Welcome, this is Melissa Armour with a Stock Swoosh, and I'm reviewing the play of the day today. And here's me. If you'd like more information, you can email me at Melissa at thestockswoosh.com or call me at 929-3200 Gap. And feel free to follow me on YouTube. Today's Golden Gap was an amazing trade. In fact, it went to the Dream Target instead of the room this morning, 38, 38, 38. I actually broke 38 then in the after hours, 37 something, this is trading out right now. Really, this is lower, couldn't even continue tomorrow. Amazing Gap. So I use my system every morning before I begin to trade or decide what I like, and I like this this morning immediately, then I rated it. It was SIG, 23 points is the Golden Gap rating. And if you decide to do my class, you'd learn how I figure that out, and that's really how I know to watch SIG or whatever I'm watching on that specific day. Anyways, something I wanna point out, this is a daily chart of SIG, is that the stock closed up here the night before. Okay, this is what a gap is for those of you that don't know. The stock closed up here around 47, whatever. Open in the morning today, gap down. Closed here, open here. So gap down overnight and open around 42, whatever. It was right under 43. Anyways, the point is that the difference between the close and the open is what a gap is, and that's what this did, a gap down. Then I rated it. I rated it this morning in the pre-market, or you can rate them in the post-market at night if you see them, but either way, you do it in the gap, you rate it, and then it tells you, is this a long or short? And SIG was a short today, and it was a good one. So there was really so many, so many setups in this. I called the first trade-in here, which actually went and missed the entry in this because it just went so quick, so fast, and I didn't wanna chase it. Anyways, did it then here, boom, got the drop, quick got out. This was a move. It continued, continued, continued. Some people held this in the morning down a lot. So this was a quick one here, really fast one here in the morning for $1400 profit. Then in the 15 minute, which again, you could have played this all day, because a drop broke, fell, went to 40 something in the morning, rallied back. Here's the 15 minute chart of the SIG. You could have done this here, and got the drop right in here. And this is a nice move right in here. It even went even more, actually. You could have done this even later in the day. But anyways, here was the 15 minute, boom. And it was not on TV today, so there was time to do this, and it was one of those ones, where you gotta take the good ones when they come. You gotta do the good trades, when they come, you gotta hit them. So total SIG profit today, 4100. Again, if you decide you wanna learn how to become a day trader, if you wanna do my class, if you wanna trade in the room, Monday, Tuesday, Wednesday, Thursday, Friday, the room is open for Golden Gap course students. You will learn how to pull money out of the market consistently, chunka, chunka, chunka, chunka, chunka. Some days it's small, some days it's medium, some days it's big. So today just happened to be one of those big ones, but I really mostly focus in the morning, but there are days when you know that a gap just rates so high, so good, that you wanna be able to take advantage of it. So if you'd like more information on my class, you can email me at blissofthestopswitch.com, my class is called the Golden Gap course. The next class is March 24th and 25th, nine to five Eastern time. Cost of the class is 54.99. If you wanna sign up, the class is online, you have to email me to sign up. I also offer a combo, which is the Trends class and the Golden Gap course for 59.99. Class is online, email me at Melissathestopswitch.com if you wanna sign up for that. And if you sign up on March 16th, the Early Bird Specialist will get one month free in the trading room, which is really nice, because being in the trading room, you will get these calls, you will get these trades just like the sake today, you will get my analysis, you will get my rating.  Before 9.30 today, I predicted that sake would drop on the day and it was around 43 word open and I predicted that it would go to 38, that's a $5 drop. And that is what I'm so good at doing and it's really the rating system here, looking at this, the daily chart, this is how I figured out where I can make those predictions in the pre-market. It is in a very important time of the day to be looking and deciding what to do. And if I don't see anything in life, then I don't do anything at all. So that's important too. Some days there isn't anything good, but the days you get the good ones, you just have to go right after it. So if you're interested and wanna sign up, this is the next class before second quarter earnings season starts. It's gonna be a really exciting time. Very much looking forward to second quarter and we're gonna take advantage of the good ones between now and then. Email me at melissa at thestockswish.com if you would like to sign up. Thanks to everybody. Have a great day.


For years, researchers have been using aircraft and satellites to produce information-rich views of our planet. It's called remote sensing, and although the science is not new, its applications continue to grow. Image processing software called ELAS, developed at NASA's Stennis Space Center in Mississippi, is now being used to manage and conserve fish populations in the Gulf of Mexico. These maps of sea surface temperatures allow researchers at the National Marine Fishery Service to predict where certain species of fish should be on a given day. This capability is a much-needed tool for scientists working aboard NOAA vessels, like the Chapman. Fishing is a major industry here, and information collected during these research missions is used to set critical catch limits. The temperature maps accessible with a standard personal computer on the ship make it easier to locate and sample fish being studied. Reducing search time saves man hours and fuel, and being able to gather data faster makes setting catch limits more precise. When changes in surface temperatures are too subtle to be mapped, especially equipped NASA leadership can use the means of concentrations of chlorophyll in coastal waters. The same boundaries between water types used to locate fish can be found with these displays as well. With the help of businessmen like Mike Frenett, the technology is now being moved into the marketplace. As with all charter boat owners, Frenett's livelihood depends upon finding fish for his customers. Reducing search times would also save larger commercial fishing operations thousands of dollars in fuel costs, savings which could in turn be passed on to consumers. And according to oceanographer John Brooks, the danger of overfishing will not be any greater. As long as the demand is there, they are going to spend the time and the effort in the future. As long as the demand is there, they are going to spend the time and the effort in the fuel to find that fish. And their livelihood depends on a long term fishery, not just to load the boat today. And so it's readily understood throughout the fish community. We have a beautiful natural resource here and it's up to us to govern that resource. Locating fish from above, using NASA software to manage an important natural resource.


The Twin Cities Power Boat Association made their annual stop in Crosby this past weekend. Our Anthony Scott gives us an inside look at the races and tells us why Power Boat Racing is a family tradition on this week's Northwest Adventure. When your heart's racing you're praying that it starts right away and you get off the beach fast. It's that kind of a general in rush that keeps drivers and race fans coming back to the annual Crosby Power Boat races. It's fast boats you know it's one of those things where you don't get to see it every day. Some of these boats will be going over 130 miles an hour. They turn on the dime and accelerate really fast. It's really a thrill to watch. The races are run by the Twin Cities Power Boat Association, a group that has been together for more than half a century. And Crosby has been their favorite spot to race over the last 32 years. We've raced over a hundred cities in Minnesota. This is the best. Spectators can see good. We're not fighting with other boats. It's just great. Even for drivers that race every weekend they still get that same feeling they had the first time they got behind the wheel of one of these power boats. Every time we go out when you're sitting at the start dock and ready to fire the engine everything else goes away other than that moment racing. You always have butterflies typically if you don't there's maybe something wrong. And Power Boat Racing is something that has become a family event in most cases. But it's really family oriented sport you know families get involved you know dad started the kids start or sometimes the other way around sometimes the kids start and then dad got into it. So after we 12 years old to race so as soon as I hit 12 I was in a boat kind of just natural thing to do in our family. Meetings are held at the start of each race day with safety being at the utmost importance. The most important thing about it all is that the fans are safe. That's where we the first start from there and then it runs through the pits of course with our own rules in there. Racers are not only racing for fun but they can also score points within the club and within the American Power Boat Association with a chance to go to nationals but at the end of the day they just want the fans to leave with a smile on their face. That's the reason why we do it too you know is to definitely put on the best show that we can for them. Reporting from Crosby for this week's Northwoods adventure Anthony Scott Lakeland News. The next time the Twin City Power Boat Association is on the water will be August 11th and 12th in Detroit Lakes at Detroit Lake. If you enjoyed this segment of Lakeland News please consider making a tax-deductible contribution to Lakeland PBS.


QuickBooks Online 2023. Adjust opening balances. Get ready to start moving on up with QuickBooks Online. Here we are in our get-gray-guitars-practice file. We started up in a prior presentation using the 30-day free trial. We also have open the free QuickBooks Online Test Drive sample company which if you want open at the same time. We recommend using the incognito window or another browser. You can get to the incognito window if using Google Chrome with the three dots in the browser. New incognito window and then search for QuickBooks Online Test Drive. We'll be using the sample company to look at the difference between the accountant view, the view that the get-gray-guitars will be in, and the business view, the view that the sample company will be in. You can switch between the two views by going to the cog up top and switch the views on down below. Support a counting instruction by clicking the link below giving you a free month membership to all of the content on our website broken out by category further broken out by course. Each course then organized in a logical reasonable fashion making it much more easy to find what you need than can be done on a YouTube page. We also include added resources such as Excel practice problems, PDF files, and more like QuickBooks backup files when applicable. So once again click the link below for a free month membership to our website and all the content on it. We're gonna open up some tabs to put reports in as we do every time right-clicking the tab up top to duplicate it right-clicking the duplicate a tab to duplicate it again back to the tab to the middle as the tab to the right things going to the reports on the left-hand side to open up the big balance sheet report in the favorites noting that if you're in the business view by the way it's in the business overview section and then your reports can be found thusly. So let's go back on over back to the tab to the right we're gonna go down to the reports on the left this time opening the P to the L the profit to the loss the income to the statement and we'll close the hand boogie up top let's do the range change starting with 0 1 0 1 2 2 tab 12 31 2 2 tab run it to refresh it tab it to the middle doing the same thing for the balance sheet burger closed scrolling up changing the ranging 0 1 0 1 2 2 tabs 12 31 2 2 tab run it remembering that we entered the data that we imagined was from our prior system as of the cutoff date 12 31 22 so we can start entering new data as of January 31st or January 1st 2023 so we could see that prior data here this is the data as of the prior December 31st 2022 from the prior accounting system our strategy has been enter each one one at a time letting QuickBooks record the other side of the transaction to whether wherever it feels it should which could be to the opening balance account which is an equity account or it might have posted to the income statement either to revenue or expense the revenue when we made the accounts receivable the expense when we made the accounts payable though the the revenue or accounts receivable using an invoice the payable using a bill basically but because we entered those in the prior period December 31st 2022 they will roll into equity so the strategy then being we've got all of our accounts lined up the other side due to the double entry accounting system has to work if the other side is simply equity it's just going to be in the wrong equity accounts and now we're gonna have to just shuffle them around and fix the equity accounts the main one being that opening balance equity which is not an actual account does not look professional is not something you typically want to be presenting because people will will have less confidence in your accounting system if there's information in that account in general because it's not an actual kind of financial reporting account it's just a holding account that QuickBooks use so you can enter the beginning balances okay so let's go back on over and recall this so if I scroll down everything should be lined up what I'm worried about here or the only place I'm focused on is the equity section so I'm going to close up equity and just give a quick recap of what equity is and remember what we have up top what we've created is the accounting equation assets equal liabilities plus equity so the assets are what we have liabilities and equity represent who has claim to what we have either third parties liabilities like the bank and the accounts payable and equity us therefore you can also format the accounting equation assets minus liabilities equals equity equity representing in essence the book value the value of the company on a book basis so if I was to increase this then we've got the balances on the assets and the liabilities and then what does equity mean then it means what we have claim to on the assets the book value of the company then the accounts within equity will depend on the type of business that we are in in terms of what the appropriate names would be and which accounts that we should be allocating the retained earnings to in essence so the opening balance account you'll recall is just a clearing account QuickBooks is putting the money into here basically telling us hey look this is where I just dumped the other side which I believe should be in an equity account when you enter the opening balances so that no matter what type of business we are corporation sole proprietorship partnership should generally be closed out to some other account and then if we're sole proprietorship we would be using an account called owner's equity because the whole thing is basically in one account owner's equity although we might also have draws that's the money that we take out of the business in order to to help in order for our own business use or our own personal use out of the business and we could add an investment account money that we put into the business rather than money that has accumulated over the business from activity through the income statement now if we were a partnership it gets a lot more complicated because then you're going to have multiple claims to the equity it's owned by multiple people more than two people two or more people so therefore you kind of have like a similar account down here as like an accounts receivable account broken out by customer you're going to basically have to have your partners and then track their balance and their equity account which you might call a capital account and you're going to have to apply out your net income in accordance with the the agreement of the partnership and you might have separate accounts for each partner like a draws for the money that each of those partners take out of the business for personal use and possibly each partner's investment account so what happens with a partnership then is QuickBooks will roll everything into net income you can you can see right here and then it will roll into some account on the balance sheet which they put as default retained earnings which is actually a corporate name but you get the idea it's the earnings from the income statement that roll in so if i go up and i change the date up top to 010123 to 123123 and run it you can see that that net income rolled in to retained earnings so that account we have to have some account that QuickBooks uses as the closing account it's going to close the income statement out into some account they name it retained earnings by default if you are a partnership you're going to have to take it out of retained earnings and put it into the applicable partner accounts in accordance with the partnership agreement so the partnership accounts are properly reflected if you're in a if you're in a sole proprietorship we can just basically rename retained earnings to owner's equity to to the capital account right because there's only one owner so that's that's an easy situation a corporation you would think would be more complicated than a partnership but it's actually generally easier because the idea of a corporation is that we're not going to break out a bunch of different owners within the equity section the way we basically do with a partnership but instead we're gonna break out the ownership of the company out into equal units kind of like dollars a measuring tool of units which are the stocks and therefore for the for the amount of money that was invested in the company that the that the owners put into the company that will be a cap the capital account because we issued the stock for that money and remember that a lot of times people buy stocks on the secondary market this would be the primary issuance of the stock the money coming into the business and then we'll track the retained earnings which represents the money that has accumulated in income that has not yet been distributed in the form of not draws as it's called with a partnership but dividends the reason it's dividends and not draws even though it's a similar kind of thing is because you can't have one owner of a corporation taking out a different number of dividends than another owner base and you know they have to have the same amount of dividends per share basically so that means that you have to have some agreement when there's going to be a distribution of dividends there also could be tax implications and so on so we're going to we're going to imagine we're sole proprietorship here so what i'm going to do is i want to change the retained earnings to an appropriate name which would be something like owners owners equity so i'm going to go back on over and do our standard process to the first tab i'm going to go down to the to the accounting our standard process in terms of the accounts that are used i mean and by the way if i go into the business view that chart that general chart of accounts is going to be under the bookkeeping and then the chart of accounts there under the business view and then the idea being that if there is an account that's in use that is similar i'm just going to change the name and not add a new account now with this retained earnings account it's a super special account because this is you have to identify the account that quickbooks is using to close out the income statement you can tell what it is because like it's the only equity account that doesn't have a register it's it's you see all these other equity accounts have a register they have a balance in it there's no balance or register here because because this is the account that they're using to roll over into so you gotta be you gotta be quite aware what that account is i can't just make another account retained earnings i can't just add a retained earnings account that will function in the same capacity there's only one account that's that quickbooks is going to use to roll over into what i can do is change the name of it so i'm going to say let's edit it and say i want to call it uh i want to call it the name owner's equity owner's equity i think that that looks right right owner's equity and it's still going to be an equity account we'll put it in equity tax section retained earnings i'm fine with that okay and so let's just save it and so that is done so now if i go back to the balance sheet now and i and i run it again we've changed the name to be more appropriate for a sole proprietorship okay so that's that's step one and then step two is to get the opening balance to be closed out and then i'm going to put it into the the uh owner's equity account so i'm going to zero out the opening balance account so i'm going to go back to the first tab and there's a couple ways you could do this you could do this with a journal entry because this would be the type of thing done with a journal entry there's no other form that is used to close out the opening balance so it's going to be a journal entry but you might go to the registers which which could you could do instead of having debits and credits and enter the journal entry that way and i like to point that out because a lot of times we only think of the register as something applicable to the checking account but you have these registers for each of the balance sheet account not the income statement accounts mind you because the income statement accounts are temporary balance sheet accounts are permanent but if i go into the the balance sheet account for opening balance equity i could say okay there's opening balance equity i can see the balance there i can use the register to enter the transaction and then i can see the balance as i enter the transaction so i then i can do a journal entry notice the options that i have here a transfer or a journal entry we don't have any check or expense form or any deposit it's just a journal entry to enter a transaction to this and then we're going to say this is going to be as of i'll do it as of uh 12 31 22 let's say which is the day before our cutoff when we've been entering our beginning balances i might do beginning balance adjustment in the memo something like that it's going to decrease by the full balance which is seven two three nine six the other side is going to go into the retained earnings account that we now change to owners equity so i'm going to type that in here hopefully it'll show up owners equity owners equity i think that is that the same one account i think it is hopefully if it's not we'll see and we'll see it on the balance sheet so what's this going to do it's going to decrease this account the other side i believe wait this is owners draws hold on a second there's owners equity so owners equity okay so the other side might say hey you're hitting the retained earnings type of account and it might give us a warning as we do that but and so that's a nice check usually because usually you don't enter a transaction to like the retained earnings or owners equity account but you might do it when you first set up the the accounts so let's save it and we saved it okay so i'm going to go then to the balance sheet and refresh it again run it to refresh it and then when i scroll down now it went into the owner's equity so it looks like it did what we needed to do so that looks good so just to recap if i take this back to 010122 to 12 3122 we entered all the data as of the date of 12 3122 scrolling down then we've got the opening balance equity is now zeroed out the the owner's equity is here and then we've got this net income that net income ties into what's on the income statement for the period january to december 2022 or we entered all the data on 12 3122 the cutoff date from our prior balance information here that we're imagined came from our prior bookkeeping system and they put in services and other miscellaneous expenses when we entered the beginning balances for our customers and accounts receivables and our vendors for the accounts payable and that's where this 5500 comes from i don't want anything on the income statement but i'm okay with it in there as long as it's prior to the the date that i'm going to start adding new data january 1st 2023 because this will close out into the balance sheet how do i know that because if i bring this up a day to go from 010122 to 1231 actually 01010123 123123 and run it there's nothing in it so i've got a clean income statement going forward as i enter new data for the year starting january 1st 2023 and if i go to the balance sheet and i scroll up and i run this up a day what's going to happen is this this week this income is going to roll into the owner's equity account so i'm going to go from 010123 to 1231 23 and run it boom and so now we've just got that one owner's equity account because it's a sole proprietor later on we might add draws the money that we take out for personal use and we might add like an investment account owner investment for money the owner puts into the business but you could account you could account for it with just the one owner's equity account if you so choose if you're a sole proprietorship okay let's now open the trial balance just to check it out in that form gonna go to the tab the right right click duplicate and let's go down to the reports on the left hand side it's down right there and close the boogie and i'm just going to type in trial balance i think that's the easiest way to do it and i like the easy way and so there we go let's change the range from if i go to 010122 to 1231 222 run it there's our trial balance as of the prior period 1231 22 the cutoff i compare that if i compare that to my data that we entered from the prior accounting system that we are imagining then there we have it we've got the 25 the 25 the 20 the 25 the inventory the accumulated depreciation furniture and fixture 15 000 for the accounts payable the visa the loan payable and then opening balance is now at zero and then equity should be at the 77 896 and it's different here because we've got those two income statement accounts in our trial balance remembering that the trial balance is the income statement on top of the or the balance sheet on top of the income statement 72 396 plus 205 00 minus the 15 000 there's the 77 896 if i bring this up a day what's going to happen is these income statement accounts are going to roll into the equity and we should match so i'm going to change that it'll be magical it'll be a magical moment get ready for 0012 3 to 1231 23 and then run it and then there we have it so now it's all in the one report and we were matched up we're good to go as of the new period so as of as of january 1st 2023 we're good to go so if you're following along you can check out your numbers they should match our numbers here and i'll try to write a trial balance every time we close up so you can kind of check your numbers going forward and if you're working along with the practice problem we'll just keep on adding stuff now we're going to start adding new data into the system from this point


Hi, I'm Dennis Clemo, a member of the InvestorIntel team. We're here today to meet with Mr. Stuart Rogers. Stuart is the president and a director of TerraX Minerals, TXR on the Toronto Venture Exchange. Hi, Stuart. Welcome to InvestorIntel. How are you today? I'm doing great. Thanks for having me here, Dennis. Oh, pleasure. I saw yesterday you had a press release with your company. Added some new members to the team. Why don't you tell us a bit about that? What we did is we're advancing the project in Yellowknife. We've got two first-class guys to the project that had experience developing a world-class asset with another company they were involved in, Cayden Resources, which was acquired by Agnico Eagle in 2014. They were attracted to this project as investors, and that's how we got to know them. We found that they have a lot of insight and contacts, and people can help us advance this project. So we're excited to have them join the board and work with us. They're very enthusiastic. That's great. Why don't you tell us a bit about your primary project, the Yellowknife City Gold Project? Yes. The Yellowknife City Gold Project is something that we've put together over the last four years. It started off to be a small block of claims. What we've done is taken this area that was undeveloped on strike with two high-grade gold mines, two of the highest-grade gold mines that produced in Canada, the Conn and Giant. We took acquired and assembled a package, basically a world-class gold district. We now have 418 square kilometers. We have all the prospective ground immediately north and south of the city, and the two high-grade gold mines that are in the area. And they're in production, the Conn and the Giant? Well, they were past producers. They operated for over 60 years. They produced over 14 million ounces of gold, an average grade of 16 grams a ton. That is good grade. That's high-grade gold. Tell us a bit about Terrex, the company. Well, they were founded in 2007. Joe Campbell had just come off some success with discovering Melodyne and developing that project and introducing it to Agnico Eagle, who acquired it for $700 million in 2010. So, at that time he said, well, that was great. Maybe I should go and start a company, find a good high-quality asset and advance the project. That's when I met him and we took this company public in 2007 and went and identified this project and moved forward on it. So, the team, management team, experienced individuals. The majority of the people working on the project, Yellowknife City Gold, work with Joe and Melodyne in developing that asset. So, we have people that know what we do. They know what they're doing, a strong team. And this is the same type of potential as that project. Stuart, you're based in the Northwest Territories. What are some of the highlights of your project? Well, the highlight for us is being with the access to infrastructure we have, with power dams, with the train workforce at the city of Yellowknife, right adjacent to where we're operating. It gives us, it dramatically reduces our cost to operate. It will reduce the cost to put a mine in, because they have the infrastructure that was there for the past mines. And it allows us to work all year round, which people are quite surprised to hear because most projects in the North, remote projects, you can't access in the middle of the winter, where we can. People live in the city. They drive to work. They drive home at the end of the day. We just completed a 15,000-meter drill program. We'll be back again drilling in the end of June. So what are your plans now for the next six to nine months in terms of advancing the property? Well, we've just completed a winter drill program about a month ago. We've released half the drill results just over a week ago. We have another 7,500 meters, 21 holes to release when we get the assay results back, which will be over the next month. And then we'll be back drilling again in probably the end of June, the first part of July, with more drill programs to follow up on the results from this and some new areas that we've identified on the project. This is a district. We have multiple zones of mineralization. It's one large system. And we have a plan to develop it. There's lots of targets there. People think maybe we're not focused, but we are focused. Well, that's great. That sounds like a terrific story. I love the grades. And it's a great area. So thank you very much for coming in to see us at Investor Intel. And we hope to see you again very soon. And good luck with TerraX Minerals, TXR, on the Toronto Venture Exchange. Thank you. Thanks, Dennis.


I'm honored to be a part of this. This is the main cruise. I missed my main cruise, so I'm happy to be a part of theirs. I wasn't part of the pre-com debt, but I'm gonna try to make this the best cruise that they've had. It's just gonna be, it's gonna bring honor to him and the ship's namesake, and it's also gonna bring honor to African Americans around the world, so. It's just, it's an honor. It's a privilege.


Everybody forced here ripping 2021 select football hobby three bucks number 35 random team style Results can be found at late and sports cards.com at any time, but you can check them out here for another minute and there's Lincoln chat, too Let's get started y'all. Good luck. All right start out of the fields based concourse Blue concourse rookie Jets Michael Carter Never 120 of 199 Kenan Wang Wu to 75 rookie auto Minnesota. That's cool Dallas. I don't know a whole lot about NASCAR Redemption coming up And it's a Justin Fields jumbo rookie signature swatches prism for the Bears Starting off big with Chicago He's got the Bears Jose M Grats Jose Jersey auto Justin Fields redemption Mac Jones base silver selections Jalen waddle concourse rookie silver Joe try and show yanker for the books Where's that expire? Another year Schwartz leatherwood concourse purple Ezekiel Elliott Cowboys 70 of 75 premier level silver rookie Michael Carter Jets if you got silver club level Caleb Farley rookie for the Titans Smith-Marcette like Wilson club level base at tricolor concourse Tyree killed chiefs Number and seven of 249 premier level silver Aaron Rodgers premier level silver Eagles Jalen Hertz tricolor Justin Fields Bears again 24 of 149 Crap man Bears crushed it grads Jose concourse over Jamar Chase word of Bengals Silver sensations Alan concourse silver rookie hunter long dolphins Concourse or not Congress, but field level white Curtis Martin Jets five out of 35 Club level silver Tyler Lockett Seahawks Zach Wilson premier level base Silver LA field level Broncos Mile Sanders to 25 tie-dye patch card Eagles Nice first box there got a concourse silver Tyree kill as well For the chiefs definitely good first box there Let's keep it rolling St. Brown base rookie concourse silver Cadarius Tony phenomenon rookie the Giants concourse silver rookie terrace Marshall Jr. Panthers tricolor premier level rookie Chargers Josh Palmer 100 of 199 club level silver Dak Prescott Dallas Silver concourse Matthew Stafford for the Rams Palmer field level rookie Snapshot to Joe Burrow silver for the Bengals club level silver Najee Harris rookie Steelers Nelson Aguilar concourse tricolor Patriots 129 of 249 concourse silver rookie Kylan Hill Packers Nice silver rookie field level Mac Jones Patriots Nice Mac silver there the field level Orange concourse Mac Jones There you go Pat's 31 of 49 Concourse silver rookie Rhonda Elmore for the Cardinals That Mac service didn't look perfect, but there's no like scratches or just like little light like specs Club level silver Terry McLauren commanders Justin Fields premier level rookie Taxins Niko Collins club level tricolor to 149 Chiefs to 35 rookie Otto Cornell Powell Eli Mitchell there are two to at well to 49 jersey card Rams Silver field level Cory Davis Jets Tide eye coming up here is a Ryan Tannehill concourse Titans 19 of 25 go Tennessee Silver concourse rookie Jalen Darden for the Bucks and Michael Carter to 75 rookie Jersey auto Jets Jets again Kenneth Gainwell and last one. Good luck everybody Silver Zach Wilson phenomenon rookie club level silver Tyler Lockett Bateman premier level Chase concourse club level tricolor Zach Ertz Cardinals 87 of 149 concourse silver rookie Kellen Mann for the Vikings XRC here Nice a Desmond Ritter Atlanta XRC Nice card Falcons premier level silver rookies Teth Williams Broncos 75 he was the Vonte Smith on the rookie jersey for the birds Concourse silver Josh Allen bills I mean it could purple Devonte Parker dolphins premier level 41 of 75 premier level silver Jim Kelly bills Alice you've got Nick Bosa at a 49 on the Jersey auto for the Niners Solid Bosa there doesn't have autos in a whole lot of products. Oh nice a tie-dye AJ Brown Titans two tie-dyes for the Titans 2425 on the Brown an Andrew Luck auto 20 to a 25 Colts Andrew Luck now well, okay Silver snapshots Stefan digs Buffalo silver field level John L. Way Broncos Premier level I blue now trail ants 31 of 99 Pretty nice rookie quarterback color in this in this three boxer Silver concourse Ian book rookie for the Saints Silver premier level rookie Elijah Moore for the Jets now Smith Mar set Ojalari rookie concourse blue for the Giants This one 61 of 199 Concourse over to a dolphins and silver club level rookie Caleb Farley Titans. Why is that chair squeaky all of a sudden? Donnie do to it. I'm sorry to figure that out We're just for the whole chairway. All right, nice three boxer here. We had an XRC desmond Ritter Jersey card Devonte Smith Eagles 2 to at well Rams Miles Sanders patch Eagles tie-dye Jersey auto Jets Michael Carter Ken and Wong Wu Vikings auto Jersey auto Nick Bosa Niners Cornell Powell auto chiefs Andrew Luck auto Colts 25 Like blue trail and Shrucky numbered to 99 for the 49ers Premier level orange concourse Mack Jones for the Pats This one's numbered to 49 Club level tri-color Justin Fields bears numbered to 149 and Bears redemption Justin Fields jumbo Rookie signatures watches prism. That's gonna do it. Thanks again, everybody mixers coming up next


how to multiply two binomials. What is 2x plus 3 times 5x plus 4? It might help us to consider the product 2 plus 3 times 5 plus 4. Obviously this is just a decomposition of 5 into 2 plus 3, and this is a decomposition of 9 into 5 plus 4, so this is just 5 times 9, which is 45. How can we use that to help us understand how to multiply two binomials? Another way of multiplying these numbers, notice that these are also binomials. They are not simplified, and they are constants, but they are binomials. If we multiply 2 times 5, and we add to that 2 times 4, and we add to that 3 times 5, and 3 times 4, we get 10 plus 8 plus 15 plus 12, which is equal to 45. So let us follow the same steps. This first term times this term, that is equal to 10x squared. This term by the second term here, that is 8x. This term by this guy, that is 15x. And this guy by this guy, that is 12. So this is equal to 10x squared plus 23x plus 12. And now we see that the multiplication of 2 binomials is something that is not so strange after all. It is something that we can do with integers or with any other numbers. It is just a property of numbers. We decompose them and we multiply them in that way. So now it doesn't seem as an arcane method of multiplying these 2 objects, but as something that is very natural.


What's up everyone take down here welcome back to another video today I'm gonna be sharing my thoughts and predictions on all elite wrestling's double or nothing pay-per-view it is right around the corner and I'm pretty excited for it because it is the first official all elite wrestling pay-per-view and I am familiar with a lot of the wrestlers that are on all elite wrestling right now however there still are a few that because they came from the Indies I'm not familiar with them because I haven't seen much of their work so the ones that I'm unfamiliar with I'm gonna try my best to give a prediction on their match but with more all elite wrestling pay-per-views and the more matches I see them in the more I'll be familiar with them and in the future I'll get I'll be able to do better prediction videos of their matches but I'm gonna try my best for the first pay-per-view right here so up first we have hangman Adam Page versus Paak now I am familiar with Adam Page I do like his gimmick like the wrestler is pretty good but Paak is basically Neville from WWE now whenever Paak first came up from the NXT to the main roster in WWE I didn't mind him then I thought he was pretty good but his last run in the WWE as the king of the cruiserweights I didn't like that gimmick at all it just to me it held down the rest of 205 live because there's a lot of great cruiserweights in my opinion and have somebody that was already established in WWE get put onto the cruiserweight division to me it kind of held the rest of the division down but it is gonna be exciting to see him in all elite wrestling as Paak and it's gonna be interesting to see what kind of gimmick he's gonna bring to all elite wrestling so for this match here just because I don't know what Paak is up to these days since he left WWE I'm gonna say Adam Page is going to win this match but I would like to see a new gimmick for Paak that be very interesting because I think I could definitely get behind him I just didn't like the king of the cruiserweights gimmick he had whenever he was last in the WWE so my final prediction for this match is Adam Page next is the over-the-budget pay-per-view sorry over the budget battle royale they keep changing the name and stuff like that but I'm not familiar with a lot of the wrestlers that are gonna be in this battle royale and it's uncertain right now what the winner is going to get I'm assuming a future championship opportunity because all elite wrestling is gonna have their own championships in the future but the wrestlers that are gonna be in this match are gonna be Sunny Kiss, Kip Saban, Brandon Cutler, Ace Romero, Glacer who I know a little bit of, Brian Pillman Jr. who I think just started wrestling last year he's following in his father's footsteps, Sunny Day's MJF Joey Janelli who I know a little bit about Chuck Taylor and Trent Beretta there's apparently gonna be a few more names added to this but this is all the ones that are available as of the time I'm recording this so in my opinion I only know three of the wrestlers so it's gonna be hired for me to make an honest prediction right now but I'm gonna say Trent Beretta because I did like his work in the WWE he was not given that good of chance in the WWE in my opinion but I did like his work and after leaving WWE he got a lot better in my opinion so he has a good chance of winning this match Joey Janella also in my opinion has a good chance of winning this match but those are gonna be my two predictions one of those two I'm not familiar with the rest but this is battle royale of course anything can happen next is gonna be a six-man tag team match for so-called uncensored I hope I pronounced that right so I'm not sure we have Christopher Daniels who I'm a huge fan of Frankie Kazarian and Scorpio Sky versus the strong hearts Kimma or Sima Teahawk and L Lindemann I'm not familiar of those three but so-called uncensored uncensored sorry I'm gonna be doing this the whole freaking video I can tell but I'm a huge fan of Christopher Daniels whenever he was in impact or TNA if you remember the whenever it used to be TNA Ring of Honor him and Frankie Kazarian did great in that too I like their work I think Christopher Daniels for all late wrestling is a scout for talent but he also does in ring stuff hence why he's in the match so this one right here I think it's gonna go to Christopher Daniels Frankie Kazarian and Scorpio Sky I'm a huge fan of them I do like them and I have seen a lot of their work in the past and they do great work in my opinion so that's my prediction for this match next is gonna be the triple a all sorry the triple a world tag team championship match who the current champions are the Young Bucks which is Matt Jackson and Nick Jackson if you're unfamiliar with them they are the current champions versus the Lucha Bros Pentagon Junior and Ray Phoenix who are still part of triple a wrestling the Mexican wrestling promotion honestly I haven't seen much of the Lucha Bros until they came over to I think it's impact and then I think they did a little bit in New Japan I'm probably probably wrong with that Ring of Honor and now they're part of all the wrestling but they still are part of triple a and I think Ray Phoenix is a single champion over on triple a right now but I do like their work I didn't see them until they came to impact the first time I think that was over a year ago now but now I'm familiar with them so this tag team match is gonna be pretty epic in my opinion honestly I was gonna say the Young Bucks we're gonna win this match because that'd be interesting but something tells me that all elite wrestling in the future are gonna have their own tag team championship instead of having triple a so since the Lucha Bros are still part of triple a as well because that's one thing that I like about all elite wrestling is whenever you sign a contract the rest of us who sign a contract with all elite wrestling they can still work for other companies they can still do other projects work the indies work other events as long as they can still make the all elite wrestling events and press and everything else that all elite wrestling is doing as long as they can still make that they can work at other companies which I think that's a lot better than WWE where if you're in WWE you're restricted to only WWE with that being said since the Lucha Bros are still part of triple a I see the Lucha Bros winning the triple a world tag team championships the next match is one that I can't honestly give a prediction on and that is the triple threat female match it is Dr. Britt Baker versus Nyla Rose versus Kylie Ray I just am not familiar with either one of them and it's just not fair to me or through those wrestlers for me to make a prediction on their match when I'm not familiar with them the more all elite wrestling pay-per-views the more matches I see them in of course I'll get more familiar with them for the future but for this match I can't make an honest prediction for it unfortunately and the last two matches in my opinion either one could be the main event and I would be okay with it up first is Cody versus Dustin Rhodes it has been confirmed and announced for the pay-per-view a while ago that Dustin Rhodes aka Goldust in the WWE was leaving the WWE and was going to be facing Cody at the double or nothing pay-per-view now a lot of people are wondering why he isn't using the gold dust name in the gimmick which he's used for about 20 years now reason for that is WWE owns the rights to it and to me I think that's crazy I think he given it that he's been the character for 20 years he should have the right sports but regardless he doesn't so he's gonna be appearing as Dustin Rhodes with from what I see from the posters and everything it's basically gonna be the gold dust gimmick except it's gonna be red and black I guess it looks like it's gonna be his face is only gonna be half painted so it looks pretty cool not gonna lie a lot of people are questioning why that was announced to be Cody's opponents and why Dustin Rhodes is gonna be facing Cody once again brother versus brother kind of like they did a few times in the WWE and for me personally I have a backstory for it and a theory on why they are facing each other for this match for me I think it boils down to the age of gold dust or Dustin Rhodes which is gonna be how he's gonna be in all elite wrestling it's not certain if he's gonna have other matches in all elite wrestling it's not certain if he's staying if he signed a lengthy contract with all elite wrestling or if he's gonna be a backstage producer or something like that it's just unfamiliar at this time but given his age I honestly feel this is gonna be Dustin Rhodes or gold dust hanging up his boots at the end of this match I think that's something that's gonna be iconic and a lot of people are questioning why if that is gonna happen why didn't he retire in WWE and let's face it out of everybody gold dust has faced to retire to your brother and have your last match be facing your brother I think that is special and that's why he wants to do it if that's what's gonna happen I would be okay with that honestly if it is gonna be his retirement match which I doubt they're gonna be announcing until after the match takes place I think it's gonna go to Dustin Rhodes either way I think it's gonna go to Dustin Rhodes now Cody Rhodes is epic since leaving WWE he really shot up in my opinion and he's pretty much a top star but something tells me because Cody Rhodes his opponent was Dustin Rhodes and it was announced to be Dustin Rhodes something tells me it's gonna be a gold dust or Dustin Rhodes last match and he's gonna be retiring after this so it's gonna be interesting to see what happens but my final prediction is gonna be Dustin Rhodes and the last match is going to be a part two and that is gonna be Kenny Omega versus Chris Jericho of course they fought to New Japan wrestling that was a huge match an epic match and to me that match was awesome I watched that match it was pretty good it went to Kenny Omega he won it that back then now that they are both on all elite wrestling both of them were the top free agents whenever they were signed from all elite wrestling honestly I see this match going to Chris ain't Chris Chris Jericho I almost said Chris angle I don't know why I say this match going to Chris Jericho and it's unfamiliar if Chris Jericho what gimmick he's gonna have I don't think he's gonna be doing his face paint which he did in all a New Japan wrestling I don't think he's gonna be doing that but he said he doesn't know right now what he's going to do but he wants to do something new or something that's pretty epic he loved his New Japan wrestling character but he's not sure if he's gonna bring that over regardless this is the second time they're gonna be facing I see this match going to Chris Jericho that to me is an ideal situation and then in the future it in a different pay-per-view down the line maybe the two of them can face for the all elite wrestling World Heavyweight Championship I honestly see that being in the works if not it might be a triple threat or fatal four-way match with other top guys because my opinion all elite wrestling has a lot of top guys but these two are among the top right now in my opinion so the fact that they're facing each other I honestly see it going to Chris Jericho and they might have a third match in the future to basically break the ice to see who is the better man that's my prediction for this match but like I said some of the people I'm unfamiliar with right now but the more I watch all elite wrestling the more pay-per-views they have the more matches they're in in the future if they're in matches and I do another prediction video on the pay-per-view I'll know them a little bit better and I'll be more familiar with them and be able to give a better prediction on the ones I'm unfamiliar with but they still have a lot of guys that I'm familiar with whether it was them on Impact or Ring of Honor or WWE I did I did know quite a few of the wrestlers that are in this pay-per-view and that are part of all elite wrestling so this these are my predictions on all elite wrestling's double or nothing pay-per-view that it's coming out very soon I hope you guys did enjoy I know I did stutter a lot in this but that's because some of the names I'm unfamiliar with right now I'm just reading them off paper so I probably butchered a lot of them but I'll see you guys in the next video please take care peace


In the last lecture, we had introduced the subject of vector calculus and we discussed fields in particular scalar and vector fields. We also brought out the meaning of gradient of a scalar field and I emphasize that the gradient of a scalar field is a vector field. Today, I would be spending a little more time in discussing special functions of vector fields or operators which act on the vector fields. But before that, I would introduce what are known as the line and the surface integrals. We are all familiar with normal integrals. As you are all aware that an integral is always regarded as a sum. Now, what we want to do is this that let us first talk about a force which is acting on a particle and is in one dimension. We are aware that integral f d x if it is a one dimensional problem gives me the work done in let us say moving the particle from a point a to the point b. Now, in computing this work done, we do a simple integration. However, suppose my force acts on a particle and takes it along a curve. It could be two dimensional or it could be even a three dimensional path. It could be an open curve like what is being shown here or it could be a closed curve for example, like this. Question is how does one compute the work done in taking the particle let us say from a to b in this case or in a circuit in the other case. Now, we will represent the work done let us call this curves as c, c standing for curve. It could be a closed curve or an open curve. The work done as we are aware is given by dot product of f with d l where d l is along the path. Now, in particular the vector d l is directed it is an infinitesimal element directed along the tangent to the path.  Now, so what we want to do is to generalize the concept of computing work in one dimension to determining how much of work has been done when a force acts on a particle and takes it from one point to another or in a circuit. Now, the point is in general the this integral that is integral of f dot d l from a to b depends on not only on the end points as we are accustomed to while we compute integrals, but it also depends on details of the path taken. Of course, there are exceptions to this for instance if the force is a conservative force exact details of these we will talk about later in the day to day, but if it is a conservative force for instance a gravitational force or electrostatic force then it can be shown that the work done is path independent. However, in such a case it would always depend upon the end points. Now, let us look at what how does one write this. So, what I have is along the curve f dot d r let us say because d r is a an element along the curve and what we do is this that let me write this as f dot d r by d t d t where I have introduced time as a parameter time as a parameter and you notice that this is nothing but the dot product of force with the velocity with the velocity and I take a time integral. So, in other words what I have done is to parameterize that with time. Now, along a curve along a curve I always have a single parameter and this is what we are going to exploit in computing the work or in general an integral like f dot d l along a given path. Now, let me start with an example. So, for instance I have given here a force which is written as f is equal to now I have taken in two dimension one could easily extended the curve to three dimension, but let me illustrate this by taking a force whose x component is x square minus y square and whose y component is two times x y. Now, what I am going to do is to compute the work along three different paths in both the cases I am going from a point which is origin let me draw it here again origin to a point two one. So, supposing this is one along the y axis and this is the point two along the x axis. So, from origin to the point p which is the point two one first I will calculate the you know if this is the force the line integral of this force along a straight line joining the point o to b o to p. Now, notice since this straight line joins origin two two one I can easily find out what is the parameter or what is the equation of this line. The equation of this line obviously is x equal to x equal to 2 y alternatively y is equal to x by 2. Now, let us look at how does one compute this?  Now, f x is x square minus y square, but we have seen that y is equal to x by 2. So, therefore it is x square minus x square by 4 d x the limit of integration over x is obviously from 0 to 2. I write the y part again. So, f y is given to be 2 x y, but x is 2 y. So, as a result I get 4 y into y is y square d y and limit on y is 0 to 1. You compute this, this is this trivial integration you will get this to be equal to 3 by 2 plus 4 by 3 which is equal to 17 by 6. So, that is the work done along this line. Now, let us compute the work done along line along this curve 2. Now, I have taken the curve 2 to be a parabola I have taken the curve 2 to be a parabola and my equation of the curve 2 is y is equal to x square by 4 I have given this. You can see that the it is of course an equation to a parabola and when x is equal to 0 y equal to 0 and when x is equal to 2 y is equal to 1 because it is 2 square by 4. Now, how does one do this for the same force field or same f the vector field. So, once again I will compute this. So, this is now 0 to 1 I have got x square I have got y square, but y is equal to x square by 4. So, y square is x to the power 4 by 16 d x plus 2 times integral x y. Now, x is 2 root y. So, I have got 2 times 2 square root of 2 root y and I had already a y there d y and limits are once again the x limits are from 0 to 2 and the y limits are from 0 to 1. So, if you compute this now this very straight forward x square is x cube by 3 x to the power 4 is x to the power 5 by 5 which gives me x to the power 5 by 80 limits are from 0 to 2 and this is 4 integral of y to the power 3 by 2. So, I get y to the power 5 by 2 divided by 5 by 2 limits are from 0 to 1 and you can easily compute this. This happens to be 34 by 15 plus 8 by 5 and is equal to 58 by 15. Now, you already noticed something that the value that we got for in the first case was 17 by 6 whereas, the value that we got in the second case is 58 by 15. Now, you can see that the value of the line integral indeed depends upon the path that we have chosen. In the third case, I am taking a path from origin to again to 1 by a little complicated path. I am first going from here 0 0 to 0 1. So, let us draw that path here.  So, this is 0 1 this is 0 0. So, I go like that first like this then like this. Recall my force is x square minus y square x component and the y component is 2 x y j. Now, look at this. When I go from origin to this point, let us call this O P Q. So, as I go from O to Q, my x remains 0 the value of x does not change. Now, if x remains 0 further along this path along this path d x is equal to 0. So, notice that if I now compute this integral from O to P integral of f x d x that is x square minus y square d x that gives me 0 because d x is 0 f y d y namely 2 x y d y gives me 0 because on this line along the entire path my x is 0. So, therefore, contribution to the line integral from the path O P. So, let me write that as integral I 1 is equal to 0. In the second path I go from P to Q. So, I will go from what I meant is I go from O to P and then I will go from P to Q. Now, when I go from P to Q, when I go from P to Q, on the P to Q my value of y remains constant y is equal to 1. Now, once y is equal to 1, d y of course, becomes equal to 0 because constant have differentiation as equal to 0. So, since d y is equal to 0, I am only going to have to compute f x d x. So, this is simply 0 to 2 x square minus y square which is 1 d x and this is equal to x cube by 3 minus x evaluated from 0 to 2 which is 8 by 3 minus 2 which is equal to 2 by 3. So, you notice one thing that I calculated this integral f d l f dot d l in three different paths 1, 2 and 3 and every time I got a different result. Now, I will talk about a slightly different technique now. I will return back to the same integral with a slight modification as I go along. Now, look at this field, remember we are talking about a vector field. Now, this vector field is f is equal to minus y i plus x j. Remember last time we gave a prescription for drawing the vector fields, catching the vector fields. So, you could see this vector field, this is and I am going to compute the integral of this vector field in along a quadrant of a circle, along a quadrant of a circle taking me from this point that is 1 0 2 0 1 in the anticlockwise fashion 1 0 2 0 1 in the anticlockwise fashion. Now, if you look at the sketch of the vector field that I have made, the sketch of the vector field that I have made you notice that in this quadrant, in this quadrant the because x is positive, because x is positive there is always a positive y component of the force field of the vector field. Now, as a result loosely speaking these vectors are all directed somewhat in the upward direction, very loosely speaking. Now, another thing that you notice is this that the vectors are directed in the upward direction and I am going to compute my line integral along this path and in this path the direction of d l is like this. So, therefore, the angle between the direction of the vectors the which we have shown in this sketch and the direction of the tangent to the curve always makes an acute angle.  Now, it so therefore, what I expect it is a purely going by geometry I expect this line integral to be of positive value. Here I will use a slightly different technique the technique is that along the curve I need to have a parameter I have already said that a along a curve I have a single parameter. Now, notice that if I have this curve which is quadrant of a circle. Now, I can use a parameter like this using this angle theta as a parameter. Now, if theta is a parameter notice that in term this is a quadrant of a circle of radius 1 because I have said the equation is x square plus y square equal to 1 and I need to compute minus y d x plus x d y. In terms of this theta since the radius of this circle is 1 I can write x is equal to cos theta and y is equal to sin theta. So, d x is minus sin theta d theta and d y is cos theta d theta. So, I notice that this integral now which is over the curve C minus y is minus sin theta d x is another minus sin theta d theta.  So, I get cos square theta d theta. Now, I can simply add them and get it as sin square theta plus cos square theta which is equal to 1 of cos d theta. So, which is nothing but integral of d theta and the limits on theta we know is from 0 to pi by 2. So, as a result the result of this integration is simply pi by 2. So, this is the parameterization. Now, in general as we said in the beginning the line integral depends on both the path of integration and the n points. Now, if the line integral f dot d l depends on the path of integration and if f is a force field then we say that this force is a non-conservative force. That is a non-conservative force is one where the line integral depends upon the path of integration. As we had seen in the first example that we gave you that there were three paths along which we calculated the line integral and found the values to be different. However, f dot d l only depends upon the n points supposing I am going along some path from a to b and it does not depend upon exactly which path do I take. I do I take this path or for example, that path or this path I do not care result is always the same. In other words this line integral in such situation depends only on the two end points a and b. Such a force is called a conservative force such a force is called a conservative force and you can see that if I have a closed path supposing I am calculating the line integral like this and closed path integrations are written like this with a circle over the integral sign it is also called contour integral closed contour. For a conservative force for a conservative force you can show that this is equal to 0. Now the proof is really straight forward supposing this is the path you are going starting from the point a you are returning back to the point b. Let me take any path arbitrary point b on the way. So, you notice my integral is from a to b by path 1 then once I reach b I am following a segment of the path which is 2. So, we will say b to a by path 2 now what you have said is this that this integral from a to b depends only on the end points depends only on the end points. So, in other words this integral f dot d l so a to b f dot d l along path 1 can be written as something like phi b minus phi a just I evaluate this and then subtract from the value of that integral standard definite integral result at b minus result at a. When I am going on path 2 I am going from b to a f dot d l now once again we have said that it depends upon only the end points. So, as a result this is going to be phi at the upper limit minus phi at the lower limit which is b.  So, for a conservative force the integral line integral from a to b depends on only the end points and if you take the force field through a closed contour the line integral is identically equal to 0. Now, let us spend a little more time on the conservative force field the reason is that the field which is of importance to us in this course namely the electrostatic field is a conservative field so is gravitational field. Now, suppose I have a force field vector field which can be written as a gradient of a scalar field force field can be written as a gradient of a scalar field. Now, remember we had said the gradient of a scalar is a vector now I am not saying that all force fields can be expressed as a gradient of a scalar field. But supposing it can be done now let us see what it means what it means is this supposing I am going from a to b again I have to calculate f dot d l. Now, since f is a gradient you remember the expression for the gradient of a scalar function in Cartesian and we had seen that this is equal to I partial phi with respect to x plus j partial phi with respect to y plus k partial phi with respect to z dotted with dotted with d l which is obviously I d x plus j d y plus k d z. Now, take the dot product remember I dot I is 1 j dot j and k dot k are also 1, but I dot j I dot k etcetera are 0.  So, in other words this is from a to b just d phi because this expression includes changes due to variation in x y and z. Now, this is the an exact differential. So, as a result the integration is very trivial it is phi b minus phi a. Notice what we have done we have said or we have proved that if a force field is expressible as gradient of a scalar function then integral f dot d l only depends upon the two end points. In other words such a field is conservative field such a field is conservative field. Now, we make a statement at this stage that if I have a conservative field of force for instance an electric field or a gravitational field I can always express the force as a gradient of some function. Generally because of various reasons instead of writing f as equal to gradient of phi we define f as equal to minus the gradient of phi and this phi is then given a name called potential this is conventional, but in principle a mathematician would call phi as a potential whether you have a minus sign in front of it or not, but we will of course in electrostatics we will be using this convention namely force is negative gradient of a potential function and the potential is a scalar function the potential is a scalar function. So, this is about line integrals. So, we have seen line integrals in general depend upon the path that we take in some cases when they do not we call the vector fields as non-conservative field in which case the line integrals only depend on the end points line integrals only depend upon the end points. So, line integrals was essentially one parameter integral though I am not saying it is in one dimension, but I go along a line I go along a curve. So, there is a single parameter in such a situation. Now, let me now extend this let me now extend this to define what is known as a surface integral. Now, in surface integral we will write this surface integral as integral of f dot n I will explain what it means d s. Now, in order to understand the surface integral look at this situation let me take an open surface for example, a hemisphere which is an open surface a better situation is to have a net for example, a butterfly net as it is called you notice that there is a rim here there is a rim here and there is a net which is bounded by this ring and this is an open surface in the sense that this surface is there only to one side of the boundary. Now, this is exactly the picture that we gave in the other one. So, I have an open surface and so what we do is that whatever is the bounding curve of the open surface. So, imagine I have a cup imagine I have a cup as hemispherical cup or even an ordinary cup it does not matter. Now, this is my bounding surface this is my bounding surface and a curve which binds it again going back to my butterfly net this is what it is this is the rim this is the rim and this net is the surface. Now, when that happens we say that direction of this n it is a unit vector which is perpendicular to the surface. Now, it is not a constant vector it changes from point to point it changes from point to point. So, for example, you are at this point. So, draw a perpendicular at that point and draw it outward. Now, what is meant by outward because after all it the outward needs to be defined the. So, what we do is we adopt a convention which is called a right handed corkscrew rule which is with which you are all familiar that is if I am rotating or if I am moving suppose this curve is being turned along the direction of a corkscrew then the direction in which head proceeds is the direction of my normal to the surface the outward normal and. So, that is why I said a butterfly net is very clear. Now, there is some issues connected with it. Number one that I need to have what are known as two sided surface in order that my definition of a surface integral makes sense I need a two sided surface. What is a two sided surface for example, an open hemisphere it has an inside surface and an outside surface you take a cup you have an outside surface and an inside surface. Now, on the other hand there are certain surfaces which are one sided surface and for example, one of the examples is what we call as a mobius strip it is diagrammatically shown here, but let me try to explain what is a mobius strip. Now, I have this strip of blue strip of paper I have taken a small strip with it. Now, notice that if I just fold it like this making it a ring then I have two sides in other words supposing I am on the upper surface and I want to go to the lower inside surface. Then I have to always cross an edge I have to always cross an edge and this is known as a double sided surface. This is true of for example, if I take this glass at this moment there is water in it, but this is the outside surface and there is an inside surface of the glass. If I want to go from the outside to the inside I will need to cross this ring I will need to cross the edge, but on the other hand please look at it this way supposing I take this again this is diagrammatically shown here supposing I take this strip again and instead of folding it like this I will give it half a turn I will give it half a turn and fold it in this fashion fold it in this fashion that is the same side does not get stuck I will get. So, I have actually already stuck it with a glue and this is the type of picture that I get. Now, notice what is so special about this this special about this is I can come from any point on this surface and traverse both sides both inside and outside by ever having to cross this ring. So, in other words at a given point the outward normal is not uniquely defined the outward normal can be both outward like this or inward on the inner surface this is a one sided surface. So, for such surfaces this example this is called a mobius strip my definition of a surface integral is not valid. So, let me talk about normal surfaces which are two surface two sided open surfaces and in going from one side to the other side I need to pass through a rim need to pass through an edge well here the same thing the this is the hemisphere surface is A and the rim of the hemisphere is the curve C and I have to sort of define that if I am going in a particular direction then the direction in which the head proceeds will be the direction of the normal outward normal. So, that is about two dimensional surface integrals. Now, one can now go and if I now remember the surface integral is dot product of a vector field with the outward normal and integrated over the surface this is usually called a flux of the force field F through this surface flux of the force field F through the surface. Now, we can now extend this concept to a three dimensional integral, but this is much simpler because the integral is simply a volume integral over a scalar function is phi and. So, this is nothing, but normal triple integration which you are familiar with. So, volume integral phi d v v here is the volume is nothing, but triple integral over x y z of phi which is a function of x y and z now I can define another type of triple integral or volume integral which is F d v that is an integral of a vector field over a volume.  So, this is nothing, but i integral F x d v now remember F x is a scalar function plus j F y d v plus k F z d v each one of them we have just now learnt what it means because this is just like any other volume integral of a scalar function which is nothing, but a triple integral. So, therefore, this is i times 1 triple integral plus j times a second triple integral plus k times a third triple integral. So, this is all about the line surface and the volume integrals. Now, let me concentrate a little more or spend a little more time on vector field once again. Now, I define what is known as divergence of a vector field the reason for the name will become clear in a little while. Now, I define divergence of a vector field as a limit of the surface integral per unit volume I will explain this with figures as I go along as the volume enclosed by the surface becomes smaller and smaller becomes infinitely smaller. Now, let us look at this relationship a little more carefully and now divergence of the vector field is written as a dive dive F one also writes it for reason to become clear as del dot F both of them are identical ways of expressing divergence. So, what I have said is this that now this is a point relationship this is a point relationship in all the words you define divergence of a vector field at a particular point. Now, supposing around that point around that point I think of a small volume infinitely small volume described by a surface d s. Now, so what we do is this take this volume to be infinitely small let us call it delta v. So, I define the del dot F as limit of delta v going to 0 of the surface integral F dot n d s. So, I have taken a closed volume very small closed volume at the point where I am computing the divergence of F. So, that is that is why this circle it circle essentially means it is closed. So, over a closed surface I have taken and divided by the volume enclosed by that surface take this as the volume becomes smaller and smaller I reduce that volume essentially to that point and that is why I say it is a point relationship. Now, this picture is what explains what happens now supposing I have a volume bigger volume. Now, what do I do let us consider a volume now I can think of splitting that volume into a large number of infinitesimally small volumes. Now, let us look at what am I trying to do in the process now look at this supposing I have a volume of this type this is a rectangular parallel pipe and I have a volume and notice I am at this moment concentrating only on two surfaces. Though it has six surfaces I have a upper surface whose direction is outward normal is this direction and a lower surface the this surface the bottom surface whose outward normal is directed like this. Now supposing this surface I am to split into two smaller surfaces. So, in other words this big surface this big volume gets cut into two I take a section and just split it now when I split it remember nothing has happened to the direction of the normal to this surface nothing has again happened to the normal to the bottom surface, but once you cut it in this section in this section the normal is directed like this and in this section the normal is directed like that I do a mental exercise. So, if I just cut it since the vector field is the same over this surface which I am going to cut then the contribution due to the surface integral from this surface in this surface because they are directed oppositely will become 0. So, what it means is that if you have a volume if you have a volume and you split that volume into in. So, supposing I have a volume like this and I split it up into smaller volumes etcetera. Now I can reintroduce talk about the same argument once again and say that the contribution from things which are inside the pairs will always cancel and what will I be then left with I will be left with only contribution from the outside surface only contribution from outside surface well. So, let us now try to compute we have said that divergence of a function is the ratio of the limit of surface integral over the vector field at a point I am saying it is a point because I have an infinitesimal volume there bounded by an infinitesimal surface. I divide it by the volume and I let the volume and the surface go to 0 and this ratio is finite and that quantity I call as the divergence of the vector field at that point. So, what is this divergence of a vector field? This picture here sort of gives you an idea of if here we have drawn the pictures of the vector field you notice as the name suggests the field gives an idea of how much the vector field diverges from a point. The situation becomes clear when we look at for example, a velocity field in a vector we know that if there are if I look at a closed region then if the fluid is flowing in to a point then of course, the density inside would increase and similarly if the fluid is flowing out the density would decrease. So, here is an example where the field is spreading out from the origin. So, here the divergence is positive in this case however the divergence is negative this is the physical meaning of a divergence of a vector. So, to summarize what we have done today is we defined a line integral and emphasize the fact that in general the line integral of a vector field could be path dependent. In case it does not depend on the path it is called a conservative field and we have seen for a conservative field I can write the force as a gradient of a potential we defined a surface integral and also defined the divergence of a vector field through a ratio of a surface integral of a closed small surface at a point with the infinitesimal volume that it encloses. Next time we will be obtaining an explicit expression for the divergence in Cartesian coordinate system.


A good question here is how close or intimately should we allow our clients to be to be with us. At what's point should we set boundaries. You know I have some thoughts on this and before I even get going I know a lot of you have watching this have thoughts experiences, whether you as a client trying to get close to a service provider. Or you as a service provider getting close to your clients. What are your thoughts on it. I, I, I'm asking not just because I want more comments on this video. No, I am genuinely curious because I'm just one person with some thoughts so please feel free to comment below I would be helpful to others also. Well, when I saw this question, kind of the first thing that came to me was about sort of time and energy expenditure, because I care so much about joyful productivity and so that's one element the other element we'll talk about is more qualitative boundaries and sort of like intimacy. But let me first talk about the time, time and money boundaries. And for me, I'll talk from my own perspective because that's what I know best so like I said please feel free to share your your perspective below, others will benefit. My perspective about it is, I am very strict about I should say I'm relatively strict strict I think compared to most heart based or spiritual service providers I'm relatively strict about my time and energy boundaries and money boundaries.  You know, we would, you know, I tell my clients listen if you're ever in town, I would love to take you out for dinner, take you out for lunch and I mean that and it's like if they're in town for for a month I probably take them out to dinner or lunch two or three times. I pay for it because I want to hang out with them and spend time with them. And, and, but, but, but when it comes to them paying for my programs, or my services, they pay just like everybody else pays, because I only have a limited amount of time to work to serve clients just as you do. Now, if if I for some reason want to talk to that person on a personal level like we're talking about other things. That's not about the service I provide for them. Then I might set up a time to talk with them about that. If they want to talk to me about other things, not not related to the service I provide. They can ask me and if I feel like I want to spend the time to do that. I will do that just like with a nut with any friend right with any friend if a friend wants to talk to you about something wants to go see a movie with you wants to talk to you about something in their their life. You decide whether you have that's that's now the question of friend boundaries whether you want to spend your Saturday evening talking to your friend about such and such it's up to you and you probably have some way of negotiating with friends. You know, I'm really tired, really, really tired this week I can't talk to you but maybe next weekend or something like that right so. So, when it gets to that kind of time boundaries, it's like, well if you like you, if you like that client, you want to talk about non business non service oriented things. Then you negotiate as a friend would, but when it comes to service oriented things they, they, they want to talk to you about the thing you provide a service for, if you're a language teacher, right, and they want to talk to you about how do you describe the particular situation and that language. It's part of your service.  This is really something we should do a session on, because this is something I, you know, really should can be considered a client relationship and let's book a time for that right something like that it. That's my preference is to be really strict about that because. I really wish you was she and it's like to. But again, there are always exceptions that you decide you can make if you want to, but I want you to notice whenever you feel when you when you start to feel a little bit. Resentment start to crop in or the possibility resentment starts to crop in. That is your mind body system. Signaling to you that a boundary has been crossed or could be crossed, leading to a negative relationship. So, when you start to notice resentment, I think you need to practice saying, of course practice graciousness. And if you don't know how to say it graciously, these days there are things like chat GPT, they can go to this is, I need to set a relationship with a client. This is a situation. How might I say it gracefully and chat GPT and other tools like that will give you a wonderful email to send them and you could, you know, and you could get multiple versions anyway. There are tools like that these days that that help you with with the etiquette. But yes, I would encourage you to to notice a resentment possibility that set boundaries. It's good practice for your own. As a service provider and your own confidence and your own sense of integrity in your business. I've been really strict about boundaries for years now and I think that's part of the reason that my business has continued to grow and thrive and I've been able to keep a lot of balance in my life productivity. Okay, so that's that's time energy element. The other element about it is more qualitative in terms of how much should you share with your clients about your personal life. The fact that you have a cat who is randomly yelling at random times in the day. That was my cat and you didn't hear. And how much did you allow your clients to share with you open up about their, their private life. If it's not related to the service you provide obviously if you're a therapist. Right, that's different but if you are, you know, or or or if you're if you're like, like, I'm a business coach. Now my clients start telling me about their personal relationships and things like that. Well, it depends. It's up to you. You know, some people are their service is like really kind of mixed like they are a business coach but also a life coach and also a spiritual counselor and if they allow themselves to to hold that as part of their service. Well then fine. But for me, I don't really want to get into coaching people about the relationships or their spiritual growth or whatever. If I do that would be a separate service. So I would be like, you know what. Yeah, you know, that's that's really tough situation but let's get back to what I can help you with. Right. And so and in terms of you sharing like, you'll notice that I'm, I'm pretty pride, especially on social media I'm very private about my life I don't say anything about my private life. And with my clients sometimes in small group someone sometimes on calls I might share a few things but just just as a, maybe icebreaker, but I but then I kind of am pretty private after that. So it's really up to you. I think it takes your experience of this kind of thing to start noticing when are you feeling a little bit too attached to that person. And if you are, then it's your decision whether or not you want to, you want to do that or not. I think it's, I think it's a fine experience to have, and then just notice whether it feels healthy for you. Anyway, that's a few thoughts. I welcome you to chat below and, and last thing I'll say last thing I'll say sometimes a client might say, oh, can we make an arrangement about payment. That's different or about how often I see you that's different than what your service usually is. And you, you might say well I'm feeling close to the client and so I feel like I should make this arrangement with them. This is an exception. I mean, I don't think it's rude that's I don't think that's an issue of how close you are to a client that's just an issue of whether or not you want to make exceptions for your service. Because sometimes a client that's brand new to you might even ask you for those exceptions. So it's not about closeness it's just about whether or not you make deals. You make exceptions and make custom deals with your clients. It's up to you. I know a lot of service providers do make custom deals and you usually see clients you know for this package you see them every single week and they pay this much. But this client says can I see you every other week or, you know, and can I pay this much instead of that. Well it's up to you. For me, I keep things simple and I basically say no. Okay, but when my business was much smaller and I just wanted to work with more people. I was willing to make a lot more exceptions and make custom deals for a lot of people whether or not I was close to them. So it's up to you because you just have to remember these different custom deals and and then maybe right now, you have the band what to do that but as your business grows, you'll probably have more, you know, less of a bandwidth and you'll get more strict about your packages and things like that. So I hope this is helpful. Again, please comment below and share what your experience is with getting too close or intimate with your clients or as a client trying to get close and I'll tell a quick story. But right now I'm learning Spanish, you know, and I have a service I use I actually love the service called italki I recommend it it a LKI. And it's just it's like fiber except to hire, you know, different language tutors. And one of my language tutors I really enjoyed and when I went to visit Mexico this time. And I take you out, you know, my wife and I can we take you out to dinner. And she said yes and we took her out and we just became like friends like it felt like that and we're on what's app together we chat we send each other funny stickers. And as a client here, I'm starting to feel like oh I hope I hope, you know, I hope she'll see me as a friend, but I'm also very aware as a service provider myself I'm not going to take advantage of that right like if I'm going to have language questions I'm going to hire her properly. And so, yeah, I think it's interesting I think we should notice how we treat service providers and notice that kind of boundary, as well as a service provider so anyway I look forward to seeing your comments below. Thank you. I hope this is helpful.