people with police they're often so shocked that the person at the other end wasn't who they thought they were or they were taken advantage of in a negative way. So you know we are also really a connector between community and police as well because often kids don't feel comfortable to come to somebody in authority so often we provide that at Bridge and we say to kids it's okay, the police are there to help you, your counsellor is there to help you, there are people in your community there that can assist you and you need to, all of us at some point in our lives need to reach out for that help and to take it and to try not to deal with these problems online alone. For us it's really good that somebody else is pushing the message out as well it's not just us waving the big scary stick, they're talking about it as well. They're a good port of call just to have that conversation. It affects one in five of our children, 59,000 children every year. This is an absolute scourge and it's a crisis facing our kids and we can prevent it if we just pull our head out of the sand and just look it in the eye and deal with it and that's what brave hearts are fighting to do and you know we've got amazing people around us like Amtas, Argos, Queensland Police, Police Forces right across the country, Federal Police they put themselves in the firing line every day they go out there, they're dragging children out of these horrendous situations it's an enormous toll, personal toll on them and emotional toll but they are driven to protect our as a community our children and we just really need to respect that and just applaud that you know we're all on mission to make sure our kids are safe. And if you are concerned that something might have happened to your child or even if something did happen to you there are very caring support services that can actually sit down and have those conversations without considering going to the level of a prosecution and all of that what that might face they might help you work through the issues and take you that way and support you through that if that's the way you want to go.


Hey everybody, what's going on? I am Greg Susman and welcome to a very special edition of the Fan Del Harriette. Today I'm joined by Joe Ranieri of Sports Grid as he takes a look at some betting 101 tips that'll help you become a better wagerer. Joe, how are you, sir? I'm excited. It's a good time of year. It's a fun time of year right now. Absolutely. So much going on, so much money for you to win as long as you're smarter about it. So let's take a look at just exactly how you place a bet. We'll begin with placing a bet against the spread. What does that mean? So a spread bet is just simply a bet against the margin of victory. That's really what a spread bet is. For instance, if you had, oh I don't know, the Kansas City Chiefs taking on the San Francisco 49ers and they were a three-point favorite, then as long as they won by more than three points, that being the Kansas City Chiefs, that's a winning bet. Or the San Francisco Giants, San Francisco 49ers rather, not losing by at least three or more points, well that would be a winning bet as well. So you're basically betting simply on the margin of victory. If it happens to tie and let's say they win by just three, that's what's called a push. And that means there's no action and your ticket will be refunded. That's how you bet against the spread and that's how you'll make some money. Up next, Joe, what is the difference between a bet against the spread and a bet on the money line? So betting on money line is the most common way to bet, certainly on, let's say, baseball and the NHL, where the margin of victory does not necessarily, when the spreads don't come into play nearly as much, you are simply betting on who you think is going to win the game. And the bigger the favorite, the more money it's going to cost you to back that favorite. And of course, the underdogs, we all know, in sports betting, especially if you're new, if a team is an underdog, chances are there's a little bit more money to be made if they win. So a money line bet is simply betting on who's going to win straight up. On the money line, all you got to do is pick the winner straight up and you'll be a winner. All right, Joe, up next for us, we have an over or under. We see a game total, but what does over under mean? Basically the same thing. The total or over under, like you said, Greg, is simply how many runs or points are going to be scored combined in the contest. So in football, how many points are the Giants going to score? How many points are the Redskins going to score? Is it going to be over whatever that point total is, or is it going to be under what that point total is? So when it comes to totals and over unders, you're simply trying to figure out that number of the amount of points or goals or runs or whatever it is. How many are going to be scored and whether you think there's going to be more or less than that? You're giving a total. Do you believe the total will be higher or lower? That's how you bet on an over under in today's market. Joe, what is a straight bet? Straight bets, you see these a lot when it comes to prop bets or proposition bets, where you are betting something outside of the, like we just said, the spread, the side, the total. You're betting something else. Like for instance, how many yards will Patrick Mahomes throw for? Will it be over 300 yards or under 300 yards? How many rushing yards will somebody have? How many points will an NBA player score? All of these types of things are very straightforward bets that have to do with that individual player or that individual team that's outside of a side or a total. So no winner or loser in these bets, straight bets when it comes to proposition bets, that's simply on making the right play for that particular prop bet. Just helps you out making a prop bet, understanding what you have to wager in order to win money. That is a straight bet. Now you combine all of these bets, you get what's called a parlay. Why don't you explain that here, Joe? So parlays allow you to combine multiple wagers into one bet. Now these are often much better odds if you hit it. So a parlay is like, let's say for instance, tonight I like the Brooklyn Nets, I like the Miami Heat, and I like the Lakers. And let's say I want to combine all three of those into a parlay. Well that will pay seven to one, eight to one, depending on the book. But more importantly guys, each one of those legs has to actually win. So it would be great. There's a lot of money to be had in a parlay if you pick each individual game correctly. You have one loser, then you've got a losing parlay. And you can win more money when you combine multiple games into a parlay and hit them all right. Rather than betting one each game by themselves, a parlay combines it all, gives you an opportunity to win more money. And then we have what's called a teaser, which is similar to a parlay, but just a little bit different. What's that, Joe? So teaser betting involves two or more different point spreads or total wagers together. So for instance, again, we'll use the Kansas City Chiefs. Let's say the Kansas City Chiefs were a 10 point favorite going up against the San Francisco 49ers. I could let's say take a six point teaser and take that number down from 10 to minus four. And then I can do the same thing for the San Francisco 49ers is take that plus 10 and make it a plus 16. So it allows you to go up or down however many points, six point teasers, seven point teasers. You take that amount of points and you just add or you subtract it to whatever the total happens to be or the spread happens to be for that game. But much like a parlay, you need every bet to hit in order for you to cash. Bigger payouts, little more risk. That's right. Bigger payouts here when you combine all of these games, but there is the risk there. And of course, when you're teasing games up or down, the amount of money you can win changes as well. Make sure you understand that when you're placing your bets. That's going to do it for us here on the Fandal Harrier. We hope you learned a little bit more about wagering and betting. And hopefully you'll win some money soon. Joe Ranieri, thank you so much for joining me. Appreciate it, Greg. Good luck, guys. Tomorrow we'll be joined by Jim Sonnis and Davis Maddock, because we take a look at the PGA Tour here for this week. Have a great night. We'll see you tomorrow.


So let's take a look at the addition and subtraction of radicals when the radicands are different. So remember that we can add and subtract radicals if the radicands are the same. So five square roots of three and three more square roots of three will give us eight square roots of three. Or if you have eight square roots of two and you remove five of the square roots of two, then you'll have three square roots of two left over. But what if the radicands are different? So for example, let's say we want to simplify square root seventy-five plus square root of twenty-seven. So the radicands are not the same, so we can't add these two directly. We have to simplify them a little bit. And so we can try to simplify them. So remember that when simplifying a radical, a factor only matters if it's a perfect square. So we can try to remove the square factors. So square root of seventy-five, well I can factor seventy-five as square root twenty-five times three. But wait, the square root of a product of positive numbers is the same as the product of the square roots of the numbers. So we can rewrite this square root of twenty-five times three as square root twenty-five times the square root of three, and the square root of twenty-five is five. So this becomes five square roots of three. Similarly, square root of 27, I can factor this as square root of 9 times 3. The square root of a product is the product of the square roots, so this becomes square root of 9 times square root of 3. And square root of 9 is 3, so we'll replace it, and everything else stays the same. Square root of 27 is 3 square roots of 3. Equals means replaceable, so instead of square root 75, I'll write 5 square roots of 3. Instead of square root of 27, I'll write 3 square roots of 3. And now our radicans are the same. They're both square roots of 3, so I can add 5 and 3 square roots of 3, our total of 8 square roots of 3. How about a subtraction? Once again, the radicans are different, so we can't do anything immediately. But we can try to simplify. So square root of 150 is square root of 25 times 6, which we can simplify, to 5 square roots of 6. Square root of 96 is square root of 16 times 6, which is square root 16 times square root of 6, which we can simplify, to 4 square roots of 6. Equals means replaceable, so instead of square root 150, I'll write 5 root 6. Instead of square root of 96, I'll write 4 square roots of 6. And so I have 5 square roots of 6, and I'm going to remove 4 square roots of 6. So I'll have 5 minus 4 square roots of 6, otherwise known as 1 square root of 6.


Okay, good morning, yesterday we concluded the lesson with this important relation. If z is x plus i y, e z is e to the power x time cos y plus i sin y, okay? Which is also e i y, okay? This is Euler relation which is very important one. This tells us that the complex exponential is in fact periodic, something new compared to real exponential. In fact, if you take e to the z plus 2 pi i, you obtain that this is e x plus i y plus 2 pi and this is the same as e to the power z. This is easily seen directly using this relationship since cos and sinus are both functions which are periodic of period 2 pi. So that in some sense what happens here, this is the, imagine this is the gas plane and take this sub domain is repeated in strips, horizontal strips, okay? Furthermore, as I noticed last time, the exponential of z is never zero and actually this becomes evident when you consider the modulus of e to the power z which is e to the power x, all right? So if you use the polar coordinates, this is the modulus and this is, so e to the power x is raw, the modulus and y is the argument of them in the exponent, in the polar representation. Then from a plus b equal to e i times e b which was proved yesterday, we also have that we can obtain conditions for the sin and cosine sum and difference of angles, right, of complex numbers. So you actually are asked to prove the following, okay? Which clearly resemble the real case but they have to be somehow verified. I suggest you to use, to heavily use this relation without going into the definition of cos and sin. But also the relation with the cos, the complex cos and the complex sin with the exponential. So that we have found, we have already established this cos z is one half e i z plus e minus i z, correct? So use this and you conclude this. So this is another exercise for you, okay? Of course when you have sin and cosine it's natural to define by analogy with a real case the tangent can use this or tan of z to be the ratio of sin and cosine of z, good? And also for the tangent function, complex tangent function, there are similar expressions for the sum and for the difference of c w or a b whatever. So without wasting our time in these calculations, let us focus our attention now to possible vertebility of this value. So what we know from the real k is that the exponential is an inverted, the real exponential is a monotone increasing function or monotone decreasing function, this exponential we are considering. So the extension of the real case is monotone increasing because e is greater than 1, right? In any case it is invertible and the inverse function is called logarithm. So now if we try to do the same we want to solve an equation like this in the complex case. So given w, find z with the property that e to the power z is w. Of course w cannot be chosen arbitrarily because w is zero but we already know that there is no z with such a property. However if w is not zero we can use the expression we have found so far and maybe w can be also written in polar coordinate like this but theta is the arc of w which makes our research much easier because here we have on the left hand side that this expression is e x times goes to y plus i sin y so by comparison we obtain the following that modulus of w is e x where e x is the standard real exponential because x is a real number and here the exponential residue on the real numbers coincides with the real exponential. So we can solve this problem until we are considering w different from zero. So x is the logarithm, real logarithm I put R here of modulus of w and it is completing meaningful because w is not zero. The problem now is related to the determination of the arc or vice versa determination of w because this is not uniquely this is not uniquely determined because as we already observed this depends on the this is this depends on a periodicity of 2 pi. So if we want we can say well okay take w to be arc sorry take y to be the arc of w but this implies that the inverse function of this is what we are looking for so the possibility of defining an inverse function for the complex exponential depends on the choice of the determination of the angle okay so in general this is not an equation because this depends on the determination of arc correct. So there are infinitely many possibilities to define logarithm complex logarithm so in any case you solve this equation this different this difficulty is related to the definition of to the uniqueness of the representation in polar coordinates depends on infinitely many choices of the arc up to a constant times 2 pi. Similarly we have infinitely many functions it's a multivalent function in general unless you restrict your choice and in this case there is a determination of a branch of the logarithm okay. So the complex logarithm one of possibility is to find as follows so w is logarithm as I determine the arc so as for the arc there is no unique determination so for logarithm complex logarithm. However as I said if this is not a multivalent function sorry if this if a choice of the interval where the arc the range of the arc is assumed to be so we can also prove the following fact take the logarithm of a times b what is this this is should be and by to verify this also comes from the relationship found yesterday eab a plus b is equal to ea times eb okay. And finally oh yes sure let me observe observed that from cos z one half e i c plus e minus c we also have possibility to invert cosine complex cosine call this number w and look for z such such okay as before w is given you look for a possible z such that this is true cos z is equal to w if this is the case we normally define this function to be arc course okay like arc complex arc course like in the real case alright what do we have here notice that here we have the opposite of e i c sorry of the opposite of i z so that this is the inverse of e i c therefore on the left hand side so in this expression here we have a quadratic relation of e i c because we have we can also write this in this form e i c squared plus one over e i c correct and this is w so this is quadratic e i c so to have z we have to find e i c and then to look for the the complex logarithm in some sense right okay so from here we should obtain that e i c is w plus or minus square root of w square minus one I invite you to make this calculation these are very easy calculation because well you multiply here to e i c okay and you solve in e i c substitute e i c and well this is the solution which depends of course plus or minus well depends well of course there are two solutions in the complex there are always two solutions okay but look this number here so at this way w plus square root of w square plus minus one sorry and w minus turn out to be one the reciprocal of the other do you see this well multiply this number times this okay yeah w square minus this square okay so w square cancelled minus one okay so these two numbers satisfy this relation yes sure thank you yeah yeah okay so from this we have e i z as minus one and hence as we are looking for z these what i z is the logarithm complex logarithm of this number so times i you have if you multiply by time by by i have minus z is i log okay then minus one then I put minus i log then since this number this pair of numbers are in fact reciprocal the logarithm of the reciprocal is minus the the the logarithm of one over w one over k is minus logarithm of k right so at the very end of the story this becomes minus minus logarithm of the w plus and this defines the function arc cos of w to define arc sine of w we don't have to repeat the same procedure using the other relation with the exponential but remember that this is by over two minor arc cos of w here previous one okay here we have a minus in front and here we have oh sure sure thank you yeah sure i is here sure sorry yes what we do have and it is quite surprising that of course there's no counterpart in the real case is the following all say elementary transcendental function so these functions are the ones you start studying when you are in a course of calculus right and complex analysis are referred or to the inverse of x okay logarithm so the exponential is a very important function like in the in the real case even more I would say okay because in the real case the trigonometric functions are not related to the exponential in any in any in any way okay good and this is one fact so the next task is to continue exploring what is the plan of this part of the course in some sense we have the following say diagram we have defined complex differentiability or sorry or holomorphicity and we have seen that this is guaranteed for power series functions which are convergent of course right so if you define a function in terms of power series expansion convergent you know where it is defined then it is automatically complexity differentiable conflict differentiable or holomorphic and this is equivalent to cr equation satisfied for the real and imaginary part so for u and v now I want to use another another equivalent definition of conflict differentiability and then that then characterize geometric property associated to this characterization okay so this guarantees that the set of completeness rangeable function is not empty now that we have examples and here we have another equivalent condition to guarantee complex differentiability now I give you another one and then study the property and as I said this long somehow long list of equivalency will close up in a in a circle because we will prove that then any complex differential function is in fact complex analytic so it has power series expansion okay so to introduce something new okay let us come back to notation like this and remember that x is one half z plus z bar z bar being the conjugate of and similar y is 1 over 2 i z minus z bar z bar is x minus i y correct okay so yesterday we started by considering a function complex value function and in some sense focusing our attention to the real imaginary part of this function so the components and using the real derivation with respect to x and to y of u and v real and imaginary part so instead of f of z with the some abuse of notation but of course I consider this quite natural consider f depending on x and y and this is u of x y plus i v of x y and of course is also u of z plus i v of z with the same abuse of notation okay but now we have also an expression of x and y in terms of z and z bar what if we take that is to say x depends on z and z bar the derivative of f with respect to z bar of course we are only applying the chain rule right but what is the expression of this new operator it's differential operator right d over d z bar something new okay we apply the observation is correct the remark is correct we apply the chain rule how do we apply the chain rule well we use this we use that well this is the f over dx times dx over d z bar correct plus the f over dy dy over d z bar and we have the expression of this and this oh we can calculate actually this is one half right because from this we calculated the derivative with respect to z bar one half and here we have pardon me i over 2 right in other words we have one half the f x plus i dy the f over dy let me copy it in a new page so this is what we have obtained one half the f dx plus i the f dy if you want to use formalism of operators complex differential operators we can say that d over d z bar is in fact one half dx plus i dy right so you plug in the function f and you see what happens okay the sum of two real operators i okay plus i the second operator well this operator is known as Cauchy Riemann operator or von Neumann operator in some books why is it important well assume that f as a solution that is assume that f over d z bar is zero assume that f is a solution of the problem d z bar d so sometimes it's also indicated by d bar this annotation d bar f equal to zero what does imply why we come back to our standard notation and we observe that the following we have here one half u x times plus y plus i v x plus i v sorry u u y minus v y correct this is the d over d z bar f and since it is zero this has to be equal to this and this is to be equal to this so u x as v y and v and the u y is minus v x so Cauchy Riemann equations and vice versa if Cauchy Riemann equation are satisfied then obviously repeat the inverse path and we obtain that d f over d z bar is equal to zero so in some sense you can characterize or homophicity in terms of Cauchy Riemann equation or solution of the Cauchy Riemann operator okay now this important fact I will summarize it in a proposition f is holomorphic of course when I'm not very precise and I omit to say at z naught okay if I say holomorphic and I don't say where I mean either at one point which means in a neighborhood actually of this point so in that at that point if you restrict the Cauchy Riemann equations or you take the limit of the incremental ratio and so on and so forth everything is defined locally okay so f is holomorphic and well at z naught if and only if f satisfies and here I mean what is this I take I take I have taken the derivative with respect to z bar and found an expression for this complex differential operator which is related to Cauchy Riemann equation so I invite you to prove this is an exercise and just the exercise that formally this is the conjugate of the d bar operator so this is also indicated by df and the exercise is not this but here okay one half df over dx minus i df over dy that's why it's the use of the contrary so d bar is the other one because it reminds that you are differentiating with respect to z bar and here we are differentiating with respect to z okay but well you see that the imaginary part changes its sign so this is the exercise and another exercise is the following assume f is holomorphic because of course this can be defined for any function in any complex value function it has meaning okay so assume that f is holomorphic which means that d bar f is zero so if this is the case df of this is f primacy and this is another exercise for you so please I invite you to use the notation of f in terms of real imaginary part and use Cauchy Riemann equation this is one easy way to solve the exercise so first find this and then substitute f with u plus IV right expression using this operator use Cauchy Riemann equations and conclude that in fact this is ux plus IVx it is the derivative okay so the two operators d and d bar are very important complex analysis because they provide you well as solutions in one case d bar f equal to zero a solution which are the class of holomorphic complex differentiable functions and the other operator gives you the derivative of complex differential functions if you are restricting your operator to the solutions another interesting exercise is to calculate this you apply first the derivative with respect to z bar and then the derivative with respect to z or you compose these two operators so I think that you have enough exercise not for tomorrow not for Friday but for next Friday say no no not for the day after tomorrow but for Friday the first Friday of October okay so I will however collect the exercises and send you the list of exercises for okay just but in case you have time you want to think about and just if you well it's a good idea to repeat this stuff by making some calculation this is a very simple calculation I invite you to to well and by the way this is found in any textbook of the subject do you have just a vague idea of the answer to this question well it it is not sure to think of a second order of operator so up to a constant in fact this is the plush okay well enjoy your yourself with the calculation good now with this basic fact recalled let me continue by giving you some other interpretation of complex differential ability one is the following so again we consider this complex value function f to be to have components u and v okay and if we're if we use the our glasses in the in the real sense okay this is a function from an open set of r2 into r2 so typically what you do if you know some regularity of the components you look for the expression of the differential right and the differential is calculated like this the first component the first component is the derivative of first component with respect to x then the first component with y the second component so first component first first variable first component second variable so gives you exactly the position and then do by two matrix all right of course this then has to be evaluated for z equal to z naught or x naught plus i y naught the pair x naught y okay but in general we have this but assume that f is holomorphic all right at z naught so your colleague is suggesting me that well this is simply the modules of the derivative at z naught the module square right you said this what is okay what is the assumption we are taking f is holomorphic at z naught so that we can replace using the koshiriman equation instead of u x for instance you put y and u y minus v x so for instance i substitute u y with minus v x and v x here instead of v y i put u x okay and these two matrices are the same because we have this condition when you take the determinant the determinant is u x square plus v x squared so okay the determinant of the matrix u x minus v x v x u u x is u x squared plus v x square which is precise the modulus squared of f prime at z naught of course this is in sequence and this number turns out to be of course no negative it is zero if and only if the derivative of f is zero otherwise it is strictly positive and this has an interpretation and real analysis do you know the interpretation of the fact that the determinant no but i'm only talking about the the the the fact that the determinant is positive let's say not not not negative to be correct so when well i'll assume that the function the holomorphic function we are considering has derivative with never vanishes do you have a do you have an example do you have an example of holomorphic function with the derivative which never vanishes we do have we do have the exponential the exponential right well good so or restricts to your function to the set where the derivative is never vanishing in this case when consider as a function from our art from an open set over two into our two this function is orientation preserving because the derivative of the differential is positive sorry derivative the determinant sorry the determinant of the yes yes conformal maximum mappings are in fact somehow related to this but all this is not sufficient it's just orientation from serving conformal mark we mean nothing comes out from from something we are we are going to see in in a minute so conformality is a property of preserving angles here we are just talking about preserving orientation not angles okay not the amplitude of the angles okay so this is another geometry if you want this is the first geometric consequence of an analytic property of complex differential functions so complex differential functions turn out to be orientation preserving or well the rotation not determined because the derivative is zero right now I want to investigate in fact conform conformality and I'll do it making this very very simple considerations now let us make a kind of picture of the situation we have an open set u and the plane where f is defined and assumed that f is holomorphic so take z naught you take a curve passing through the knot so gamma is a curve for the sake of simplicity well okay take minus i a okay real number at the time zero well we can also take the open interval minus a a in r at the time zero the curve passes through z naught okay so and then I consider another curve call it w of t to be f of gamma of t this is well defined because gamma takes its values in u and f is defining u correct in order to prove conformality we have to talk about tangent vectors to curves tangent vectors or say vectors passing through a point applied to a point so the natural thing to think of is well I take this gamma prime of t naught and since I'm I want to have a vector here I note the zero vector I may assume that gamma prime of zero is not zero okay so I take a curve passing at time zero through the knot with velocity velocity different from zero okay this is the physical interpretation I'm assuming that derivative is different from zero in zero now with this assumption we have that this is another path is a curve of path okay w is a function from minus a a into c and w of zero is f of z naught what about the velocity of this curve at the time zero well we use a chain rule because well we can first I'm saying this is c one it has some derivative otherwise we are talking about nuts we assume that this is not zero because we want to have the tangent vector but we have to have we have to require that the function has a derivative first okay so it has a derivative and the derivative is not zero so if f is holomorphic then automatically we know that it has also it is complex differential also it has a derivative so w is complex differential and the derivative at zero is f prime of z naught times gamma prime zero right remember that gamma of zero is z naught so in fact this is gamma of zero and you are we are using the chain rule precisely the chain rule okay therefore we have gamma prime of zero and double prime zero being f prime of z naught once again we are interested in consideration about angles of vector so we are not interested in vectors which can be that zero vector so we already assumed that gamma prime of zero is different from zero but this does not guarantee that w prime of zero is different from zero because this is not zero but this can't be zero so we have to assume that the derivative is not zero for the function f at z naught all right so if if we have both conditions so function holomorphic at z naught the derivative at z naught is different from zero then we can say w w prime of zero is different from zero and to see conformality we have to measure the angle what is the tool we have to measure an angle the argument the argument of a complex number gives you the angle with respect to the real axis now what is the arc of w prime of zero well I cannot calculate I don't have any information but I can say that this is the arc of f prime at z naught times gamma prime of zero and remember that one of the property of the arc is that the arc of the product is the sum of the arc so this is the arc of f prime of z naught plus the arc of gamma prime of zero which is a very interesting interpretation so the difference of the two angles depends only on the value of the arc the two angles of the two vectors we have vector tangent to gamma to sorry tangent to z naught to f of z naught corresponding to the the velocity vector of the curve gamma and of the curve w the angles between them is the difference of the two angles depends only on this which is independent of the choice of gamma you say this well arc of w prime at zero minus arc of gamma prime at zero is arc of f prime of z naught arc of f prime of z naught is independent of the choice of gamma which means that if you take another curve passing with the same speed at the same time through z naught and then you consider the image using the function f holomorphic at z naught whose derivative z naught is not zero nothing changes about the arc of the the curve the image curve okay in other words we have the following so arc this difference is independent gamma now some consequences take two curves passing through z naught and having two vectors gamma prime of zero and alpha prime of zero okay the corresponding image curves both pass through f of z naught and if f prime of z naught is different from zero we have two vectors here one is as I said w prime of zero and here maybe I use another another letter is better prime of zero while you consider the differences and the differences between the arcs is the same which means that the function f is conform so they preserve angles not so only the amplitude of the angles not the they're not isometrics they are just preserving angles and conformity is a very important property in geometry and in the past it was extremely important for for sailors huh you know these famous cartography results well just let let us spend two minutes to say something it's not possible to represent the sphere or a portion of a sphere on a on a portion of a plane because of curvature reasons okay this is a result known and well it was um empirically it was known quite far away for quite a quite a long ago but it was proved definitely proved by Gauss if you want to keep an isometry between two surfaces well then curvature is preserved it's not possible well the only possibility is to use a sphere with the same if the if earth is considered as a sphere you cannot even represent isometrically a sphere with another sphere with a different radius so you have to use the same radius so it's not very useful okay as a cartography tool a sphere which is as large as the entire but you can have good conformal representation so representation of two surfaces preserving angles between points between so between vectors and this is the case for many many maps used by sailors the famous representation is the following so this is assume that this is earth this is the equator okay the idea was the following take a cylinder wrapping the equator and take a projection from the center of earth and at any point of of earth and project it on the cylinder so of course this is not an isometric representation but this angle preserving so it is conformal see any any point on the same with the same longitude is on the line and as you can see here the presentation is very good but when you move to the poles north or south poles actually become distorted terribly but for sailors it was not important because they couldn't even sail near the pole so they were interested in in a band around the equator this is the the so-called projection of Mercatore Kauffman it's very important and even today for for sailing it is important to have the angles instead of using any other indication angles with respect to some reference line can be the meridians or sorry meridians or the equator or latitudes right so knowing these angles you can make a you can you can say okay and using of course some GPS system as a system okay good so it turns out that conformal mappings are sorry that holomorphic mappings are conformal good now what about the the embers of this so let us consider the up the this this problem to characterize conformal mapping mappings uh complex value to conformal mappings okay i take f from u again into c and i assume some regularity assume f as components which are c1 i cannot make any consideration in terms of derivatives without this this assumption and then i notice that gamma of t can be also represented in this way and the plane or with some abuse of notation with a pair of real numbers x of t y of t which represent respectively the first and the second order in the plane so we are assuming that well well this function here is such that gamma of zero is x zero plus i zero is z naught as before and i also denote this to be x naught plus i y naught okay um and and moreover i'm taking f to have components so u and v who's derivative with respect to x and y are continuous now i take w of t which is f of gamma t so i'm restricting my function along the curve gamma and looking for the curve coming out from this composition when i have to consider w prime at t this is and this is as i said a function of t in this way right once again i use the chain rule you already observed this and i have that this is df over dx times at t at sorry at the point x y which i probably better omit to write times x prime plus the derivative of f with respect to y times one prime of t correct this is real analysis real calculus now i have x t y t and then i define sorry i use gamma prime of t is x prime of t plus i y prime of t and this is in fact what parameterize our variable in u it is like z of t right and gamma prime bar is x prime of t minus i y prime of t correct now keep in mind that we have this we have this w prime of t is i copy it the f over dx x prime of t plus tf over dy y prime of t and that gamma prime of t is sorry i have to copy it x prime of t plus i y prime of t and it's conjugate sorry gamma prime bar of t gamma prime bar of t is the conjugate then i make this one manipulation on this expression all right it's one half df over dx over t and then i put a one half in here and i use well minus i one half the i sorry i df over dy y prime of t plus one half df dx x prime of t and then i have minus i sorry plus i i'm sorry minus i one half i yes sure df dy y prime of t so instead of putting minus and minus i write minus and i squared and i use i a split i squared into i and i it's a trick effect but this becomes okay one half df over dx then i put plus i first minus i df over dy and then i have a gamma prime of t plus one half df dx plus i df dy gamma prime t bar do you see this okay one half and here and i df of dy is here but then i have x prime i y prime with a minus in front minus is here so i y prime and x prime and minus prime from here and the other part the part comes so i invite you if you if you're not convinced by me that prove this verify actually it just a matter of calculation so substitute this and this to this expression improve it okay last effort is a following so as i said this expression represents w prime of t which is one half df dx minus i df dy gamma prime of t plus one half d half dx plus i df dy gamma prime of t bar now you notice that i after these manipulations the operators the differential operator koshiriman operator is as appeared okay here and it's conjugate so i have d bar and d operator here i'm not assuming that the function is holomorphic i'm just making the consideration in general so i have some regularity for the the derivatives and this is the expression of the derivative of the image curve of the tangent of the image curve sorry in terms of the derivative of f of the function with respect to x to y and with and with respect to gamma prime and gamma prime bar what is this this is the difference over the two arcs of course because remember that now because because of this condition right as i said gamma prime of t is not zero so everything here is well defined on the other hand since w prime t as this expression w prime t over gamma prime t turns out to be one half df dx minus i df dy plus one half df dx plus i df dy of the portion gamma prime bar t of gamma prime of t and here comes the observation if angles are preserved by f so this arc the arc of the the portion has to be constant this difference to be constant okay but on the right hand side here i have something which is not constant and if t varies okay this vector here is modules one so it is a vector plus something and the vector of modules one so it is a circle so the arc varies with p but actually well this is not possible unless this number here so the radius of the circle is zero so the only possibility to have the arc preserved so sorry the angle preserved so the difference of the arc the arcs to be constant with respect to t as this number is zero are you with me so this expression here is one thing which does not depend on t plus something times a vector which has modules one remember that gamma prime is not zero and gamma prime bar is not zero as well but the the ratio gamma prime bar over gamma prime is back is a vector of modules one is a unitary vector so we have something plus something else and times a vector of modules one as t varies the only possibility to have this independent of t is to kill this part and if we kill this part then necessarily f okay i write it here so this is so with this in mind i write it on a new on a new slide which is the last slide for today also because i don't have any sheets of paper so in order to have the arc of its constant we have require one half the f over dx plus i the f over dy is zero because the difference of the two arcs is the arc of the quotient and the argument in the quotient as this expression so the arc varies okay unless this is zero so that we conclude that the function turns out to be conformal in the real sense if and only if f is satisfying the bar f equal to zero or Cauchy-Vaman equation or it is holomorphic so this is the important conclusion that's why in complex analysis sometimes to say well complex analysis is it's a topic which covers conformal mappings because in fact those functions which represent the main class studied in complex analysis holomorphic function turn out to be also conform in this in the sense in the classical real analysis sense all right so this gives you also another motivation to study complex analysis if i can because it provides geometric interpretation using analytic definition or calculus tools so it's a good mixture and a good training subject where you can mix your your attitudes in analysis and geometry and as you will see it it always requires some basic knowledge but some knowledge some some basic facts from topology because all sorts of positive will play an important role in complex analysis for instance we would say that a condition for existence of some branch of the lower is related to topological properties of the domain and this is also important to know okay now in the in the very last part of this lesson let me just okay give you kind of plan of the next two lecture lessons i want to introduce very basic fact about complex integration right so in some sense we will not make a general theory of integral functions and so on but we refer complex integration to real integration and use many many properties from the theory which i assume you know so what i will consider in particular will be the integral over a function of two variable real variables along a path in the plane so what is called typically line integral and some of the properties related to function and forms actually for line integrals in particular i assume that you are familiar with the fact that the line integral depends only on the end points of integration if and only if there exists a potential function or equivalently if the integral of a closed path is zero and we will characterize also a normal function using these integrals and then obtain cushy integral formula and with cushy integral formula we will show that any normal function is in fact complex analytic and this is what i consider somehow compulsory for a course in complex analysis to know then there are lots of other theorems of results concerning class of function properties general properties and in particular i want to show you properties of the zeros of homomorphic functions and some singularity for homomorphic functions okay at that stage of the course i believe it will be in two weeks we can decide if you are really bothered of this stuff and you already know everything and what and what follows we can make the big step and start talking about several complex variables okay otherwise we continue with some other topic which can be interesting or not but however more standard but definitely we need cushy integral formula and also in in the case you decide to to have some basic information about the several complex case okay so for the time being please those who already know everything wait for for two weeks okay all right in any case what i i i assure you now is that i'm not going to to stress too much the course about calculations so that in case you are worried about possible exercise in integration and so on well this is not the main part of the course right just i need to have the tools from complex integration to define a formula to define to have a theorem and define and obtain the results from the theory which is essential okay so i stop here and see you next Friday see you in two days right


Gilead, a point, how are you feeling after that? I'd say first half, I'd probably say we weren't really at the levels we needed to be, then second half I think we did well, we created some good chances, a bit frustrated but a point against a good team who were up there and we'll obviously move on to the next game. Not the worst point, I think we come away here and I think they're probably a bit happy with it than we are, obviously we were aware of how big the game was tonight and I think that we can take some comfort from how we played, I think coming off after Saturday I think we were so disappointed in that result because I think we did more than enough to win the game. Coming into tonight, they're one of the form sides in the division, from 2023 onwards I think they've picked up the most points of any side in the league so they're always going to be a tough challenge and I thought we stood up to it well and it was a battle but we did enough, a point was probably fair reflection in the end but yeah better than losing for sure. A bit frustrated, obviously we need to start picking up these points and we've had too many draws but I thought I'm quite up to Carlisle, they made it a really difficult game and made it a bit long like long balls and hard to break down and yeah they frustrated us a lot so we didn't really get as many political chances we had in the last game but yeah we're just frustrated that we didn't create a bit more. Is there a sense of frustration in the dressing room though given the amount of possession and the chances that we had in the second half? Yeah we're disappointed because three points would have been absolutely massive you know it would have really closed the gap on them but as I say if you can't win don't lose and I think we had chances to win the game but you know so did they, you know last second of the game they hit one off the post and that goes in and then we're you know then it's really really disappointed for us but yeah if you can't win don't lose so that was that was more important you know I'm beating record maintains intact but we need to turn these draws into wins really and you know we've got mind games left and we're all aware of how massive everyone of them is and you know I think we've probably got a target of what we need to get to and and that's important that we need to turn these draws into wins. Yeah I think obviously like I said first off I weren't really up to them to them levels the second half I got there created some good chances but one of them games you know if we could have nicked the goal I felt we would have won the game but yeah there's another point in the right direction and we'll move on. Yeah everyone we know what we need to be doing now we need to be picking up points so for the last run of games obviously we think we put ourselves in good position to do so and we haven't so yeah we need to look at that and regroup and really get get better quality at the top end of the pitch obviously like I said credit cut to call out you made it really difficult I don't I can't think of any click on such as I add and we really don't have many so now that they're a good team and and at the end of the day picked up a point against them so we move on. Four clean shoots in the last six now so we could take a lot of pride in that especially when we're pushing for the promotion places. Well that's the thing there you know I would say from the start the season it's probably changed in that you know earlier on I was probably having to make a good save in the game a good save or two but now you know tonight I don't really think I've made many saves you know and I think that's testament to how solid we've been and you know Stubdy's come in and you know we go to the back three tonight and and Roms comes in and you know but there's so many quality defenders here that I'm not really having to do much you know and I've said it for a long time you know the goalkeeper's job was just be there when he's even easier and if the team don't need you that often and you have quite nice and you know I'm not about to complain about that I'm happy four clean sheets out of six you know I've got targets I'm aiming for and and yeah we're a solid outfit the minute I think that's that side of things we've really kicked on it's it's now the other end of the pitch that we need to sort of figure out a bit there. I think you know the lads are showing that we're doing well defensively kept a lot of clean sheets recently but we just need to keep going you know trying to get them goals at the other end. A full 90 minutes for you today how are you feeling after that? Yeah obviously delighted to get back in it's tough obviously not played a lot of late so I'm happy to get back in and just make sure I do everything right and I recover right and body's not used to the 90 minutes as it has been so just like I've been doing everything I can away from the pitch to stay as fit as possible so just to crack on with that and be ready for the next game. We have 10 days rest now I guess it's a sense of regroup and reassess ahead of Grimsby in 10 days time. That's the thing there you know when we come back it's going to be an absolute battle from Grimsby onwards you know it's everything you've got you know you've got to be you've got to give every single thing you have and this is that this is the last weekend where there's no game today in the season so it's time to switch off a little bit relax press the reset button and prepare yourself for a battle as we've said it's nine cup finals nine you know do or die and yeah it's up to manage that so yeah I'll be I'm gonna I'm gonna go home see my mum and press the reset button and then come back ready to go for the last nine so I think just mentally for a lot of the lads it's been a lot of games a lot of time a lot of traveling and so I think a lot there'll be a lot of tired legs in there that'll be happy to get that recovery so for all of us it's a good chance to just relax a bit and and then get the mind off it for a few days and then get right back on it when we come back in and obviously fresh lads fresh minds at it and and then yeah we just go do what we say how to do you know obviously the Gaffer and Glenelgallis on the training pitch and getting into the stuff the one has to do for that Grimsley game you know it's a massive game and I think when it comes around well we'll be ready for it but a good time just to get some rest in the legs you know like I say next week we'll just be full focus on what we need to do in that game on Saturday and like I say the stage of the season we just need to be winning games perfect thank you


okay so today's session is on free body diagram okay so I have taken free body diagram just to you know introduce to you how how can we use the concept of vectors in the free body diagram okay just like you have taken example of projectile motion to understand how vectors can be used to analyze the projectile motion okay similarly I'm going to take free body diagram to make you understand how can we use the concept of vector that we have just learned on to this okay from where we work from like where we will then integral part of this chapter laws of motion okay have you seen that equation force will do mass demonstration okay so we are going to learn how to apply force will do mass time acceleration okay that force is net force okay so we're going to write that equation now just like g is equal to u but then when we use g is equal to u plus a t in projectile motion we have used in x-axis and then long y-axis yes or no right similarly we are going to use force is equal to mass demonstration along x-axis and then long y-axis and if it is required along z-axis also okay but before you would write that equation before you even write that equation you need to do some basic analysis as in what are the forces acting on the body which directions there the forces are acting so that basic analysis what we are doing 90% of that is free body diagram only fine so if you do this properly physics okay because laws of motion is the basic chapter of the physics if you're good at laws of motion your entire physics becomes very easy but if you're bad at laws of motion it becomes extremely difficult fine so you cannot afford to be bad at you know drawing free body diagram or this chapter laws of motion okay so we are going to do this law of motion again when we start class 11 syllabus we'll be doing laws of motion again but right now I'm just introducing to you a section of laws of motion which is free body diagram so that we can understand how can you use vectors there okay but before we even get into free body diagram alright we need to understand what are the forces what are the properties of forces what are the laws which forces or any object in motion follows okay so write down the motion I'm not going to complete this chapter I'm just taking the relevant portion of this chapter which in which we can use the vectors law of motion as a name suggests it is not which an object in motion follows understood if object is under motion that object must follow these laws which we are going to talk about okay now there is the concept of object at rest doesn't exist actually fine so you might like you know any object at rest can be thought of moving with respect to some other object like all these moving you might think that your address you're not okay so when I say laws of motion these laws are also valid for an object which is at rest fine so you might have the motion since from your class 9th or 7th also at times Newton Newton or the person who has formalized these laws in a systematic way and he has put across it as first law of motion second law of motion and third law of motion right can you tell me what is the first law of motion what it is anybody remain in rest but we haven't see the thing is I know you haven't you have this concept of force already you know it but the way I started teaching in class 11 that I'm not assuming you know anything okay I'm not even assuming you know what is force till we introduce what is force let's not use these terms fine so can you define the first law of motion it introduces tendency of an object isn't it it tells you the behavior of an object and what is the object what is the definition of an object anything that has mass anything that has mass first law talks about the behavior of it Newton's first law of motion and I'm just writing first-law motion it talks about behavior what is this behavior all about the basic behavior of anything any mass it doesn't want to change that and that is also like you know for any you don't want anything to be changed in your life also at times okay the same kind of behavior every mass has whatever the mass is doing the mass wants to do it okay but and that is its behavior all right the object right now the object or mass are we using object and mass interchangeably okay object doesn't it's state whatever state the object is in the object wants to be in that state okay now when I say state two kinds of states all right state of rest right on the concept but then you remember you know that when Newton was formulating laws you know he was treating rest an object at rest means the object is at rest with respect to earth we are sitting we are standing on the earth okay we are at rest so that that is what it means object at rest okay so object at rest is a state of rest and state of uniform motion these are the two states so if these two states are there in an object object doesn't want to change are you getting it so you don't have to do anything for example it is like this okay if you do not make any effort object that's what it means okay and if uniform motion means uniform velocity not even direction is changing getting it direction and magnitude should be constant uniform motion means uniform velocity right now direction is unchanged and constant speed okay constant speed motion in a straight line is a uniform velocity okay so if object is moving a straight line you don't need to make any effort in doing so why because that's what if some some object is sliding on the earth support on the floor some object is sliding and then stops after some time why that happens this object is moving velocity V this object will stop some after some time all that wants to move isn't it but why it is stopping because somebody is making effort what should happen with this block they keep on moving the friction a little bit the masses for further move away that tells you the behavior of the mass that you know if ground lessens its effort of stopping this mass or it completely removes its effort then the mass will keep on going fine so this tells the you know behavior of the mass getting it now let's talk examples okay and you have any examples of a day-to-day life where you observe Newton's first law one example I tell you for example where when you have a dust from a drug what you do beat it beat it right so when you beat the rug what happens is that the loose particles of the dust they want to stay wherever the rug suddenly shifts when you beat it the dust particle they fall down this is the example of Newton's first law give me another example like when you're sitting in a like car which I'm which are not move and start suddenly starts because you want to stay in that position but you are moving back now you said because you can't stay there for you are you correct so car moves ahead but your upper body so your lower body is moving with the car but upper body wants to remain there itself okay so that's how you feel as if you're falling back okay what else what are example one example smart phone only last year last week when you play the car I stacked the coin one after the other and you hit the bottom of the coin with a very high velocity what happens to the upper coins they stay like that only and they fall vertically down so the upper coins wants to maintain their position wants to remain at rest also remain as it is okay so that's why they they don't fall down but if you put the striker a little slowly the upper one also falls down why so because then there's like enough time to transmit that force to the top correct so there then the lowest lowest coin gets sufficient time to affect the motion of the upper coins getting it but if you don't give it time what else but if there is no friction suppose between the lowest coin and the upper coin then it doesn't matter yeah what else think of tell you there's so many examples so many don't think of okay tell me one thing suppose one example is this when you when you shake a tree I mean not a branch of a tree better example when you shake a branch of a tree the the fruits which are I mean the fruits which are heavy or which are right they they fall down right but the fruits which are like not right they will not fall down why they want to stay right they want to stay there but then the pump to stay there yes or no but the heavy one wants to stay more that's kind so this behavior these two behavior is stronger for heavier mass density of behavior also I mean these two be here will be definitely there for each and every object but the intensity or behavior is more for heavier mass so this behavior is proportional to the mass yes or no to check this you can think of a light object at rest you can easily make it move but if there is a heavy object when you push it it is difficult to getting it so that tells you that the heavier object has more intense behavior okay and since this behavior is very common across the universe this behavior is okay so we name this behavior and the name of this behavior is inertia so inertia is more for the heavier mass but mass is not inertia mass is a representation of the inertia and you think of another example give me couple of balls get off the bus then so suppose you have this bus so bus is moving don't do that okay but suppose you are out of your mind and you jump okay which side you will fall forward or backward why because you are moving forward when you jump your feet come to rest keep on moving so you fall forward so if you don't want to fall then what you should do keep running keep running in this direction only what will happen then is that your upper body lower body all are moving with the same velocity okay so you should run with the velocity of the bus only for for a couple of seconds okay but if you run with more velocity of velocity of bus you'll fall backwards okay so you should calculate first and then jump this is not safe you might be safe that you you know you're not toppling but from behind some some other vehicle might be coming okay so you have to do a lot of other calculations as well don't know that any other example check out there are stacks of books stacks of books and there's a piece of cloth that is at the bottom you have to pull it very fast one more like they set the plates on the dining table and the dining cloth is there if you pull the dining cloth really fast the plates but the clothes should be you know completely flat and okay then what else that's it I mean let's move forward so there's so many examples of this okay in fact you know how to take motion that we were doing some of my fighter aircraft velocity of the bomb initially same as the ball wants to continue moving with the same velocity with which it was like when it is in the plane getting it so there's so many examples and anyway so when we proceed with this chapter we will come to know ways other examples and last one that is coming in my mind is this suppose an object is moving in a circle circuit motion okay and there's a string attached to it this mass is moving a circle and a string is attached to it suppose the string breaks how the muscle go and gently why tangentially that's because that was its velocity at that point in time it tries to maintain that velocity but if there is a string that way so it's velocity changes and it makes a kind of circle but if string is cut nobody is there to pull and gently it tries to maintain its velocity at that point in time fine so like this you know the first law of motion but you should be more aware of it you know you should know that okay fine I'm using first law of motion fine then you'll feel a lot more in control when you solve numericals getting it down the second law of motion second law of motion is the most important law in the physics class 11th and 12th okay which looks very simple force is equal to mass and excitation but don't go by how simple it looks it will make your life for two years okay so lot of tricky questions can be made out of just application of this simple equation okay so be very watchful and be be very observant of whatever is discussed in this chapter okay so second law of motion tells you that there's a behavior and then in state what is the state two states okay the second law of motion tells you that how much effort you should make to change to change the state it quantifies the amount of what you should be making to change the state of the object here first but if you want to change the state how much effort you should be making that Newton's second law tells you getting it now Newton's second law is force equal to mass immaculation if mass is constant but that's that law is not God has a mass immaculation it is our own creation getting into the power 3 nobody's going to in the simplistic equation possible more simple equation which is force equal to mass immaculation okay but whatever in sync with what is the observation you should not say that force is equal to acceleration divided by mass why because you know that more is the mass more force you need that kind of basic observation must be taken care of so let's talk about few observation then we can arrive at the Newton's second law okay observation number one two masses this is smaller mass this is bigger mass moving with the same velocity moving with the same velocity so where you have to make more efforts to stop the bigger mass simple now since we are talking about the effort let's try to let's give this a name a formal name because effort is a very subjective name fine so we call effort in physics as force okay so going forward when I say for when I say force it quantifies the amount of effort okay so force required the masses more what is the assumption you are stopping both the masses in same time same time okay in two seconds suppose you stop this and in two second you stop that also so effort required here is more okay so from this you can say that force is directly proportional to mass times a you can say mass to the power a isn't it even that is correct a can be one a can be two a can be three but a cannot be minus one minus two understood so this is observation one observation two suppose you have two mass I mean two equal masses this is moving with hundred meter per second it's like a bullet and this is moving with one meter per second which will take more effort to stop of course the higher velocity okay so what the force is proportional to velocity here according to observation what it is velocity you so you can like change with change in velocity change in velocity suppose I change hundred meter per second from hundred to just 99 then it's the same then it is same okay but I am changing more the velocity of suppose I change one meter per second this side to hundred meter per second backwards so the change is 101 getting it so it is changing velocity the force is proportional to yes or no okay so force is proportional to change in velocity to the power b you can say observation three two equal masses two equal velocities this year stopping quickly this year taking time to stop where you take more we have to make more effort here so so lesser time you take to change the velocity more is the is when when a critic catches the ball he does like this he follows the motion of the ball okay so that the effort required by his hand to stop the ball is less okay so is inversely proportional that's a delta t raised to power c but it so these are the three observations which must be true whatever equation you come out of delta t is see t equal to 0 is a hypothetical concept t is equal to 0 you can say the beginning of time or whatever you can say delta t is amount of right now time is can become 10 a.m. 20 minutes so delta t is finite time managed initial time that is delta so that's a common way of writing delta t if I just say t it means why it cannot be one by delta t if you say you're you're making it specific why it should be one by delta t if you say one by delta t then I should ask why one by delta t I am saying that it can be anything b c can be anything one two three four five why it cannot be one by delta t square all you have determined by the observation is if delta t is less four should be more that's all I really you have what it is first of all you don't even know how to calculate force force is an effort you're quantifying the effort but what it is it is creation of our it is our own creation that one Newton is this much so these are the three observation if you combine see you know if any of these three suppose m is 0 what is 0 if delta t is infinite effort is 0 fine so if any of these three if m is 0 this condition happened condition I will be able to write force as proportional to m to the power delta v to the power b divided by delta t raised to power c getting it must be true okay now it is up to us to take whatever a b c you want to take so what we do we assume a simple possible most simple possible relation what is that constant proportionality one b is equal to one c is equal to one fine all right so we have assumed that force which is quantification of effort is m times delta v by delta t and by the way this represents the average force the on an average this much force is required any doubts nothing so where's the constant constant is one you have assumed it to be one you assumed it to be one fine it is up to me to define what is force is equal to find force equal to this m into delta v by delta t nobody can say something is wrong with that but everybody has come to this understanding that let's assume a simplest possible relation which is this getting it this is the average force suppose you want to find out force at that instant or what does it mean instantaneous force you have to find out instantaneous rate of change of velocity which is acceleration understood if you combine everything you will get force is equal to mass times acceleration acceleration is instantaneous rate of change of velocity and these are vector quantities acceleration force they are vector quantities capital M which is mass is a scalar so scalar times vector is another vector force okay now here throughout we have assumed mass is constant getting it mass need not be constant for example rocket it is fired it moves up but force applied is down gravity is acting down still it is moving up why because mass is not constant when your balloon inside balloon there is air if you release the balloon it should come down because of gravity but it goes up the mass system will not talk about it right now okay that will deal in when we talk about laws of motion chapter again right now our focus is whatever is required for application of vectors okay so we're assuming mass as constant and 99% of the cases mass will be constant only 99.5% you can say fine all right so copy this down all of you quickly let me know if you have any doubts no doubts okay now what is the equation force vector is equal to mass times acceleration vector right this is a vector equation just like v is equal to u plus 80 all the vector equation from x axis and x axis along x axis like that we have divided the equation into two along x axis equation along y axis the equation okay same thing over here we can say that force in x axis is equal to mass time acceleration along x axis what is good if you write like this I mean how this will help this is a vector equation this is a scalar equation scalar equation we have been solving since our childhood x plus y is equal to 2 x minus 3 y is equal to side so what is x and y okay so we'll be able to solve it like a scalar equation if you write like this that is why fine otherwise you'll get a vector equation which is a little tricky to deal with okay where fx and fy will be equal to m into a y in this chapter it will be two dimensional in nature okay so everything can be described but by just taking two coordinates x and y okay there will be rarely a chance that along z axis also something happens fine so whatever needed to be covered will be covered in x and y only there will be cases where z axis is also involved that will take up later on right now our focus is again application of vectors okay so we are taking only two dimensional problems fine now tell me if multiple forces are acting suppose multiple forces are acting something to write then we can add it up simple if you add it up split it into their components and then add the x and the y correct good so this is summation of all the forces this is summation of all forces along x axis summation of all the forces along y axis okay but experience only net force yes or no consideration can object has multiple forces so why it can't have multiple object doesn't know multiple forces are acting on it already multiple forces similarly acceleration can be split into two or three things okay there are cases in which you have to add acceleration for the same object along x axis and then say that this is the equation along x axis I'll just give you an example for example if this is the case that you know this object is moving with acceleration some acceleration all right even this object is moving this object's acceleration is a1 with respect to this object so total execution will be a1 plus a5 so we'll come back to this no need to worry about it but as I'm telling you that there exist such cases where you split the acceleration also and then when you write equation you add it up fine so don't think that acceleration will be always one vector in the diagram there can be multiple vectors which represent the acceleration then you have to take component of acceleration also along x axis and along y axis now these things which I'm telling you you'll only understand when you solve questions which we'll be doing it later okay but that's just that I'm telling you for our information okay fine so this is the second law of motion this gives you an equation the most important part first law and third law doesn't give you equation second law gives you an equation okay so this is about this second law of motion now let's talk about the third law of motion what is third law of motion about reaction okay but then have you thought about like what does it mean what do it mean so you are you have the basic understanding of what does it mean so let's try to you know get into the depth of it and try to understand more about the third law of motion because the third law of motion is where the most of the student makes error okay they think that entire mechanics is just forward to mass acceleration and they forget to apply Newton's third law usually okay so third law of motion says I mean the statement could be many you know the way you explain it the the most common explanation of third law of motion is every action has equal and opposite reaction okay what is action action we thought see when I say action action has a wrong connotation also action when you say it it signifies some motion some movement but even if there is not a movement the force which is applied is called the action it is not a movement which called the action every force if you repeat the statement every force has equal and opposite force simple every force has equal and opposite force so so if there is a mass if there is a mass for this mass if somebody applies force F so this force has equal and opposite force so there will be any opposite force applied on the mass also is that true the mass correct so the person applies F on the mass mass applies opposite force on the person let's try to put it in a more simple manner draw two masses two objects object A and object B object A applies a force on B as FBA let's say F only then B will apply force on A F simple getting it if A applies force on B in this direction B will apply force on A in opposite direction this is what Newton third law is getting it so if there is a force in this universe somebody might be applying it somebody something will be applied that force will be applying that force yes or no so on that object or person there will be equal and opposite force also applied so in this universe total forces zero zero zero okay so say about third law of motion is that right now everything exists in pairs all the forces exist in pair car engine generates force like car engine where is the opposite car engine generates force where on the wheels like the reason why I can't say over it is because of the opposite force no no no he's saying something else he's saying something else you car engine applies force on the wheel wheel applies force on the engine trying to let's say one example I'm trying to twist this remote okay I am experiencing a force you you will experience a force when interested yes or no but you are applying force how come you are experiencing the force you are experiencing the force because this is applying force on you simple a force you should ask yourself a force if you're not able to find the pair you're doing it wrong you'll get a wrong answer getting it simple thing when you draw a force in a diagram that this is the force yourself where is a pair of this force oh there is this pair so I'm ignoring it let's move forward now if you're not able to answer that question you're doing it wrong simple okay now tell me if I'm standing here what are the forces acting on me gravity is acting on me so where is the pair of that force no who is applying gravity on me earth earth so I should apply a force on the yeah other applying force on me right me or mg on me so I should apply mg in a poet that's standing here on you there is mg force so on earth also there will be mg force this is a pair things are very very simple so keep it simple don't complicate in your head so where is applying mg simple whatever we have learned nothing else okay I'm standing on the floor floor is applying force on me upwards yes or no and you are also applying force on the floor downwards the floor applies and on you and you are applying normal reaction or the force on the floor also yes or no floor it's very weak so I will break because you're applying force on it these are action reaction pair mg mg n a 9th equal opposite pairs what about the air here air is pushing you down you can say that you're also pushing the air up if you don't push the air up air will come down okay but we are ignoring these side effects okay we are ignoring the air drag and all those things and any of the doubt anything else no examples of third law of motion it is very very common so you shoot a bullet what happens yes the gun the gun is applying force on the bullet so the bullet applies force on the gun so you feel the push backwards okay so that like when you're launching a lock it like you're like spelling all the gases downwards so the force is downwards so pushing the gas downward gas is pushing out okay okay ball is applying force on the wall and wall applies force on the ball the boat moving in water the boat exerts the force on the water and the water exerts the reaction force of the boat how you're exerting force on the water the boat exerts the force on the water body how with the no with the weight okay very correct correct so weight weight is balanced by the force due to the water when you're playing cricket ball is coming fast then you can see the back like when you walk you use that you push the ground and the ground push the ground correct when you're walking you're actually pushing the ground backwards and ground is pushing you forward and you're walking so when you jump then then you will create a force on the ground when I'm jumping which direction I'm pushing the floor down floor pushes you up what else so if if you if you were on a frictionless surface and you were to throw something really heavy forwards you would start sliding backwards like I saw this in this yeah it is possible yeah it happened in fact in the space now people do it yeah yeah like what else when you swim you're pushed the water back water pushes you forward okay so Newton's third law is very very common very common okay so when we will solve problems we will keep that thing in mind okay so when you solve problem we'll be representing the forces so every time you represent the force ask yourself where is the pair of this force and then you proceed forward okay right so these are the three laws of motion which I mean which you should understand in a great depth okay more understanding will be developed when you will solve problems in class 11 it will never happen that you understand everything by reading theory and then you solve problem that will never ever ever will happen your 70 to 80 percent of learning will happen when you solve problems so do not I mean what what happens is that when we go back home we start reading book let me complete the chapter then I'll start solving problems no start solving problems and then read the chapter when you're stuck somewhere reading it that's the way you have to learn that's the only way to learn your school exam your ut is or whatever it is do not read the chapter again and again solve problems after the problems after problem because 11 12 is about application of knowledge it's not about knowledge just they're not going to tell you they're not going to ask what are the returns three law of motion lays down no they're going to ask you how you apply that so you need to learn that that will be learned only when you solve problems okay