So I hope, yeah, I think we can start the session. Once again, welcome back to the live session on bookmap tools. Basically, we are going to discuss about the tools that are offered by bookmap and how to utilize them in order to effectively trade the market. So basically, we are all, we're going to discuss about the nifty and bank nifty in the first place. And after that, I will also discuss the same concepts on the present condition of market, which is crude oil, which is presently running. So I will also discuss the concept on that instrument. All right, so if that is clear, let's quickly get into the session. So before I proceed, it's a quick disclaimer. Anything that I'm going to share from now onwards is going to be an educational information. Please don't consider it as a recommendation or by our self-recommendation. Please consider it as an educational information. Any information or any decision that you take in stock market trading and investing are subject to market risk. My name is Nataraj Malwade, a full-time trader. Basically, a rule-based intraday and swing trader. This is about myself. So I have also published my book called Mastermind of Trade Trading, all right? So let's quickly get into the subject, just a moment. Yeah. So the first important thing is, let's analyze the nifty in the first place. Before we jump into the concepts of book map, we look into the information of the price in terms of the historical behavior of the price actions and then we will move on to the book map charts, okay? So nifty continuation future, presently I'm using AmiBroker software by the data is powered by TrueData, all right? So this is a nifty presently it is set to monthly chart. Let me change it to 30 minutes. Since we are, we trade in a very short term market, I mean short term trading, so we use 30 minute chart. So this is how the entire 30 minute chart of nifty looks like. So the first thing that we're going to do is basically to identify the trend. That's the basic framework I follow. So I basically call it as T-L-E, right? So I will quickly explain this and I will quickly move on to the next subject. So basically T means a trend. So you will identify the primary trend of the market, whether my market is moving in upward direction or downward direction or sideways direction. This is point one. And then you will identify the locations where you can possibly look for an opportunity, right? So for example, support and resistances, basically if you're a day trader, obviously previous day high, previous day low, and obviously if intraday trading, the first one hour high and low also matters the most, all right? So those are the key levels where we can monitor our price action. So that's what we call it as the second step, which is location. Once these two steps are clear, then we will move on to the third step, which is getting in and getting out of the trend. That's like we can call it as entry indexing. And in the third step, we will be utilizing the order flow information offered by Bookmap in order to take effective decision. So don't worry, I will explain the entire concepts in the upcoming chapter. So right now let's try to focus on first two steps, which is identifying the primary trend and the locations. So what is the importance of it? Basically when you know where the primarily market is heading. So all it takes us for us to align with that particular trend. Assume if the market is moving in upward direction by making higher rise and higher lows. So we can deploy the buying strategy whenever there is an opportunity. For example, you can buy the break off last swings or you can get into the pullback entry when it approaches to the last support. So that's also one of the strategy. So ultimately, when we are aligning with the primary trend, the possibility of we making profits increases. This is the importance of identifying the trend. And what about the location? See we cannot take random decisions just because you know it is offering. Even though we identified the trend, we need to wait for the price to approach to a certain level. That's what we can call it as support and resistance. When price approaches to those level, the significance of our entry and exit increases. And hence we look for trading opportunity when price approaches to a certain level, okay? So if these points are clear to, then we will move on to the third step. So let's try to focus on the first two steps which is identifying the trend and the locations, all right? So as we can clearly see that after breaking off last support from this scandal, market is trending in the downward direction. No doubt about it at the present moment, market is presently trending in the downward direction. However, today's condition is little bit different. As you can see, till this point market was smoothly moving in the downward direction, but today the volatility kicked in the market. So that's what we can see after formation of one hour. There is a huge volatility that you can see in the first candle itself, there is a huge volatility and market stayed within that particular range, all right? So there is a involvement of the volatility at the present moment. Even though market is presently moving in the downward direction, the volatility is high. This is point one. And with respect to this, the immediate locations that we can look for the price action or basically order flow information is basically this high which is an immediate swing high and the immediate swing low. How beautifully it has taken the support that you can clearly see that's why I said the first one hour high and low plays a critical role, all right? Once this is done, we need to wait for a trade opportunity. For example, in this case, according to this context, we can either look for shorting opportunity when price approaches to last swing, which is let me mark that level. Around 17,830 level is the key level that we can monitor when price approaches to this level. We can look for shorting opportunity based on the triggers that we get from order flow information. I will explain that in the upcoming section. And then we have one more important level which is basically the low where we have seen the immediate bounce in the ending session of the market. That's the bounce we have seen. So that's basically 17,550 price level in nifty. So these two levels are absolutely critical for tomorrow's market. Considering the primary trend is in downward direction, we can look for shorting opportunity when price approaches to this level based on certain conditions. Suppose if the price starts to accept above this particular level continuously, then the sentiment changes. Instead of short, we can look for long-term. So for tomorrow's market, this particular level is going to play the critical role for nifty futures, okay? So this is about nifty. Let's add bank nifty and see what is happening with respect to the bank nifty. So at the present moment, considering to the nifty, bank nifty is still struggling. So because from this particular candle, after breaking up this particular support, price is continuously turning in the downward direction. The same context applies here. The volatility has kicked in and this low and this side. They basically today's high and today's low is going to play the critical role. We need to monitor these levels for tomorrow's market. So for example, let's mark that level here and here. So at any given point of time, if market approaches to this level and takes certain patterns, so we can look for shorting opportunities until unless price sustains and accepts above this particular range, we don't look for long opportunities, all right? So we look for shorting opportunities primarily because primarily market is moving in the downward direction, but we look for long opportunities only if the price accepts above the 41,000, around 41,150 level, somewhere around 200 level, okay? So this is what I call it as a basic context. If this is clear to you, we can move ahead and we can move on to the next part which is getting enter and getting out of the trades using the order flow information. So I hope this is clear. If you have any particular questions, please take a note of it, I will ask them. So you can ask the doubts and queries in the Q&A section, okay? So this is about Nifty and Bank 50. Let's quickly analyze crude oil as well because it's presently running. We will utilize the order flow information in that case. So it is beautifully trending in the downward direction and some major activities are happening in the crude oil. So with respect to this also, okay? So last range, this was the range which was, until this point, from this point, market was trending within this particular range and yesterday in this candle, it has broken that particular pattern and again it has took the resistance to the same level, polarity level and then market is again falling. So this is a clear technical pattern that we are seeing. With respect to this, obviously we should have shorted at this particular level based on some triggers. So I will explain that. And at the present moment, we can align with this particular trend whenever there is in the microstructure of the price, if there is any pullbacks and if it gives any particular information to short, we can still look for shorting opportunities in crude oil. So this is an important of understanding why the basic structure, understanding the basic structure of the market is essential, okay? So if this is clear to you, then we can quickly move on to the next point where we look for entry and exit opportunities, okay? So for that, I'm opening a book map chart. So this is how the book map chart looks like the moment you opened it. And there are so much of information, so don't worry, I will be simplifying things to you. So basically today I'm going to explain about what this exactly, so whatever the information that you are seeing. So what exactly the new information does. So just give me a moment, let me disable everything and yeah. So the moment you open it here, you'll be able to see the bubble kind of a chart and also the lines within them. So try to focus on the lines that you are seeing here. So basically that indicator is called as BBO indicator, best bid, best offer indicator. So what the software does smartly is that it historically collects the data of the best bid and best offer available at any given point of time. So if you open a depth of any particular instrument, you'll be able to see the best bits and best offers, the five best bits and five best offers in that the first best bid and first best offer will be recorded historically. So what is the use of it? What can happen with respect to this? So basically if I enable the candlestick charts, okay. Right now the information that you are seeing with respect to the candle, we will only be able to see open high low close. That's the 2D information, but if you look within the candle, all right, the best bid and best offer indicator, it is providing us an indication or the pattern within the entire 15 minute candle. So if you just look at the candle, you will be able to see only OHLC, but if you are seeing a BBO indicator, it gives you the structure, the price microstructure within the candlestick chart. So what is the use case of it or what is the advantage of it? The main advantage is that you don't have to wait for the candle confirmation, point one. And point two, you will also able to see the price pattern within the candlestick or within a specific period of a time. So candlestick reliance on the specific period of a time, but this particular indicator reliance on the present activity of the market with respect to the best bid and best offers, all right. So if this is clear to you, let's try to understand one more indicator that you are seeing here, the basically the bubble, okay. So bubble is nothing but an indication of delta, okay. So now you can ask, what is delta? In simple words, at any given point of time, there will be a market buy orders and market sell orders. So that's what moves the market up and down, all right. So assume for, you know, 6,000 level, any particular level, there is five market orders placed, which market order placed and executed and 10 market order got executed in terms of the selling orders. So delta simply means market buy orders, executed market buy orders minus executed market sell orders. So which equals to minus five. So for that particular price level, minus five is the value of the delta. So the red bubble indicates the negative value. That means at this particular price level, sellers were in control in terms of the aggressiveness, market sell orders placed more in this particular price level. Similarly, the green market buy orders placed more at this particular price level. This is one of the information and most importantly, you will also be able to see the relative difference. For example, even though they are same in the color, you can compare this particular two bubbles, even though they are same in the color, relatively, we can see this particular price level attracted more market sell orders. So don't worry how to utilize this information. I'll definitely come to that point. So try to understand the concept. Basically this bubble indicates the executed orders details and it by persecutes the buy and sell orders and it also gives the relative difference between the executed volumes, okay? So if this is clear to you, then let's try to understand one more indicator called absorption indicator, okay? This is one of my favorite indicators. So just give me a moment, just a moment. Just give me a moment, yeah. So what absorption indicator does, okay? So I would like to show you this in the here. So this is a nifty chart and this was the day low, okay? So assume this was the day low until price approach to this particular level. As I said, the day low is going to play the critical role. When price approach to this level, how beautifully it has reacted? So basically this was the day low and it approached and we saw the absorption. So the question here is what exactly the absorption is all about? If you open the setting of absorption indicator, this is the setting that software considers to calculate the absorption. So within 2.5 seconds, you know, this much of orders executed, then the software will consider that condition as an absorption. There is a reason behind it. The main differentiator here is the time. So within a fraction of second, many orders are getting executed. The question here is at any given point of time, the software calculates only or software considers the executed orders. So in that case, for example, assume market is moving in this direction and someone placed around 5,000 quantities, okay? 5,000 market buy orders, aggressively, okay? Market buy orders. And the question here is who consumed those orders? So obviously there has to be an opponent party. Someone is sitting with 5,000 quantities of limit sell orders. That is the order which consumed these particular orders within 2.5 seconds. If it is happening within a fraction of seconds, then it is an absorption of these buy orders. So the one who entered here, we can assume, maybe a smart player who was already waiting for price to approach to this level to consume this much of orders within 2.5 seconds. So that's the significance of absorption indicator. So if it happens in the buying set, that means absorption of the buying volume indicates the presence of passive sellers. If it happens here, for example, market moving in the downward direction and suddenly the selling orders that placed here, the huge orders consume, which indicates the presence of passive buyer. So at most of the situations, in most of the cases, absorption happens at a specific key levels and most importantly it gives us an opportunity to divide or understand the difference between normal structure and an important structure. What I mean to say, for example, this is a BVO indicator and if you just observe it, there is so much of fluctuations going on. So there are, even though it is a structure, so there is a lot of uncertain structure which is going on. But when you consider, when you include the absorption indicator along with this particular pattern, you will be able to spot the real swings. So for example, in this case, here is where we saw the absorption, you know, presence of passive sellers, basically absorption of buying volumes, here also the same thing, here also we are seeing absorption. So basically when we include absorption, we will be able to filter out the unnecessary swings that is given by the BVO indicator. So in that particular case, it will help us to identify the clear pattern of the microstructure of the price. So in this case, what I mean to say is that, for example, from here market took a reversal here because we saw the absorption and again without any major pullback, again market fell back and again the same thing, here, here, here and here. So if you just rely on a BVO indicator, you will be getting lot of these patterns, but along with the absorption indicator, we are now able to differentiate the clear swings with respect to this. So it will help us to identify the real things, possible real swings and with respect to that, we can identify the pattern of the structure and we can simply align with the trend. For example, assume you are looking for a shorting opportunity in today's market, Nifty, okay? So you are looking for a pattern where you can look for shorting opportunities. How we can do that? So basically the swing lows, right? So basically last swing low, assume market was moving here upward direction and suddenly there is a pullback and again there is a pullback. So again, there will be a second pullback. The break of the second pullback, the break of the second swing will be the confirmation to get into the trend. In this particular case, here is where we saw the first pullback and second pullback. Either we can short the market at this particular level or we can short it here. This is one of the use case of absorption indicator, point one and point two, we can also use absorption indicator to keep trailing our stop losses.  In that case, you can keep your stop loss for the last absorption and same pattern continues. Again, again, if there is one more break with an absorption here, we can simply trial your stop loss to the next absorption level. So absorption indicator can be used either to enter to the trade or also we can utilize this particular tool to, you know, keep our stop loss and trailing our stop loss whenever trade works in our favor, okay? So I hope this is clear. Yeah. So if this is clear to you, let's try to understand the same concept, okay? I have a few more indicators to discuss but before that, let's try to understand in the crude oil itself, okay? So this is a crude oil information and yeah, I forgot to inform you that in absorption indicator, there are two settings. One is the automatic setting that is presently enabled. If you don't want it, you can disable it and manually input the value. For example, in this case for crude oil, assume that I'm giving 20 orders, okay? So the number of volumes decreases more the signals will get. So number of volumes increases less the signals that you'll get. So instead of keeping it as a manual, so it is always recommended to keep it as automatic mode. How does this work? So basically you can see a HD interval. So basically a look back period of 30 minutes is recommended. So basically if you keep it as 30 minutes and five. So what it does this, it look back last 30 minutes and it averages the volume. Assume the last 30 minute average volume is five, okay? Last 30 minute average volume is five. And provided you have given as an HD multiplier input as five. So it multiplies with five and gives the value of 25. So this 25 will be inputted to the, sorry. Just a moment. So that 25, just a moment. So that 25 quantity will be entered here automatically. So the advantage here, one thing is that it will be dynamic. You don't have to manually interrupt here and automatically according to the dynamics of the market, it will consider the volume that needs to be given as an input in the order window. So this is a setting that I'm presently using, 30 minute look back and five multiplier and one more information. So this 30 minute make sure that you are giving here 30 minute after completion of the 30 minute of the market activity. So for example, assume that our market opens at 915. If you want to give it as 30 minute, then you have to give it after 945. So if you are within the bracket of 945, then you can reduce this particular number according to the present timeframe. For example, if you are looking the absorption indicator after 15 minute, so 15 minute is completed. In that case, you can consider 15 minute. Once after 30 minute, you can simply keep it as 30 minute because that gives most of the time the ideal value. So in the live, it looks something like this. So we saw the absorption here, absorption here and continuous absorptions here, okay? So basically in that particular case, there is not much of an absorptions that are happening with respect to the crude oil at the present moment. So let's try to understand one more indicator then we will try to include the overall information here. So there is one more indicator called market pulse. Last week also we have discussed about it. So basically, sorry, before this, we need to understand, sorry, price level we need to add it. So this will, yeah. So what this price level indicator does is it will help us to identify the present day high and low dynamically. So whenever there is a change in the low, so you can see till this particular point, the low was the same, okay? But after breaking up the low, market is continuously making lower lows, okay? So break of this particular low was the day high and day low. So it will dynamically change according to the market dynamic. So at the present moment, at the present moment, the high is at around 6,750, somewhere around, okay? So instead of manually drawing, you can simply enable the price level indicator to identify the dynamic high and low of the particular day. And also there are a few more levels. So basically you can add a 50% level which will help you to identify the day's 50% level or you can enable, you know, custom level as well. For example, in this case, assume I want 70 level, you can simply add 70 level. In that particular case, 70 percentage level will be just change this setting. You'll be able to see the, you know, 61.7%. So this is the white line that we are seeing here. So this is the custom level based on the percentage. This is a percentage level based on the present day range. It will calculate automatically and plot the line. So ideally I use day high and day low to identify, you know, day highs and day lows. Presently it has already broken the day low, which is a good point considering that the presently market is moving in the downward direction. Assume that if you have taken an entry over here, you can keep your stop loss just above the absorption. So basically here are the last absorption.  So break of this level, assume that market takes a pullback and again breaks this particular level by keeping, by giving an absorption, then simply you can try your stop loss from this particular level to this level. Okay, this is the use case of absorption. And let's enable the, another indicator which is called a market pulse. Okay, so just a moment, it's getting loaded. Just a moment, it's getting loaded. So I think it's, it's the moment. It's also available for global markets. So basically the book map is from the global markets only. So recently they have, they have entered into the Indian market. This tool is available in the, I think there seems to be some issues. So I'll just check with that. Just a moment. So by meantime, if you have any particular questions with respect to any particular indicator that we have discussed so far, if you have any particular questions, feel free to ask them, any particular questions. So there is the one more tool that I want to tell you. So basically market pulse, I think there seems to be a little issue. So I'll just check with that. Meanwhile, just a moment guys. So just give me a moment. So it's fine. So let's try to understand. There is one more tool called footprint charts. So the tool that is recently been introduced. So by if you are, any chance, if you are in a global market, you can visit bookmap.com. You'll be able to see the details, bookmap.com. Right now it's selected for India. You can go with like this. So you'll be able to see the overall information of the global markets over here. And if you visit more and go to the knowledge base, you'll be able to access the knowledge base of bookmap where you will get the data of almost all the add-ons that we have. So if you go to the add-ons, you'll be able to download the instruments. So basically, you know, absorption indicator, any other indicator, we need to download that. That will not be by default installed in the software. So you need to download that from this particular indication. For example, absorption indicator. You need to visit this and you have to download the plugin. And then in the software, go to the configure add-ons and then go to the add. And you have to add them manually. So once you add, you will be able to see the information over here. So see at the present moment, we are seeing that the market is presently, you know, approaching to the, again, days low. And there is not much of an absorption here. So in that particular case, yeah. In that particular case, you know, we can wait for another absorption to keep our stop loss from the initial level to this particular present level. Okay. So, yeah. So according to the Nifty, let's try to understand what happened with respect to the today's market. So since we were looking for a shorting operation, even in Nifty, you know, even in Bank Nifty considering our initial discussion. So you can utilize the combination of absorption indicator and Vivo indicator to sort out the patterns that happens in the Vivo indicator and simply try your stop loss according to this.   There was two particular trades according to the present structure in the Bank Nifty today's market, according to the absorption indicator. Okay. So just a moment. So similar to that, I have also explained about the price levels that you can enable to identify the day high and day low. So this will help you to automatically identify the day high and day low. The last close was not reached to the day high. So market entirely traded within the initial hours of the market. Okay. Yeah. So now with respect to this information, okay. There is one more tool that I wanted to show you which is called as a footprint chart. Okay. So just a moment, I'll disable the price levels and show you that you'll be able to see it. Yeah. The column that you are seeing here, this particular column is nothing but the footprint chart. So which will help you to identify the market's volume behavior. For example, in this particular case, the entire 15 minute, okay. So this is based on the specific period of time. So if I show you the setting here, okay, footprint chart, it's based on the time interval, 15 minute of footprint chart. In that particular case, it is identifying the delta plus and minus information. Okay. Whatever we are seeing here is a delta plus and minus information. And it is showing us in terms of the text, okay. So the setting belongs to the text which is showing like, for example, in this particular case, okay. So this entire 15 minute, this is where we are seeing maximum number of orders, okay. As per the profile of volume, this is where the maximum number of orders happening. And in that case, it is also showing us whether it was a market buy orders or market sell orders. So if you can observe here, 150 quantities were executed as a market buy orders at this particular price level. 88 orders was market sell orders. So ultimately as per the delta concept, delta is more towards market buy orders. So this will help us to identify a specific period of time, support and resistances according to the volume profile. And also it will help us in terms of this particular level. For example, let's say this was a market buy orders, right. So someone placed a aggressive buy orders that too when market was falling. So this is a context we need to understand here. Someone is placing a market buy orders when market is falling. That's a caution sign for us as a short sellers. In that particular case, we can keep our trailing stop losses or stop losses very near to the, you know, the risk. For example, in this case, we can simply trial our stop loss to the last swing and product our stop loss. So this is one of the use case of, you know, footprint charts which will help us to identify the dynamic behavior of the orders and also help us to identify the profile dynamically in the live market itself. So market is falling, but still people are aggressively buying at the lower side, which is a caution sign for the sellers. That's the basic concept. So at any given point of time, price has to sustain below this particular low to make this particular theory nullified. So in that particular case, market has to sustain below this particular level to continue its downward direction. So if that happens, if it breaks this particular level, you can simply trial your stop loss to the last act. So this is also one of the tool or the indicator that we can use. This is called as a footprint charts, which will help us to identify the volume profile of a specific interval and also identify the volume behaviors according to the context. Okay. So any questions with respect to today's discussion? Okay, there is a question from 007. So basically it works with the MBO data or it makes, so there are a couple of connections. So if you visit this, you'll be able to see the settings over there. So if you just go to the add-ons, okay, so if I disturb it, so the setting will be disturbed. So there are many data providers available for the global market. See this is exactly the dynamic pattern that we are seeing. When we are seeing a market by orders at the lower level, price is pushing. So that means the one who entered here aggressively as a buyers are a strong buyers. And as a result, we are seeing market is moving on the opposite side. So at any given point of time, if you want to short the market, the break up last low is highly recommended. Or if we are in any particular short-term, it is better to, you know, get out of the trade and wait for the right opportunity. So this is how we can utilize the footprint charts on a timely basis. About L2 data hit map. So that will be explained by another person. So he will take care of that particular part and he will explain that in the detailed manner on a specific session, okay? So today's session was about absorption indicator and basically about footprint charts and, sorry. And you know, the basic structure of the book map. So I want to recall the overall information that I have shared so far. Yeah. So I want to share you the information that I've shared so far is basically we have gone through a specific framework where we have discussed about the trend, identifying the trend and location and then entry and exit. So first step is to identify the trend. With respect to that, we will try to align in the market and then we identify the locations where we look for trading opportunity to get in and get out of the trade. And similarly, we look for entry and exit using the order flow information and trigger. So in simple words, you can utilize the BVO indicator pattern and get into the trend. For example, if you are mainly looking for a long trades, so basically break of higher rise and higher low, pattern of higher and higher low in the BVO indicator is recommended. In that case, you will enter into the break of the swing high and simply trailer stop loss to the absorption swing lows. Similarly, if you're looking for a short trade, the lower low, lower high format in the BVO indicator is recommended and you will short the market at the lows and keep the absorption highs levels as the stop loss level. So that's the basic information. I hope that's clear. Let's try to monitor the market in the live. So this is bank nifty. We have included absorption and similarly the footprint jumps. So at the present moment, this level where we have seen a maximum number of market buy orders is a critical level according to the footprint and market is presently taking the reversal from this particular level. So at any given point of time, if you want to look for a short trade, the break of this particular zone is absolutely critical because the possibility of them depending, even though price approach back to this particular level, the possibility of this buyers depending that particular action is quite high. Okay, so this is how we read the footprint information. So now market sell orders are getting in the market, which is relatively more than this particular information. Let's try to understand and we will get any particular absorption here or not. You'll see that. Meanwhile, if you have any particular questions on today's discussion, please feel free to ask in comment section. So let's try to monitor the market. If you have any particular questions, feel free to ask them in the comment. Okay, footprint charts are not visible, okay. You can change that setting. So basically intentionally zoomed out. So when you sir, you can change that setting here, okay. For example, in this case, you can simply increase the tick multiplier. For example, three, in that particular case, you'll be able to see the bigger numbers. Is that clear? You can change the settings here. So for Nifty, I'm presently using three. For Bank Nifty, it is as the six. So the 15 minute is completed. And in that case, so you can clearly see that here there are maximum, you know, aggressive buyers. And here also there is an aggressive buyers. At the present moment, the price is presently sustaining about this particular level. The point to be understood here is that this particular volumes got accumulated when the market was falling. So who has the courage to, you know, go against to the market? And again, market is also accepting that particular correct. So in that particular case, this buyers are going to play the critical role as in when price sustains above this particular level, that's not a good sign for any short trades, all right. And similar thing happened here as well. You can see this is where we saw the maximum selling aggressive sellers and break up that particular level bought more selling into the market. Hope that's clear. If you have any particular questions, feel free to ask it. Okay, how to set threshold levels for absorption? There is a question. You can do that, for example, if you visit this particular information, okay. So configure add-ons, go to the absorption indicator, right now it is set to automatic mode. You can control the setting here. So if you enable, if you disable the automatic setting, you can simply, for example, I want absorption to be considered for every 10 orders. So we'll get more signals, for example, 20 orders. So by default, it is better that we keep it as automatic because it works according to the dynamics of market with a little visual help. Yeah, with, you know, this changes from instrument to instrument, you have to try it out. Basically, I don't, you know, actively trade crude oil. So that's why I don't know this particular number. So if you know this particular numbers, for example, in this case, assume, instead of, you know, five, you can keep it as three, as multiplier. In that case, you'll be able to see more of absorptions, okay. Three or four somewhere. So that plays to be good. So this changes from instrument to instrument, so that we have to work it out based on the appearance and based on the comfortability that you get with that particular instrument. Now, more of market sell orders are getting executed in the market. Let's see what happens. To initiate any fresh starts, we need to wait for the break-up last low, which is very critical because we know the reason that there is more of agree to buyers are sitting here by depending the price two times. This is first time, this is second time, and second time market is pushing and forming a reversal pattern according to the dynamics of it. So that doesn't mean we need to get into the long trade. Since the price is primarily training the downward direction, we try to align with that until unless we get the strong confirmation. So with respect to this for any short position, initiation of the short portion, we need to wait for the break-up the low. As I said, you can use the price level to identify the day highs and day lows. So that's simple as that. Okay, is this color coding in footprint by default or modified? So is it default one or modified one? It's a default one, I haven't changed anything. So color you can work out with respect to your needs. So basically the positive and negative color is the same. The only thing, the shading difference that you are seeing here is based on the shades. So I have enabled an option of use shades. So in that particular case, I'll be able to see the shared information. So that will help us to, even though it is in a green color, it will help us to identify the significant levels more importantly. So that's the point. Now, as you can see here, we are seeing an absorption, right? So the lower level absorption, which is the presence of passive bias, right? So the break of this particular level can bring the upper momentum actually. So in this particular case. So in this case, how we are defining it. So basically, like, you know, this is a basic price structure information, right? So market was moving in the lower low lower eye format and suddenly something has happened. The low that has created in this particular case was not able to break the last swing. In that case, it is, instead it is moving in the opposite direction. So the break of this particular swing is basically the pattern change. So the trend change itself in terms of the microstructure of the BBO indicator. So either, you know, aligning with the trend is absolutely important.  You can directly utilize the information of the BBO indicator itself to identify the patterns of the price. Okay. And more importantly, if you include up-drop indicator, we'll be able to identify the most significant swing, swing highs and swing lows. Yeah. Shares are really good because that will help us to identify the significant price levels just by, you know, looking at them. Yeah. I hope that is clear. By the chance if you have any particular questions, feel free to ask with respect to footprint, up-drop indicator and, you know, volume dots, any information, if you have any particular questions or if you are stuck at something, so you can feel free to ask. Why I can't see heat map. So it's presently in the trial mode. So you can contact book map chart. Rohan, you can contact book map chart. So they will help you to set it up. So I think it's in beta mode presently. So they will help you out with that. In India, yeah. See with respect to the data limitations, they were working on it. So maybe it's in a beta mode. So we can contact them. They will help you with respect to that. Okay. Most welcome Rohan. By the way, how many of you are already using book map pro charts? Renus sir is using Markandeya, who is using Superb. The break of this particular low will be the important point for the present moment. All right. So which means this is a present range of the market in the microstructure or the BVO indicator information. All right. So anything, the break of these two levels are very critical. So in that case, break of this particular range, in that direction, we can expect the momentum. Ideally possibility of the break of this particular level is quite high because with respect to the primary trend, it is presented any in the downward direction. However, with respect to this, it is forming a kind of a reversal pattern over here with respect to the footprint chat. So let's see this entire range is critical at the present moment. And we have shared a bookmap India Discord channel. So we can join that particular channel to get immediate access to the information. So most of the traders who will share their views and discuss about markets in the live and also offline markets in the Discord channel. So the link that we have already shared in the live Well, I think I hope today's information added value to you. And thank you so much for your wonderful time. And just a moment, if you have any particular questions, so feel free to ask. I think I have cleared all the doubts. All right. Well, thank you so much for your wonderful time. I hope you have learned something new and interesting today and keep attending these live sessions. We'll be getting a lot of knowledge about the dynamics of market and also the bookmap tools on a regular basis. All right, so thank you so much for your time. See you in the next session. Until then, take care. Bye-bye. Jai Hind. Let's see whether the break up this particular level breaks or not. So I will stop the streaming just most will come.


All right, first is Elemental Hero Thunder Giant. We got, oh, that's a good start. We got a Dark Bright. Okay, let's see what we got. Do we get the 10? Ooh, a mint nine. Okay, we have a Blade Edge. This would be a really cool 10. Ooh, a nine. Okay, Neo Space and Air Hummingbird. Gem Mint 10. Okay, very nice. Almost turned it over. Come on here, Nios. Forbidden Yu-Gi-Oh card. Get a nine at least. All right, okay. We can take that. 70 million start. Oh, the Dark Nios. The R10 Dark Nios. Come on, here we go. What? I jinxed it. No. Aqua Nios. This is not a counterfeit one, which is nice. Let's go. 10. Oh, a nine. Okay, Aqua Nios. This would be pretty sick as a 10. Flare Nios. Looks amazing. Let's just see what we got. A nine. Okay, so I think it had some edge worth. We have Flame Wingman. Uh-oh. Very off-center. Don't give us the eight. Come on. I knew it. It's such a minty card to get an eight. I think this might be another Flame Wingman. Let's end it with a bang. Let's see Flame Wingman. Ah! Two, eight.


Okay, how are you guys doing? Are you ready for Thanksgiving break? All right, yeah, me too. So, well, we still have one hour and a half to go. One hour, 20 minutes to be precise. So the initial motivation for the material which we are discussing now was our desire to generalize Green's theory so as to apply it to more general regions in the three-dimensional space, okay? And last time, we made the first steps towards that goal. We introduced the curl of a vector field. So now we are almost ready to state Stokes' theorem and it's going to take the following shape. On the right hand side, we're going to have something we've learned before and understand fairly well by now, I hope. And that's a line integral of a vector field over the boundary of a surface in a three-dimensional space. So we are talking about things like spheres or maybe a better example which we discussed before was a hemisphere, an upper hemisphere. Like this. So a sphere, as we discussed already many times, is a two-dimensional object. I'm not talking about the interior of the sphere, I'm talking about just the surface of the sphere. It is a surface hence a two-dimensional object but it's a kind of two-dimensional object which really lives in a three-dimensional space and not on the plane, not in a two-dimensional space because it's curved. So this surface has a boundary. So this surface, M, M will stand for membrane, you'll see why, we'll talk about it later. No cell phones. I'm actually, I'm impressed how rarely we see this, we hear this sound, so keep it up. This surface has a boundary and that's what I call here B of M. So in this case, if it is really an upper hemisphere, if you really think in terms of the upper hemisphere, it is a circle. In any case, it's a curve and over that curve we can take this line integral where F is a vector field. So that's the left-hand side. The left-hand side is kind of easy in the sense that we already know what it means. And the left-hand side is going to involve a double integral over M itself and now we can say the integral of what? It's going to be the integral of what we learned last time, the curl of F. So if you have a vector field F, you can define a new vector field which we call curl of F, okay? So this has already been defined. And now we have to learn how to integrate vector fields over surfaces. So that notation here represents a surface integral, an integral of a vector field over the surface, also known as a flux. And today we will define integrals of this type so that finally we'll be able to say what we mean by the left-hand side and at least we'll be able to make the statement of Stokes theory which we will then discuss next week, okay? So that's the plan. We would like to learn surface integrals, okay? Now this is, by the way, about the final exam which again will be on December 19th at Hearst Gym. No more jokes about it. And now I put the materials for the final on my homepage so you can see the review problems and various other things there, okay? I will put the solutions of the review problems probably sometime during next week. All right, so let me raise it now. So what we'd like to talk about is surface integrals. And as always, when we talk about things like that, it's always good to think in terms of analogies, okay? And an analogy to a surface integral would be an integral over a curve, over one dimensional object. Something which we now understand fairly well. So let's revisit first line integrals. Revisit integrals over curves. One of the aspects of the story, something that I know you have found puzzling at first is the fact that actually the way we thought about it, the way we talked about it, there are two types of integrals over curves. One has, one type is the integral of a function. And this is a kind of, it's arc length type integral. In this case, we have a curve C and we integrate a function. And the expression for this is FDS. So F here is a function on this curve.  So what is the meaning of this integral? The meaning of this integral is that if F is equal to one, we get arc length of the curve. Arc length of the curve. And if F is not equal to one in general, but it is a non-negative, takes non-negative values, we can interpret this function as a density function. For example, mass density function. We can think of a curve as being a metal wire like this. And that function would then represent the density of mass. Some parts of it could be heavier, some parts lighter. And this integral would correspond to the total mass. Total mass. That's the interpretation of such an integral, okay? So that's the first type. And the second type, which actually we have studied more and which turned out to be more important for our purposes, in particular for formulating things like stock's theorem or Green's theorem, or fundamental theorem of line integrals for that matter, are integrals of vector fields. Line integrals of vector fields. So let me recall that. So in this case, we have a vector field F, we have a vector field, and we are computing its line integral, which we do not like this. And that also had a nice physical interpretation. It's not a mass, but rather it is work done by force. If F represents a force vector field, then this represents the work done by this force. Examples of such vector fields are the force of gravity, which we talked about before. Also the electric field and the magnetic field. When we talked about Maxwell equations last time, you had those two vector fields, E and B, right? So this would be examples of such vector fields. And such a line integral for such a vector field would actually have a very concrete physical interpretation as the work done by this force. For example, electric field, the work done by it in moving an electron from one place to another along this curve C. So those were the two types. The reason I am reminding you of all this is because surface integrals, which we are going to study today, can also be broken into these two different categories. One is the analog of the first type here, and the second one is the analog of the second type here. In other words, the first one, the first type of surface integrals will apply to functions, will integrate functions. And the second type will apply to vector fields. So we'll be integrating vector fields. And in both cases, they will have nice interpretation, very similar to the interpretation we've learned for line integrals. So having said that, having recalled the integrals over curves, I want to now talk about surface integrals. Now for surface integrals. For surface integrals, we'll also have two types. Now, I have to say from the beginning that this looks a little bit disconcerting, the way we talk about this, that there are two types and they kind of seem unrelated. When we talked about these guys, integrals over curves, I actually tried to explain that in a way, these are two special cases of kind of most general integrals over curves. So from the point of view of the big picture, these are actually not so unrelated. They're actually quite close to each other. But for the purposes of this course, for the purposes of the kind of results we want to obtain and kind of applications we have in mind, it is just easier to introduce them separately and kind of think about them as two different types. One applies to functions, one applies to vector fields. And the same actually will be true here for surface integrals. We'll kind of divide surface integrals into these two big groups. But I just want to tell you that actually from the point of view of the big picture, this is unnatural. And there is actually more natural theory. There is a more natural point of view from which both of these integrals appear as just special cases of some general construction. Nonetheless, we'll do fine. We'll do just fine by kind of defining them in two separate ways. Okay. So for surface integrals, we'll also have two types then. Also two types. The way we will introduce them here, two types. And so the first type will be an integral of a function, surface integral of a function. So we will have a surface M again, like in Stoke's theorem on that board. But we will be integrating not a vector field, but a function. And it will be represented in this way. So this is not yet the kind of integral which will show up in Stoke's theorem. This is sort of a preliminary integral, which is like type one integral in the case of curves. And this is something which we will use to define also line, to define surface integrals for vector fields. The reason why we define them first is because they're easier to define than the other guys in some sense. And also we will actually use this theory to define the integrals of the second type. So what does this integral represent? Well, just like in the case of curves, it represents something which is sort of inherent to this curve. The quantity which is to the surface in this case. So here there was a curve. And the inherent quantity for curve is arc length. It's arc length. Or if you have a general density function, the total mass. And likewise here, this will represent, for f equal one, this will represent the surface area. Surface area. And for f, not equal to one, but non-negative or maybe positive is better. This will correspond to total mass. If we interpret this function as a density function, is mass density function. So let's talk about the special case when f is actually equal to one. So in this case, we will denote this integral just like this. And this will simply correspond to the area of f. So let's discuss how to find the area of f. This is not entirely new. It's not an entirely new subject for us because at the very beginning of this course, we actually talked about revolution surfaces. And we talked about the areas of revolution surfaces. So we actually know one example, at least one example of this kind of integral. But now we will talk about areas of more general surfaces. Not just the revolution surfaces. Revolution surface remember was obtained when you rotated the graph of a function in one variable around the x-axis or maybe around the y-axis. But this is only very special kind of surface. It's like a vase, if you know what I mean. Like a vase is a revolution surface if it's a symmetrical vase, you know, like a round vase. Is a revolution surface of a curve around this z-axis in this case, if you think about vertical axis as z-axis. But not every surface is like this. There are even vases which are not exactly symmetrical, right? For example, you could have squares. So we would like to have a formalism which would enable us to calculate the area of any surface. So what are the necessary steps for that? Well, how should we approach this? It's the same drill as before. We have a piece of surface like this. And we would like to compute its area. And as always, what we need to do is we need to break it into small pieces and evaluate the area of each small piece and then take the sum. And then the sum, when the partition becomes finer, will actually give us an integral. But how would we actually break it into small pieces? A priori is not clear, right? So in the case of curves, in the case of curves, if you had a curve like this, you also wanted to break it into small pieces and take the small lengths of each piece and take the sum. And the way we learned to do this was through parameterization. This was in fact the most efficient way to talk about curves in the first place, right? Because a general curve was best represented algebraically if you expressed it in the parametric form. In other words, we would introduce an auxiliary parameter, auxiliary parameter T and identify an interval from, so this is parameter T from A to B. We would identify this curve with this interval, right? And algebraically, we would just write that X is a function of T, Y is a function of T, and well, in two dimensions, that would be enough, but in three dimensions, we would also have Z equals Z of T. And then we'll say T is between A and B, right? So when we write these formulas, what we are doing is we are identifying points on this interval, right? On this straight line interval, which is very simple, was this complicated curve. Once we do that, there is a natural way to break into pieces because we can just break this interval into pieces of equal length, say 1,000 pieces or 100 pieces in the beginning, maybe 10 or 100, 1,000, and then each of them, for example, this one will correspond to this part here, and then we'll try to find the length of this one, and then the next one, and so on, okay? And so the result of this calculation was an integral of the first step. Now, instead of a curve, instead of this curve, we have a surface, right? So we should apply the same approach, and we should try to parameterize this surface. This is something which we have actually managed to avoid up to now because this is something we talked about before, but we never actually got around to trying to write surfaces in the same way, or present surfaces in the same way we've been representing curves, and the reason is very simple because surface is two-dimensional. So surface is two-dimensional, so this M, let's call it M, its dimension is two, and so it requires two independent parameters, two independent parameters, but only one equation, only one equation. So whenever we could refine a surface by equations, we said it's better to define by equations and try to find parameterization because one is better than two in some sense. One is easier to handle than two. One equation is easier to handle than two parameters, okay? Notice that for curves, it's the opposite situation because the curve is one-dimensional, so it requires one auxiliary parameter or two equations. So that's why curves usually show up through parametric representation. On rare occasions, we will talk about representing curves as by two equations or as intersection of two surfaces, but now it's finally time to use the first approach and to parameterize a surface by two parameters now, okay? So how is this going to work? Well, in exactly the same way as in the case of curves, we will choose two auxiliary parameters which will, oftentimes we'll call them UNV, but again, it's just a choice of notation. You can choose whatever letters you like, UNV, and we will try to identify this piece of curved surface in a three-dimensional space like this with a flat domain in a two-dimensional space of these two parameters. What I mean is that somewhere in the background, we will have a two-dimensional coordinate system with coordinates UNV. Note that this is the analog of that line with the coordinate T which I drew on the top blackboard. And the analog of that interval from A to B now will be a two-dimensional domain on this auxiliary plane. So it is going to look something like this, right? See now, of course, there is more choice for such a domain. There are certainly more pictures we can draw like this than pictures of intervals, interval is an interval. I mean, the only things which we can play with is just the endpoints, but it looks, they all look the same. Here the domains look different, right? They have different shapes. Now, you can of course ask, why would be bothered to identify this and this? The reason is that this object is much simpler to handle than this object. You have to realize what this object represents. This object represents something which actually does not fit in the blackboard, right? It is actually, it can only live in three-dimensional space. In other words, without bending and squashing it, I wouldn't be able to fit in the two-dimensional blackboard. Here, I have drawn it on the two-dimensional plane, but that's just a projection, right? So in fact, usually to emphasize the curviness of this object, I would kind of draw a grid, this kind of a curvy grid to kind of indicate that it has this shape. And this is in stark contrast to the structure of this object, you see. So this guy is actually flat. This guy actually fits in a two-dimensional plane. And that's why it's to our advantage to try to identify this curvy object with this flat one. So that's the first idea, okay? And once we do that, we can then try to calculate the surface area. Now, since I drew it like this, it would be more, it kind of, it looks strange that I established one-to-one correspondence between this one and this one, right? So of course, it would be more natural if in fact the flat object, which was on the plane, which corresponds to it, would be something more like a rectangle. Then the picture would make a lot more sense. Here, I drew just a most general region that you could have a priori, but just to simplify our pictures, let's assume that in this particular instance, we will have a rectangular domain. This is not the most general one, but this is just for the sake of the argument, just to do an example. Okay, so if we do that, for this guy, there is a natural grid because just like for the interval, there is a natural grid because we can just break it into equal pieces like this. And once we do that, if we identify this and this, okay, we'll be able to also give a grid to this picture. You see what I mean? Just as before. So when I say identify, it means that I would map, I would try to map this flat rectangle to this curved surface in a three-dimensional space so that for instance, this interval would go here and this one would go here, this one would go here and this one would go here, but not only that, also each of this horizontal lines, horizontal intervals would go to these curves and all the vertical ones would go to those curves. So if I have already a parameterization like this, okay, then I would be able to break it into smaller pieces and then that would be the first step in getting the integral. So let's talk about examples of this kind of parameterization. Let's talk about examples. So the simplest one appears when our surface is graph of a function. If the surface M is part of a graph of a function, E equals, in this case, we can just choose X and Y as the auxiliary parameters. So, or if you will, we can say X is U and Y is V and Z is F of UV. So here is a parameterization. I have represented each of the three variables as a function of my new auxiliary variables, U and V, okay? And so this is just like parameterizing curves, the difference being that now the functions on the right-hand side depend on two variables, U and V, instead of just one variable, namely T in the previous case. So that's the first example. The second example is example of a plane because after all, plane is also a surface. Plane is two-dimensional. In fact, plane is the simplest possible surface, right? So in the case of planes, we are used to representing planes. So this is two. We are used to representing planes by equations. Exactly what I just said now, which is that surfaces oftentimes are easier to represent by a single equation. But now we're talking about parameterization, which is important for integrals. So let's talk about parameterizing a surface, which is very easy because think about, in fact, let me use my favorite surface, this one, right? So in the case of a surface, in the case of a curve, if I had a line, to know a line, I need to know a point and I need to know a direction vector. For a plane, one vector is not enough, right? Because one vector would just give me this line. If I want to get the entire plane, I should also pick a second vector, which is kind of transversal to this one. So if I do that, if I pick two vectors, let's call it R1 and R2, then any other vector on this plane can be obtained as a superposition of some multiple of this one and some multiple of this one, or some combination of this one and this one. And I can use that to find a parametric representation for this plane. So let me draw it on this picture. So let's say I would pick, so this is my coordinate system. So I would have, this would be some vector R0 and then I would have two vectors, R1 and R2. And so a general point on this plane would correspond to some vector R of UV. And it can be found by using this one plus a superposition of R1 and R2. So it will be R0 plus UR1 plus VR2. And then if I write this in coordinates, because as always, R is just X, Y, Z, that's three coordinates. So likewise for R0, R1 and R2. So say R1 is K1, L1, M1, and same for R2. Then I can actually write each of them as a function of U and V. So X would be X0, the first component of R0 plus U, K1 plus V, K2, Y would be Y0 plus U, L1 plus V, L2, and Z would be Z0 plus U, M1 plus V, M2. So here these numbers, K1, L1, M1 and K2, L2, M2 are given and the variables are U and V. So now I have been able to express each point on this plane as a function of two variables, U and V. So that's exactly what I'm talking about in general, except in general these functions will be more complicated. Here they are linear functions in U and V. In other words, the degree is a polynomial of degree one or less than or equal to one. Okay, something slightly more interesting corresponds to a sphere, corresponds to a sphere. For a sphere we could use spherical coordinates. For a sphere we could actually apply spherical coordinates. So we use as auxiliary variable, auxiliary variable will be theta and phi and we would write X sine phi cosine theta, Y is R sine phi, sine theta and Z is R cosine phi, where R is the number, R is the radius of the sphere, it's fixed. So see the difference between the spherical coordinates and this parameterization is, whereas for spherical coordinates we had rho here. So there was three variables, rho, phi and theta and those variables gave us all points on the three dimensional space. And now we are talking about points just on the sphere. So that actually leaves only two independent variables, phi and theta and rho is fixed because rho is equal to a fixed number for all points on the sphere. You have a question. When I drew, the question is what does this plane represent? This picture means like that plane. No, that's a plane in a three dimensional space. But now I am relating it to a UV plane. It's actually easier maybe to think about it in this way that so here is a plane, right? And this plane is in a three dimensional space which is our classroom. And now I would like to parameterize all points on this plane by points of another plane which say will be this table, okay? This table will have coordinates U and V and if I have a point UV here, they will correspond to a point here. Namely, that would be the point for which this vector, that would be this point for which this vector is U R 1 plus V R 2. Meaning that this is U R 1. Actually didn't draw it very well. Let me do, I have to do it in a parallel way, right? Like this, it's moving. That's why I'm not making a good picture. So this is V R 2, okay? Doesn't make sense? So once I fix U and V on an auxiliary plane, right? I can get a point here. Namely I just draw the vector U times R 1, that's this one. Then V times R 2, I take their sum, that's the point. And this way I can get to all points of this plane, of this tilted plane, right? So U and V, this pair of numbers, U and V provide me a unique address system for all points on this green plane. And that's what we are doing in general. It's just that in general, our surface is not flat, it's curved, but we are finding an address system for all points of that surface, just the way I do here for a plane. Okay, so for instance, in the case of a sphere, you know, so here is a sphere. Let me again draw an upper part of the sphere because it's easier to see, to analyze like this. So for instance, so this is the sphere of radius R, some number, let's say R is equal to five. So sphere of radius five. So this number is fixed. But now there are many points here, right? And I would like to address each point on this upper hemisphere by some unique pair of numbers. I call those numbers phi and theta. And I denote them just by using the same rule as I use for spherical coordinates. So for example, if I want the entire, if I want the entire sphere, this would correspond to the limits theta from zero to two pi and phi from zero, sorry, phi from zero to pi. This is a whole sphere. If on the other hand, I want just the upper hemisphere, this would mean that theta is between zero and two pi. And phi is between zero and pi over two, right? I put a limit pi over two because I don't wanna go more than upper hemisphere. So going beyond pi over two would correspond to the lower hemisphere. And then you can also think of, even if you think about this cone, if you just want the part of the cone, for example, the stop part. Think about an ice cream cone. So that would be the ice cream, the part of the ice cream, right? So that would correspond to theta between zero and two pi and phi between zero and pi over four. So by choosing limits in an appropriate way, I can describe different regions on the sphere by using spherical coordinates. And this is the third example. Okay, now once I have a parameterization, I can get to work and try to calculate the surface area. You can calculate the surface area. And now I have to, I just have to look at this picture. And what I need to do is I need to find a way to express the area of this one little curvy rectangle on the surface, which will correspond to a particular rectangle in the parameterizing domain, okay? So here let's say we would be delta U, the length would be delta U, and this would be delta V. And I would like to find the area of this, which is delta S. And this actually is something which we have already done. We have done this when we talked about double integrals in general coordinate systems. When we talked about that, we did exactly the same calculation. The only difference was that actually our surfaces at that time were still part of the plane, but the calculation is exactly the same. If you remember, we took the tangent vectors here, R sub U and R sub V, and we took the cross product, which represented the area of the parallel ground. So the answer is the same, answer is the same as before, as in the old calculation for double integrals is in all calculations. So you can review that if you like. Just to save time, I'll give you the answer.  This area of this little piece. So that's the formula.  So R is just X of R of U V is just the vector which position vector of the point X, Y, Z.  And if I want to take R sub U, that just means taking the derivative of X with respect to U times I plus derivative of Y with respect to U times J plus the derivative of Z with respect to U times K. So what I mean by X sub U is just DX, DU, but it's just easier to write it. It's faster to write it like this, so that's why I'm writing. And likewise for R sub V is X sub V I. So you see, and now to find the area, the area and of course I would like to emphasize, this is not the exact answer. This is approximate answer, just as before. This is an approximate answer for this little surface area. And what we need to do after this, we have to take the sum of all of this over all pieces of this partition and then take the limit when the partition becomes finer.  So this way we find that the area of this membrane of this two-dimensional surface is a double integral, but now it's a double integral over D, you see. D being this, this is D. D is a domain which is on the flat domain. The whole purpose of this discussion was to replace a curvy surface by a flat surface. And to replace a curvy integral, which we are trying to define by a flat integral. A flat integral we already know. We studied the months ago, it's been awhile, right? So that's something which we understand fairly well. So the whole point of discussion is to define the area as an integral, but over a flat domain which we understand. The question is the integral of what? And now we know the integral of what? Because we know the elementary area. And it's the same game as before. Once you know that the elementary object, you know the formula for the elementary object, which in this case is the area of this little piece and is given in this way in terms of delta U and delta V, then you know that when you take the limit, this will get replaced by DUDV and this will be the function. Yes. Why is the LS equal to this? I would love to explain this, but I'm afraid that if I do that then I won't have time to explain other things. But, so I have to make a choice. The reason why I'm not explaining this is because we actually, it's verbatim the explanation for general coordinate systems. Which actually I didn't explain in great detail either. I have to confess. But I explained enough of it to convince you that that's the right answer. But think about the fact that, remember the fact that the cross product of two vectors always represents the area of the rectangle which is formed by them. So what I do here is that I, the calculation is done in a following way. I approximate this curvy rectangle by a flat rectangle formed by the tangent vectors. And then once I have the two vectors, the area of the rectangle spanned by them is just a cross product. So the only question is what are these two vectors? And you know when you do tangent vectors you take derivatives. So if you actually look, if you look at this question you will see right away. And actually we have all the, you have all the information you need to be able to find this because we talked about tangent vectors two lines. And this actually lines, sorry lines, curves. These are tangent vectors to curves.  And the second one is r sub v delta v. So actually the answer comes out to this. Comes out to be this. But then I just take the delta u and delta v which are numbers out of the cross product. And that's how I get this formula. Maybe this is an, if you look at this formula you'll see it more, you know, it will be more convincing. Okay? So having said all of that, this is the formula I get. r sub u cross r sub v du dv. Which is actually, you can also write. It's more appropriate to write as dA.  So that's the formula for the area. And now we can actually look back at all of those examples that we have studied. Here, it tried to figure out what the answer is going to be. And the point is that, well, let's do it in this case. In this case, r sub u, you have to take the derivative of these three functions with respect to u. So this derivative is one. So it's i. This derivative is zero because derivative of v with respect to u is zero. So I'll just skip it. And then the derivative of this one is f sub u times k.  So now I can actually find this expression by simply taking the cross product of these two vectors. So what do I get?  Just use the usual rule. So you'll have i, j, k. And then you'll have one, zero, f sub u, zero, one, f sub v, right? And that's what, that's negative f sub, let me write it here.   And so that's the function which in this particular case, I would have to put here and integrate it over my flat domain parameterizing my surface. And that would give me the surface area, right? So it's very straightforward. Likewise, you can calculate the answer for sphere. For sphere, you just have to do the same kind of calculation, you have to write, find out what r sub phi is and what r sub theta is, put this together, calculate the cross product. This has already been done. So I'll just give you the answer. And in fact, the answer you can guess because we already know the Jacobian for the spherical coordinates. And this is going to be exactly the same formula as the formula for the Jacobian of spherical coordinates except that now we fix the radius of the sphere. The row is fixed. So the answer is, is r squared times sine phi, right? The answer was r squared times sine phi, but now it's r squared times sine, because it's fixed. So again, if you're working with a sphere, this is a function you'll have to insert in this integral to obtain the surface area, okay? So that's surface areas. So this completes the case when one, your function is one. So this is by definition and the definition goes and usually we write definition from left to right, but now I write definition from right to left, okay? So this is a definition for the integral over m. This was our first task, to calculate the integral, calculate the surface area as an integral over a flat region. So this is the answer, okay? And the next step is very easy. The next step is we want to now calculate not just an integral of ds, not just ds, but we want to integrate an arbitrary density function. For example, think about the mass. If it's an aluminum foil, which has some parts heavier, some parts lighter, so the mass is distributed in an uneven way, described by this function f, so then this would be the mass. How to calculate this integral? Well, very easy, you just insert this f in this integral and that's it. And this completes the definition of the first type of integral. And you know, the reason why I'm kind of going so fast over this is because I would like you to appreciate the fact that this is really parallel to our derivation, for example, of the arc length of a general curve or the formula for the mass of a general curve with a given density function. It's exactly the same derivation. What has changed is that now the surface is parameterized by two parameters, by two variables, whereas before there was only one. But the idea of the calculation is exactly the same. Okay? Any questions about this? So that completes the double integrals of the first kind. And now finally we are getting to, we're getting to something more interesting, which is the double integral, double integrals of the second kind where we will be integrating not over, not, we will not be integrating functions over surfaces, we'll be integrating vector fields, which is what is relevant to the Stokes' theorem in particular. Okay? And now, before I give you a technical definition or I give you the formal definition, I would like to motivate it by considering a real-life problem where this kind of integral shows up. And again, I want you to appreciate the parallel with line integrals of vector fields. For line integrals of vector fields, we talked about work done by force. And then we try to see what this means and how to represent it by an integral. And that's how we came up with the notion of line integral with vector field. So here we're going to do the same thing, except now of course it's not going to be work done by force, but what it's going to be is it's going to be the flux of a vector field or the rate of a flow of a vector field. Okay, this is slightly, this is slightly trickier. I have to say just slightly trickier than the other one, but very transparent nonetheless. Okay? So here is a real-life situation that you can imagine. Imagine that there is a pipe and you have the flow of water through the pipe or any other liquid. I'm not going to make any suggestions. So, and you would like to find the rate of the flow through a given, think about the membrane, that there is some, think of this invisible membrane and you would like to know how much water will flow through this membrane per second. Okay, that's what I mean by rate of flow. So for that, we have to know the density, which would be amount of water per second, or per hour or whatever, per unit of time. And I'm assuming here that my membrane is strictly perpendicular to the flow. So it is perpendicular to this pipe. I'm looking really at the flow through this section of the pipe, which is just obtained by cutting it in a perpendicular way. So what do I need to know? I need to know the density, density function, the density of water, right? And I need to know the velocity, mass, this is mass density, mass density. Then I need to know, so this is a mass per cube of a measure of length, let's say meters. Then I have a velocity vector, right? Whose length is the length of the velocity vector. So this is the velocity vector. And the length of the velocity vector is measured in what, in distance, so it's a meters per second, okay? So what do I need to do? I need to, oh, and I also have the delta S, the area of this section of this membrane. This is the area. So this is a meter squared. So what I need to do is I need to just multiply this. So the rate of flow is equal to rho times V, right? Times delta S, that's pretty clear. Because, well, because over given period of time, it will move, say the water that will move through this membrane will correspond to this part of the pipe, right? Which we find by multiplying, by taking the length of this velocity vector multiplying by time and multiplying by the area of this membrane. So I think this is pretty clear, okay? So now I would like to make it more complicated. So first we have to make things more complicated, then it will become easier. That's always the idea, right? And here's what I want to do. I would like to now understand the flow, not through this one, which is kind of perpendicular, but through another one, which is tilted, through another membrane, which is tilted like this, okay? So what has changed now? Well, the point is what has changed is that what will contribute now to the flow through this membrane is not V. It's the component of V, which is perpendicular to this membrane. Listen. So this little membrane will have a normal vector, N. Let's call it N. And what will matter now is not V, but the component of V along N. Because you can decompose the velocity vector, right? Let me draw this picture one more time. This is my membrane. This is N and this is V. So V has two components now. One is a normal component, which is, let's say, this one, and then it was intangential component. So you can think that the flow of the water is combined, flow of the water through this membrane is combined of two different effects. One is some water flows in a perpendicular way and one sort of water kind of flows is a superposition of flow in a perpendicular direction and the parallel to the membrane. And of course, what is parallel to the membrane is irrelevant, nothing flows. If the flow is just like this, there will be no flow through the membrane. So the only thing that you have that is going to contribute is just the normal component of this vector. And what is this component? So this is V. So let's remember again dot products, right? So the normal component, the length of normal component of V, the length of the normal component of V is just V times the cosine of the angle, which is formed by these two vectors, right? That's this normal component. And that's just V dot N. So my more general formula is going to be like this. So the flow rate is going to be rho times V dot N times delta S, okay? So I hope I convinced you that in this idealized situation where you have the uniform flow of this liquid, that's the velocity vector is the same everywhere. And the normal vector is the same everywhere that the rate of flow is given by this formula. And now I generalize, okay? But before I generalize, there's a question. Right, but N is the unit normal vector. I'm sorry, I should have said that. Right, the question is, I need to put here, I assume that the length of this N is one. So it's a unit normal vector. When I take the dot product, it's going to be the, oh, you're also saying this. Sorry, that's right. You're right, thank you for correcting me. So the, first of all, I should have put the length of V, the magnitude of V. And second, I would normally have to also put the magnitude of this vector, but I assume it to be one, to be equal to what? It's the unit normal vector, if you want. Okay, so now we're ready to generalize it. We generalize it by saying that it could be an arbitrary membrane, and also the velocity could be different in different places. The velocity would just be given by some vector field. So what answer are we going to get now? It's the same idea as in the case of our calculation of the surface area. We break a general membrane, we break a general membrane into small pieces, and for each small piece, we apply the idealized scenario, the idealized scenario, where we assume that approximately the velocity is the same everywhere, okay? And the normal vector is approximately the same everywhere. So then we use this formula. But after this, we have to take the sum over all of those pieces. And when we take the sum, we take the limit when the partition becomes finer. And that's what gives us the integral of the second type. And you can read off the expression for this integral from this formula. As always, just look at the formula for the elementary calculation, which we get from the idealized situation, okay, like this, and just replace delta S by dS, where now we already know what dS is, because we are able to calculate dS by using parameterization, by using this rU cross rV, right? So for general membrane, the flow rate just looking at this formula is, and let me put rho outside, because rho is just a number. It doesn't depend on, actually, might as well, might as well, might as well depend. So let me do something else. So let me say that F is rho times V. The mass density function could, of course, in general be non-constant, that's fine. And V, the velocity vector field is non-constant. But let me just isolate this, the product of the two as a particular vector field. This formula will be nicer this way. So then, well, if I do that, this can be written as F dot N dot times delta S. Times delta S. So what I end up with is an integral over M of F dot N dS, okay? But now, what is N? Now I would like to make sense of this integral by using a parameterization. You know, this is already a definition, right? Because once I take the dot product, I start with a vector field. But my surface has its own vector field attached to it. It's a kind of inherent vector field attached to the surface, namely the normal vector, unit normal vector at each point. And I take the dot product and I get a function. And functions I have already learned how to integrate. You see? Since I've learned how to integrate functions, this actually already makes sense. But let me spell it out in more detail by using parameterization. So suppose now that my surface is actually parameterized in this way, then I can actually find a normal vector and the normal vector is going to be precisely this r u cross r v, which we have used here. This actually, as a vector, it is normal to the surface at this point. It is normal because remember, as I said, in this calculation, you have to take a cross product of two tangent vectors. When you take the cross product of two tangent vectors, you get something which is perpendicular to them and therefore perpendicular to the surface. This is a normal vector. This guy is a normal vector. But in a previous calculation, we just took its norm and we saw that the norm is this. So this gives me a way to actually to get my formula very, obtain a very explicit formula because this is a normal vector. But unfortunately, it's not normalized, which is a horrible pun, right? So it's normal in the sense that it is perpendicular, but it's not normalized in the sense that its norm is not equal to one. But now I will normalize it. So we'll get a normalized normal vector. Okay, I think that this is because our language and our everyday language is too poor to express some mathematical concepts. And that's why it sounds really bad, but so let me just write a formula instead without saying anything more. The formula is that n is r u cross r v divided by its magnitude. If you have a vector and you divide it by its norm or magnitude, you obtain a vector which has norm one. There's always the case. So this is what I should put here instead of n. And also instead of ds, I have to put my old formula has already been erased, but I want to remind you maybe here that the integral over m of f ds was equal to the integral over d, where d is this region of r sub u cross r sub v dA. In other words, sorry, and I forgot the function f. So in other words, ds becomes dA times this guy. So what's going to happen now and it will have a very nice consolation. We will have double integral over d of f dot r u cross r v. Then we'll have to divide by this r u cross r v. And then we'll have to multiply by r u cross r v dA. Do you see? So it's kind of lucky because the same quantity appears in the numerator and denominator and it's non-zero. This is actually something which we have to take care of because it should be non-zero because otherwise it would not be a parameterization. This is like, think about this is like the Jacobian. It is a Jacobian if you think about it. It is a Jacobian. And in fact, not surprisingly, the formulas we get for it, say for spherical coordinates, is a Jacobian, right, it's not a Jacobian. And the Jacobian should be non-zero. For otherwise, this is not going to be a good parameterization. So that's why actually it's non-zero and we have the right to cancel them out. When we cancel them out, we'll finally get a very nice formula. Namely, we get this. And finally, so finally we have come up with the definition of the integral of a vector field, of a vector field across a membrane, across a two-dimensional surface. If the surface is parameterized by a flat domain D, this integral is simply given by this formula, by this double integral over D. What it is, on the other hand, what it is more in a more canonical way, is an integral over M, with respect to the measure of the surface area, of this function, f dot M. But since we don't want to write it each time, we will actually write it. We will actually write it. We'll use a new notation for it. We will call it f dot ds, where we put ds over ds, we put an arrow. In other words, we say that ds as a vector by definition is n times ds. If we do that, then we can combine these two guys into single symbol ds, with a vector over it. And what we're doing is we're taking the dot product of f with ds. So let me summarize. We are trying to understand what to make of an integral over vector field across a membrane. And a good analogy is a line integral over vector field. In the case of a line integral vector field, we had a different interpretation, because in the case of a line integral of a vector field, it was all about pushing something along the curve. So what mattered was the tangential component of the vector field, tangential to the curve. What matters now is the perpendicular component, because we are talking about the flow. The flow through, not along the surface, there's no such thing as a flow along the surface. It could be something along the curve, you could not have something along the surface. It has to be across the surface. If it's across the surface, what matters is the perpendicular component, the normal component, which is what we get by taking the dot product of our vector field with n. So we are trying to find this, and this is called the surface integral of a vector field or a flux of a vector field through or across the surface. And this is very important. It's not along, it's across. This is an important thing, an important distinction between curves and surfaces. For curves, you go along the curve. For surfaces, this is a membrane, you go across the membrane. Okay, that's why it's a dot product with a normal. And so this integral is given by this formula, f is a vector field, and the meaning of this integral is given by this formula. Once you parameterize your membrane m by a domain D, like here, you can actually evaluate this integral in a very straightforward way by simply taking this cross product and taking the dot with f. For example, let's suppose that we are in this situation. We are in this situation where our u cross rv is given by this formula. This, I recall, is the situation where m is part of a graph of a function f, okay? So, and your vector field f, say, is some pi plus qj plus rk. In this case, we can use this formula. So what we have to do is we simply have to take the dot product of this vector and this vector. And so what do we get?  Very explicit formula.  You are given a vector field, pqr.  In this particular case, although I have to warn you, this is a very special case when the parameterization is like this. This is the case when your surface is part of a graph, right? So you are given function f. All you need to do is to find the partial derivatives of f with respect to the two variables, right? And after this, you assemble all of this data into this formula and you get a double integral. And double integrals, you know how to do. You can use any method you like. In fact, I wanted to do an example, but I'm, okay, well, maybe, let's do a very quick example just to see how this works. And then I actually have something to show you, something fun. Know that this has not been fun. I have to say the way I kind of, the way I present this material is not the most economical and I kind of, I feel a little awkward because this is not the best possible way to present it. It looks a little bit ad hoc again because first you introduce this surface area, that was fine, surface area was fine, and the mass is fine, but defining this flux in terms of the surface area integral of the first type, that's a little bit, that's not the most optimal way. So there is a much nicer way to explain this and I will explain it if not next week than during the review lecture for sure. So I will give you a much nicer perspective on this type of integrals. And then you will see right away where these formulas come from. It will not be, it will not look anymore so ad hoc. But for now, this will suffice. This will give you a reasonable definition which you can work with and you can calculate flux within the surface integrals very easily, like in this case. So here is one example. Find the flux of the vector field F which is minus x i minus y j plus z squared k. And s is part of the cone, part of the cone z equals square root of x squared plus y squared between the planes z equal one and z equal two. So let's do this calculation. First of all, in this formula, in this formula above, let's just say u is x and v is y.  These are our x and y. So in this case, the function F is this. This is F of x, y. So I'm just substituting in that formula. So what I get is that F dot ds over this membrane. And by the way, it's very easy to draw it. This is a cone. We're talking about the cone like this. And you use the part which is confined between two planes. So it's this one. I mean just the surface of the cone, but between this plane and this plane. It's equal one and it's equal two. Okay, think about ice cream cones. So this is going to be the integral over, under this, we'll have to do a parameterization. And the natural parameterization would be by its projection onto the x, y plane. And under this projection, this will go to an analyst. Somebody said analyst, right? Very good. This goes to, but you don't have to say it's quite so loudly. Anyway, so you get this analyst. And then you have to integrate over this analyst. An analyst of course has a very nice parameterization which is that you have r and theta, polar coordinates r between one and two and theta is between zero and two pi, right? So you can always convert this into an integrated integral very easily, right? But now integral of what? We have to take the dot product of P F sub x. So what is P? P is negative x, right? And some reason, right? So when you do that, what you're going to end up with is negative x. So you first of all have an overall negative sign. Then you have negative x. That's this negative x. Then you have to take the derivative of this, right? That will take the derivative of this. The derivative of this, you know, as well as I do, it's x divided by square root of x squared plus y squared. Then the next one is negative y. Well, first of all, it's minus, overall minus because there is an overall minus here. Then you have to take this negative y and the derivative which is y divided by square root of x squared plus y squared. And finally, you have to take this times just, just take this z squared. I think that, let me make it a little bit easier. So I write it like this. So you have the last one is z squared divided by square root of x squared plus y squared. Actually, no, no, it's okay. Let's just put, let's just do it this way. So you get plus z squared dA, okay? And after this, you have to substitute, you have to substitute the square of this in here. So that would be just x squared plus y squared. So here, you got a very simple formula, which you can now, a very simple integral. You can of course bring this, take the sum and then you cancel out. You get square root of x squared plus y squared and numerator. Here you get just x squared plus y squared. So you just calculate it by using the polar coordinates very easily. So this is how you do it, okay? Now, there is one issue which is left unanswered here. And that is the issue of the choice of this vector n. And let me illustrate it in this example. In this example, we are talking about this surface, right? This part. But now the point is which n, which n am I talking about? Am I talking about this n which goes downward? Or am I talking about this n which goes upward? Okay, so there is this one issue which we have not yet talked about. And to make everything precise, we have to, in other words, to make the definition work, we have to say which normal vector we choose. And the proper terminology for this is orientation. This unit normal vector to our surface or unit normal vector field to our surface is called orientation. It tells us sort of which side is the front and which side is the back, okay? And as always, there are two different choices. It's kind of parallel to the choice of orientation of a curve. But again, for a curve, orientation goes along the curve. And for a surface, it goes across, it's perpendicular to the surface. But there are two different orientations in this case. And we have to say which one. So always on the problems which you will solve on this homework, it will have to say which orientation you have to choose. When it says downward, it means that the z component of the vector is negative. When it says upward, it means that the z component of the vector is positive. In this particular case, we use the normal vector for which the z component is one. So that means that this formula refers to the upward orientation, okay? And then, of course, the really fun part, which unfortunately I don't have time to talk about, is about surfaces which have no orientation. And this is something really cool which we'll have to talk about next week. But for now, happy Thanksgiving and see you next week.