Hi everyone, my name is Kat Hoffman. Today we'll be discussing making swimmer plots for longitudinal data using ggplot. Little bit about me. I am a research biostatistician at Wild Coral Medicine in New York City. I primarily communicate via blogging on KH stats, but I love to hear feedback on blog posts, so feel free to contact me via any of the modes listed below. So what is a swimmer plot? A swimmer plot is a graphical way to show a subject or a patient's profile over time. It's comprised of a series of horizontal lines in which each line represents one subject. And the colors of the shapes on the line usually indicate treatments or some other type of status at that subject at a particular time. This example on the right is a swimmer plot taken from cell. It's on COVID patients. It's showing the composition of the patients or demographics and the circles on the plot are indicating when the patient was sampled for proteomics. And so swimmer plots can be used in manuscripts, but they can also just be really great for exploratory data analysis or for quick presentations to show different aspects of some type of study cohort. You might be thinking, do we really have to hand code a swimmer plot? It looks sort of complicated. And the answer is that swimmer plot packages such as swimplot do exist, but it's generally just more customizable to write the N10 GG plot code yourself because these packages ultimately just output a GG plot object. For today's talk, we're going to be remaking this swimmer plot that's shown on the right, which is a swimmer plot or a treatment timeline for 30 patients. It's a cohort of hospitalized COVID patients and we're going to use the swimmer plot to show the timing of severe hypoxia, intubation if they were intubated and steroids administration, as well as 28 day mortality. First things first, we will read in a data set. And the data set that you'll probably want to start with is a pretty standard long form data set. It contains an ID column, some sort of time column, columns for each status that you're interested in denoting ultimately on your swimmer plot. So maybe it's a drug or time sampling or meeting other criteria. And you're going to have one row per subject, per unit of time that you're interested in, which is pretty standard format for longitudinal data. We can see for our example plot, there's a data set on my GitHub that contains one ID per patient. One row per day of patient is in the study all the way up to 28, which is since we're looking at 28 day mortality. We have then four columns for the four statuses that we're interested in, which is whether an occupation was intubated, whether or not they received steroids, zero if they were receiving steroids that day and one if they were. Same idea for death and same idea for severe, which is the day that they met criteria for severe hypoxia. It's an indicator variable at that day. So you would probably have a standard data set that looks like this or something like that. And you'll need to get it ready for plotting for your swimmer plot. And there's two main modifications that you'll need to make to make your data ready for plotting. And that is changes to the ID column and the status columns. The ID column change is optional, but I like to refactor or reorder my ID column by the length of time that my subjects are in my study of interest. So that it's a little easier on the eye to just look down and see like a nice descending order of length of stay or length of study time. So we can look really quickly at how we would do that. We can take our long form data set, we can group by ID, make a maximum day variable and then use factor reorder and reorder our IDs by the maximum day in the study. We'll have the same time column, but for our status column, we now want to make a status column that is the time that the status occurred, assuming you want the status to appear on the plot. So instead of it just being an indicator variable for status, we're going to make it be the time if the status occurred and you want it shown on the plot. And we're still going to have one row per subject per time, but let's look at how we can modify that status variable. So we'll take our reordered ID data set and we're going to use the statement case one from Duplier, which basically says if we're going to make a new variable called like the status underscore this underscore day and that status is going to be if the patient met the status that day, then it's the day that the status occurred. So instead of it being an indicator, we're now going to have it be day. And the default for case one, if you don't give a statement that is met logically, is just to be NA, which is actually perfect. That's exactly what we want. So for our severity status, steroids and death, we're going to do this case one statement to make new status columns to be able to show on our GG plot. And so we can see what the data set looks like. We still have our ID day column, but now we have these new columns called severe this day, for example, and this first patient became severe, severely hypoxic on day zero, one, two. And so at that day, instead of it being an indicator variable, we now have a day variable, which is two. And it's NA otherwise. They never received steroids, but they did die on day 16. And so instead of being an indicator variable, it's now 16. You can look and see that a different patient received steroids, for example, on days 13, 14, 15, and 16. So this is what we're going to save and start to plot in our GG plot. So for GG plot, we are going to high-press from our data and then set a global aesthetic for our GG plot that is the ID. And ID is going to be mapped to both our Y aesthetic as well as the group aesthetic. And you'll see why it's basically to make all the lines stay together with the ID. So this is our reordered ID variable. We're just going to take away that gray background with theme underscore BW. And we're going to add our first geometry, which is a geome underscore line. And to geome line, we're going to additionally map that the x-axis should be the day column. And when that's just the day that they're in the study. And the color is going to be mapped to intubation status. And so we can see now we have a line for every patient ordered by length of stay and their intubation, the line is colored by their intubation status. So that's a great start to the timeline. We can then add the day that each patient receives steroids. So steroids underscore this day. We can map that to geome point aesthetic and then it'll just be these default, little black circles as a shape. We'll do the same thing for the day that they met severe hypoxia criteria. And the same thing for the day that they died. And of course this plot is not very useful right now because it's just a bunch of black circles and no way to really distinguish them. So let's modify those geome layers. And to modify those, let's start over. So we'll go back to sort of our blank slate ggplot. And this time when we add the geome layers, we're going to start adding some characteristics of the geometries outside of the aesthetics argument. So here I've added, I've bumped up the size for the intubation status. I'm also going to change the stroke and the shape to be squares for my steroids indicators. I'll do a similar thing for severity of hypoxia. I'll make them circles and I'll make the death, little checker, the hexes. And so now we need to have some sort of a way to be able to distinguish these different colors probably. That's a good next step. And you'll notice that we also, since we added all of these characteristics outside of the aesthetics argument, we're not getting a legend that corresponds to any of these shapes right now or different stroke types. So when we add the colors, you're going to see that we're also going to do this in a way that gives us a legend for other symbols on the graph. So this is sort of the most important step, I would say, to be able to make nice swimmer plots with juju plot. And that is that understanding how you can add colors and the legend at the same time. So bear with me, the big picture is that we want a legend that is going to correctly map every characteristic of our statuses to like into the legend, like actually show it. And to do that, we need all of our statuses if we show up in the legend, which they currently don't because they're not being mapped to aesthetics. We're going to force all the statuses to show up by creating a new aesthetic, which is going to be color that corresponds to the name we want each status to be labeled in the legend. We'll then be able to modify this legend just for color to contain information about the shape, line type, et cetera. You'll be able to see this all in one second, but sort of keep in mind that that's the big picture is we need to force over juju plots, I guess natural tendencies of what it wants to show. So we're going to do that by taking, we'll restart with our gem point as geometries. And when we add them, we're now going to add this argument for color in the aesthetic and that this parameter that we're going to put in is going to match what we want the symbol. So like for this one, it's steroids. What we want that symbol to show up as in our juju plot legend. So we have this variable steroids this day. We want it to be called steroids. And you'll see that when we add it, we toggle back and forth. It gets added here now steroids into this legend, which now describes the colors on our graph. We'll do the same thing for hypoxia. Say we want it to be called severe hypoxia. We see it shows up on the graph. And for death, same thing. It's now in the graph. And one thing to note is obviously these are just default colors for juju plots. So we're going to figure out how to change those in the next step. Also, this is clearly just a legend for color. It's not incorporating any of the shapes that are happening right now. And this legend is in alphabetical order, which may not be actually what we want it to be in like in our final plot. Like I don't want death to be shown first because that's the thing that happens last chronologically. So we're going to show in the next step how to change all of that. So it's time now to modify colors. And to modify the colors, we're going to make a key that corresponds to the syntax that this layer that you can add to juju plot, scale, color, manual, what that takes in. So the key for that is going to be a vector and it's going to be a labeled vector. And the labels are going to match the colors that we just assigned in the previous slide. So I said the label for the legend was severe hypoxia. I'm going to put that there. I'm going to match it exactly same character strings as I have in the color's aesthetic. Those are going to be the names of my vector and the values in my vector are going to be the colors that I want to appear. And that's just the syntax that scale, color, manual takes for values, the values argument. And oh no, the order of this color key needs to be the same as the order of the statuses appear in your legend or as you want them to appear in your legend. So as I said, I didn't want death to occur first in the legend, even though it's alphabetically first. So I put that last here. And we're going to put this calls key that we just made as the values argument of scale, color, manual. So let's do that now. So we have our same plot as before. And you can see we're just really one line of code adding scale, color, manual, values equals our color key and the name equal to the name you want the legend title to be. So patient status right here. We can really do a lot with that just one line of code. So we can toggle back and forth. You can see we got all of the colors that we wanted and the order of the legend that we wanted corresponding to the labels that we want. And yeah, our plot is now looking in much better shape. The last thing that we need to do to this plot is you'll notice that it's still just a legend for color.  So we need to modify this legend using override AES. And override AES stands for override aesthetic and that can be used, well, you'll see but it's in one of the guides layers that you can add for Gigi plot. But to do this, to use override aesthetic we need to create vectors for overriding everything and it's a shape, line, type, stroke and size for this plot because I changed all of those things in the Gion point layers. So these override vectors need to be in the same order as the color key that we made, the calls key. And as a reminder, the calls key looks like this. So it's severe voxia, like all the names and then all the colors that they correspond to. And we'll just make vectors that go in that same order. So for shape override, if you remember for the Gion points we said that hypoxia should be a shape of 21 because we wanted it to be a circle. We don't want a shape to show up for intubated and non-intubated in our legends. We're gonna use NA. We have a shape of 15 and four for steroids in death respectively. Similar idea for all of the other characteristics. We don't wanna line to show up for our status indicators but we do want them to show up for intubation. So we'll put ones there and NA's for the other. So now we'll take these four vectors, these override vectors and this is the same plot as before. I'm just going to add this sort of block of code for the guides layer of dgplot. And we'll say, this basically just says take the color guides and color legend and override the aesthetic with this list that I've given you. So my list is, hey, for stroke, I want you to override the legend and use my stroke override arguments, parameters, column. And for size, same thing, shape line type. And as we toggle back and forth, you can see over here the legend is removing all of the extra, I guess, stuff that we don't want to show in the legend and yet customizing it to be our own. So at this point, we really have a plot that's pretty close to being what I originally showed you in the first slide. So we can take just a couple of minor edits. I'll go through this really quick. These are very much your own personal preference but I add labels, I remove white space using scale X continuous, do a bunch of things to the fonts and the justification, move the legend inside and then I remove these last lines although definitely personal preference but right here is the final plot that I showed you at the beginning of this slide deck. So in summary, long format plotting data needs to have one column per status per row that indicates the time of the status marker if you want it to be denoted. And once you've got your data properly formatted like that Gigi plot can be used to make very customizable sort of lots with the GM line and GM point layers. The legend can be properly configured using override aesthetic from in the guides layer and these ideas can be extended in many ways such as showing patterns of missing data. This plot is showing patterns of missing data using transparency and that continuous covariate from a 2019 blog post but these slide deck corresponds to a 2022 blog post that I have. And yes, thank you all so much for listening and to the authors of sharing and flip book are for making these slides possible. Again, we'd love to hear from you all after this and we'll take any questions now. Thank you.


Hi, and welcome to the session. Let's discuss the following question. It says A D and B C are equal perpendicular to a line segment AB. Show that C D bisects AB. So let us first understand the basic approach to solve it. We'll prove that triangle B O C is congruent to triangle A O D and to prove that they are congruent we'll be using ASA congruence criteria. So let us understand what is ASA congruence criteria. If we have two triangles ABC and DEF then by ASA congruence criteria if two angles and one included sides are equal then we say that triangle ABC is congruent to triangle DEF. So this approach becomes the idea. Now we are given AD is perpendicular to AB and BC is perpendicular to AB. Also BC is equal to AD and we have to prove that C D bisects AB that is BO is equal to AO. Let's now start the solution. Now our aim is to prove that triangle BOC is congruent to triangle A O D. So in triangle BOC and A O D we are given that BC is equal to AD angle C BO is equal to DAO angle C BO is equal to angle DAO because they are right angles and angle BCO is equal to angle ADO because they are alternate angles as ADA and BC are parallel so angle BCO is equal to angle ADO. They are alternate angles and BC is parallel to AD. So we have proved that two angles and one included side of the two triangles are equal. Therefore triangle BOC is congruent to triangle A O D. Hence BO is equal to AO by CPCT that is the corresponding parts of congruent triangles are congruent. Hence we have proved that C D bisects AB by proving that triangle BOC is congruent to triangle A O D by ASA criteria. Hence the result is proved. So this completes the question. Bye for now. Take care. Hope you enjoyed this session.


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Personal Finance PowerPoint Presentation, Income Fund. Prepare to get financially fit by practicing personal finance. Most of this information comes from Investopedia Income Fund, which you can find online. Take a look at the references, resources, continue your research from there. This is by James Chen, updated March 22, 2021 in private presentations. We've been looking at investment goals, strategies, tools, keeping in mind the two major categories of investments that typically being the fixed income, usually the bonds, and the equities, typically the common stock. Also thinking about tools we might be using, such as mutual funds, such as ETFs, possibly helping us to diversify with less of an upfront investment than if we invested in individual stocks, individual bonds. Now we're thinking about what is an income fund? An income fund is a type of mutual fund or exchange traded fund that emphasizes current income either on a monthly or quarterly basis, as opposed to capital gains or appreciation. So if we think about the type of investments that are being made, we're going to be thinking about a mutual fund. So we're pooling our money together under the mutual fund. We're restricting the fund manager to be able to diversify, but at least restricting them to where we want them to put their money, that being the income fund component of the mutual fund or ETF. If we think about the types of companies they're putting their money in and their business cycle, for example, remember that here in the business cycle, that would be kind of the growth area. Some companies go through this kind of growth area where we would expect them not to be paying out dividends as much there because they're trying to hold on to their money, put it back into the business, buy things like property, plants and equipment, which we're hoping will generate future earnings and therefore be reflected in the stock price. When you're talking about more established companies then that have already have their infrastructure in place to not buying a whole lot more stuff in comparison to their overall size at this point in time, you would expect them to have and take their earnings such as the utility companies or a phone company that already has the phone lines up, for example, and give it back to the owners in the form of dividends. Dividends to us as the owners would be the income. So that's the type of fund we're looking at. Usually these are the kind of funds that we want to be more heavily weighted towards when we're closer to our end goal, say retirement, for example, because we might be living off of the income of the income fund as opposed to trying to have them grow the company, for example, at that point. Okay, such funds usually hold a variety of government, municipal and corporate debt obligations, preferred stock, money market instruments and dividend paying stocks. So if we're thinking about income, let's pretend we're in retirement, we've got our nest egg is tucked away. We don't really want to tap into the nest egg itself, but we'd like it to just grow and we live off the earnings of it. So we might say, okay, well, then they might have government or municipal debt instruments like bond, for example, and bonds pay interest. So if they pay us interest, then we can live off the interest and that would be great. And then they might have the preferred stock, which we noted was kind of between an equity and a bond, but it kind of acts like a bond where it's pretty much fixed payments. So that would be we can live off the proceeds for the preferred stock with the preferred dividends, the money market instruments, fairly liquid instruments that could be paying the interest here. And then the stocks when we're investing in equity stocks, that's when we're thinking about the companies that are typically more established companies, oftentimes at that point, because those are the ones that are often likely to pay the higher dividends, which are going to be the income to us. So the basics of income funds share prices of income funds are not fixed. They tend to fall when interest rates are rising and to increase when interest rates are falling. So we've got this kind of kind of what's going to happen in relation to what's happening with interest rates. Generally, the bonds included in the portfolio of these funds are investment grade. The other securities are of sufficient credit quality to assure the preservation of capital. There are two popular types of high risk funds that also focus mainly on income. You've got the high yield bond funds that invest primarily in corporate junk bonds and the bank loan funds that invest in floating rate loans issued by banks or other financial institutions. Income funds come in several varieties. The primary differentiation involves the types of securities they invest in to generate the income. So if we want income fund, we're kind of in retirement. We've got our nest egg possibly. That would be a general scenario oftentimes for an income fund. Now we want to try to get some income. What kind of income will be dependent upon what we're investing in? Is it bonds? Is it preferred stock? Is it common stock? So money market funds. Money market funds generally invest in certificates of deposits, these being the CDs, commercial paper and short term treasury bills. So these funds are designed to be very safe investments aiming to maintain a low share price at all times, but they also tend to offer relatively low yields. While these funds don't carry the federal deposit insurance corporation, that's the FDIC insurance, that the bank products do, money market funds have traditionally provided a high degree of safety. So if you're parking your money in the money market fund, you're not getting a lot of exciting activity there. Even for a fixed income, you're probably not going to get a lot of the interest on that type of investment, but it's a fairly safe investment. Now, if you put your money in the fund, you might not have the capacity to get that federal deposit insurance corporation. If you put your money in actual CDs within a bank, for example, and you might stagger the CDs that you're holding in a bank so that the maturity dates are staggered and that kind of thing, then you might be able to get a little bit more security in the event that there's like a run on the bank or something like that with the federal deposit insurance corporation. Although you do have diversification with the funds because they're investing in multiple different kinds of funds. Also just note the ease that it would be to take the money out of a money market fund versus parking the money in like a CD where you might have to keep it in there or be penalized if you were to take it out. So there are pros and cons there. So the bonds funds. So then bond funds typically invest in corporate and government bonds. Government bond funds carry virtually no default risk and therefore can act as a safe haven for investors in times of uncertainty but normally offer lower yields than comparable corporate bond funds. So now we're going to say, hey, look, maybe I can get a higher return than just putting my money in the bank or possibly even CDs, possibly putting my money in the bonds. And we can think about bonds as we have talked about in prior presentations as kind of like we're lending money to the issuer of the bond that either being a government entity or a corporation. Typically the corporate government entities, I should say are usually the ones that we want to compare to as kind of like the baseline, the ruler that we're going to be comparing to because if we're talking about US government bonds, they can basically print money or tax people. So you would think it's almost impossible for the government bonds to be defaulted on, meaning they don't pay them because the government went bankrupt or something. So the risk there is very low. That means also, however, that if you invest in the government bonds that you're not going to get as big a return because the risk is so low. And then if you look at corporate bonds in comparison, you'd say even the large corporations would be more risky than putting my money into government bonds who can basically print money to pay off the bond if they really needed to. So you would expect the corporate bonds to have higher interest rates. If we're living off the bonds, we want to have as low risk as possible. We would like to have all our money in the government bonds in that case or much of it so that it would be low risk. But we're not going to get as big a return. If we want the higher returns, then we might be going into more corporate bonds that might have a higher returns or foreign government bonds and so on that might give us a higher return, which is what we're trying to live on because we're imagining kind of we're in retirement living on our nest egg here with the money coming from. So corporate bonds carry the additional risk that the issuer may not be able to make principal or interest payments. As a result, they tend to pay higher interest rates to account for the additional risk. Corporate bond funds can be split into investment grade bond funds and below investment grade or junk bond funds. Equity income funds, many companies pay dividends on their stocks. And now we're on the stock side of things. So if we're on stocks and we're in retirement, we're trying to get some money here.  So we want the stocks that are going to give dividends. Funds invested primarily in stocks that pay regular dividends are known as equity income funds. So those are typically the more established ones, right? So these types of funds are especially popular among retired aged investors that look to live off the predictable monthly income generated from their portfolio. Right? So I want my dividend fund to come in and I'm going to get that midday special at the local restaurant with it. So historically dividends have provided a significant percentage of a stock's total long term return. Other income funds, other income producing funds includes those focused on real estate investment trusts, those being the RITs, REITS, master limited partnerships, those are the MLPs and preferred stocks. So example of an income fund, you got the TE raw price equity income fund has $17 and $51, 17.51 billion dollars. In net assets as of Q1 quarter one 2021 and seeks a high rate of growth through high dividend paying stocks in combination with capital appreciation. The fund, which distributes payouts quarterly, paid a dividend of 18 cents per share on December 14th, 2020. The fund was performed relatively in line with its benchmark. An investment of $10,000 in the T raw price equity income fund at inception in 1985 would be worth around 24. I think there's an added zero here. I think it should be 24,500 about as of February 28th, 2021. The lipper equity income funds average result for the same amount over the same period would be about 25,150.


Good morning. We have had two lectures on electrostatics, but primarily the problems that many of you have seem to come from the vector calculus part. And what I will do is I will begin this session first clarifying the questions that have been raised by you which could not be taken online. So, the I will fortunately most of the questions that have been raised are all on one thing, but let me come to that a little later. But the first there was this question from SDM college 1047 center the person who asked this question did not understand the way I defined the divergence by means of a limiting value. So, let me explain what do I mean by that see the point is this that you are all accustomed to defining divergence in terms of its differential expression. What I did yesterday is to define divergence of a vector field f as a limiting value when I take a very small volume let us say delta v going to 0. And I said it is the ratio of the surface integral of the vector field normal f dot n d s divided by this delta v in the limit of delta v going to 0. Now couple of comments I will make to make this definition clear and I will also tell you how this definition is the same as what you have learnt as a differential relationship. One of the things is that I told you that the fields are characterized by having a point definition. In other words the a field is defined at every point in certain region of space and a vector field again is defined at points at all points, but having not only a magnitude, but also as a direction. So, what I told you is this that supposing you have a point and you are looking at you have a vector field at that point of course and you are looking at how do I define a divergence in that. So, the way to do it is this according to my definition now is to consider a small volume it is not material what type of shape of the volume that you take about that point around that point and calculate how much is the surface integral of the vector field f from all faces of whatever shape you have taken and divide it by the volume of this quantity. Now this is the definition I gave and from here getting into the what we called as the divergence theorem was very easy. So, we said I will come back to this definition again. So, we said this is the one of the most important things that we need to understand and this is divergence theorem which tells me that the normal component when it is integrated that is the surface integral of the vector field is given by the volume integral of the divergence of the vector field. Remember that divergence is a scalar. So, this is at this side is a scalar because of this dot product and this side of course by definition of scalar. So, this is my you know I mean simplest way of deriving the divergence.  So, what I do is the following. So, let me first draw a I will take for simplicity a cube and let us suppose this is my x axis this is my y axis and this is my z axis. Let me orient a cube here. So, supposing I draw a cube like this and let us say that this is oriented along the axis. So, that it is intercept when the x axis is delta x on the y axis is delta y and on the z axis is delta z. So, the point is this that to realize what I am trying to say is this. So, notice first let me let me look at any two surfaces opposite surfaces. So, here is one surface which is parallel to the y axis, but the normal direction is along minus y direction and here is the opposite phase which is along the plus y direction. So, the direction of the normal are different. So, let us look at how much is the contribution of flux due to these two phases. Now, what I do is this I consider the corresponding points the corresponding points where the x and the y coordinates remain the same. So, what I said is that the left hand sides normal is along minus j direction the right hand sides normal is along plus j direction that is the y direction. Now, how much is the flux from the left side let us look at the flux of the field from left. So, if you look at the flux of the field from the left which is which has the normal as minus j. So, since I am considering only points opposite points whose values of x and y are the same. So, I do not have to worry about the x and the y component I simply look at the y dependence of the vector field only y of course, there is x and z, but they are the same on both sides. So, f y of y is the combination here and since minus direction is there and it is mine. So, I have a minus and this times the surface area which is delta x times delta z. Now, if you look at the flux from the right which is of course, a outward flux. So, I have a plus this time again the y component and, but this is y plus delta y because the x and the z have been taken to be the same and I have a y plus delta y, but the surface area multiplying is delta x into delta y. Now, look at what is the contribution to the flux from these two opposite side. So, in order to do that I need I notice that delta x and delta z are the same delta x delta z I am sorry here are the same. So, therefore, I simply have this minus this. So, my net outward flux is given by f y y plus delta y minus f y of y. Now, this is net outward flux from the two opposite faces whose area is delta x into delta z. Now, since delta y is small I can do a Taylor expansion of this and I can give get you then delta f y by y times delta y and I have delta x delta z. Now, I can do the similar calculation from the other pairs of faces and if I do the for example, the x face I would get f x x plus delta x minus f x of x delta y delta z and so, from the six faces my contribution will be now notice this is going to be delta x delta y delta z which is nothing but delta v. So, from six faces the flux the surface integral is given by delta f x by delta x plus delta f y by delta y plus delta f z by delta z multiplied by the volume delta v. Now, remember this is your traditional definition of the divergence with the Cartesian expression for the delta f v. So, what I have on this side is delta v times the sorry delta where divergence of f times delta v. My definition was of the divergence was divide this expression by delta v and take the limit. Now, limit actually in some sense I have already taken when I said that when you subtract only keep the first term of the Taylor series. So, this limit has already been taken. So, therefore, the definition I gave you yesterday is the same as the definition of the divergence that you are familiar with namely the divergence of f is given by d f x by d x plus d f y by d y plus d f z by d z, but that is the original if you like the geometrical definition of the divergence and so, therefore, that is what I thought was a good idea to define the divergence here. Then done that let me go to other questions. There was a question from that I will simply answer in one line. Center number 1339 College of Engineering Bhuvaneshwar. There are a couple of questions, but let me first go to the simpler of the question. The question was is a potential a scalar field? Obviously, yes you notice how did I define my field? I said a field is a quantity which has a value at every point and it would be a scalar field if that quantity only has magnitude, but no direction. So, therefore, since potential is a you can define potential at every point supposing I have an electric field you define potential due to the electric charges at every point in space. So, it is a point function. Secondly potential is a scalar. So, therefore, there is no question that potential is an example of a scalar field. However, the derivative of the potential the gradient of the potential which we know is essentially equal to electric field that is a vector field because again it is a point function, but which is defined from point to point having both a magnitude and direction. I have a set of 7 or 8 questions after that. And in fact since so many of you have asked this question and I guess a very large number of you have a confusion on a statement made by me that I said that every vector field can be split into two parts. The one part is one for which the curl is equal to 0 we call it as irrotational and another part for which the divergence is equal to 0 we call it solenoidal. Now, most of you all these questions that I said want to know two things one is what is the nomenclature you see the irrotational is very standard I mean there is no curl. So, therefore, obviously the field does not have a rotational motion there. So, this splitting that I talked to you about it is called Helmholtz decomposition. And they so let me let me try to explain that and also along with that take questions like why. So, basically the Helmholtz decomposition says incidentally I did not mention the mathematical restrictions that are there on some of those like the fields must fall faster than 1 over r square. So, let me let me not be try to be mathematically very rigorous, but the statement is that a vector field can be decomposed into one part which has 0 curl which is irrotational this nomenclature is obvious. And another part whose divergence is 0 which is known I am sorry it is written wrongly there it should be called solenoidal that second bracket irrotational is wrong it should be called solenoidal. And so therefore, I have stated there that f is equal to minus gradient of phi which is and we know that a curl of a gradient is equal to 0. The second one is this is of course very trivial because notice that the curl of a gradient makes sense that is because we are talking about the curl must be of a vector. So, anyway curl of a gradient that is equal to 0 and the second one is divergence of a curl is always equal to 0. So, therefore, now why is it called solenoidal in order to explain why it is called solenoidal because some of these are very old nomenclature and you have to try to understand the history a little bit while irrotational is very trivial to understand. You remember the physical meaning of the divergence as I told you I told you divergence essentially tells me how much a field is spreading out. So, in other words the supposing you are looking at a point if the if the field spreads apart at that point then I called it as positive divergence if the the the fields are coming in towards that we call it negative. Now, so when I say divergence is 0 it automatically implies that the there is the the the fields are not spreading out. Now, imagine lines which are not spreading out if the lines are not spreading out and you are trying to draw the field lines basically you will have to draw them as parallel lines which remain close. Now, the another way of saying that is that it is as if those field lines can be put inside a tube because they are not spreading out. So, therefore, they will not go out of the tube and you will recall from your magnetism what is a solenoid a solenoid is actually a tube like thing. So, therefore, it is as if the whole thing can be contained inside a solenoid and therefore, the name came that the the there is no divergence there. So, it can be put inside tube the field lines can be put inside tubes and therefore, there those with 0 divergence. So, that is about nomenclature, but that is not the main question most of you wanted to know you know what is the you know wanted me to explain this theorem a little better. So, let me let me go through I need a few more I did not plan on proving this theorem, but since very large number of you have asked that the theorem should be explained better I have no option, but to do it. So, let me I need a little more mathematics, but most of you would know it. So, therefore, I would hurry through those mathematics first comment that I want to make is remember grad curl and divergence they are all first order differential operators grad is gradient of course, curl up and divergence we have been using it. Now, so what we do is we define an operator which we call as the del operator which is written like this i d by dx plus j d by dy plus k d by dx. Now, the when it acts on a scalar function we call it grad.  So, that is your grad. Now, divergence which is written by the del operator del dot there is a dot missing because of math mismatch this is an operator which acts on a vector function because del dot is a divergence.  The curl similarly is also written like this I am not going to derive these expressions these you are all very familiar with all right. Now, let me define using this operator a thing called a Laplacian. Now, Laplacian is written as del square which is nothing, but a short form of del dot del. Now, notice that the second del second del being a gradient operator has to act on a scalar function and then it will give me a vector it will give me a vector function on which the del dot or the divergence will operate. So, therefore, the meaning of del dot del which is also known as the Laplacian operator is this that is i d by dx plus j d by dy plus k d by dz dotted with this and that gives you del square d square f by dx square dou square f by dou x square dou square f by dou y square plus dou square f by dou z square. So, I define my Laplacian operator as d square by dx square plus dou square by dou x square plus dou square by dou y square plus dou square by dou z square this is called a Laplacian operator. Now, let us look at this Laplacian operator a little better, but in order to do something like this and particularly because we work with electric charges which have a 1 over r potential I need another bit of a mathematics and I will spend a bit of a time on it because most of you might have done it as a course, but it requires a bit of a an explanation. I need to define and explain what is meant by a Dirac delta function Dirac as you know was a well known physicist. Now, if you look at it strictly it is not a function in fact, many people give it different name, but physicists like to call it a function. So, let me stick with that that it is called Dirac delta function the definition of the Dirac delta function is this when the value of its argument is not equal to 0. In this example, if delta of x is if x is not equal to 0 then the function becomes equal to 0 on the other hand if x is equal to 0 I do not define that function at all I do not define that function, but I define something crazy I say if the function is 0 everywhere accepting at a point what it is at that point I do not explain I say simply that if you integrate that function integrate any function multiplying it with that delta function and the range of your integration includes this point of argument the point at which this argument becomes 0. So, any interval from minus epsilon to plus epsilon will do f x delta x dx that is equal to simply the value of the function at the point 0 and this is we require that f should be continuous at x equal to 0. Now, you would immediately ask me why is it not a function in the proper sense. Now, in order to understand it go back to your definition of an integral. Now, remember that the integrals that we do they are usually called Riemann integrals and in the Riemann integral there is a theorem which you must have learnt that supposing I have a function which is 0 everywhere accepting at one point. Now, if I have a function like this then the area under that curve becomes equal to 0 because if a function irrespective of whatever its value at one point if the width of a rectangle which it encloses is 0 then your integral has to be equal to 0. But what I am saying here the delta function is such a function for which the integral of f x delta x dx equal to f 0 or integral delta x dx is equal to 1. Now, what does it actually mean if it is not a function what is it? If it is not a function you can look at it in many ways. So, for example, you can look at delta function as a limit of sequence of functions. So, let me explain by giving couple of example. For example, look at a sequence of quantities like delta n of x which are all 0 as long as x is less than either less than minus 1 over 2 n or greater than 1 over 2 n. But if it is between these 2 values then its value is n. Now, that is these are basically rectangle within this range of n height which is equal to n. So, look at what the picture is like this is the type of picture that you take various values of n and you get rectangles of different value. Now, what happens when you increase the value of n? Now, if you increase the value of n what is happening is that your this width is decreasing, but the height is increasing because height is proportional to n. But in such a way that the area of that rectangle becomes equal to 1. So, therefore, when you take this idea and take the limit of n going to infinity. So, what has happened is the width here is shrunk to 0, but the height has essentially gone to infinity if you like, but in such a way that the area which is the product of the height and the base has remained equal to 1. So, this is one of the ways of understanding it. There are many other ways this can be looked as a limit of Gaussian functions which you are familiar with I will just because we are going a little slow I will skip it, but this is what it is the picture is, but there is just an axis here. So, I take one Gaussian function which is in the green. Now, what I do is I keep on reducing its width, but increasing its height which is given by the blue and you could keep on doing this. You keep on doing this the Gaussians have been normalized in such a way that the integral is equal to 1. So, integral will remain 1 when this width keeps on decreasing and the height keeps on increasing, but so ultimately it will be a very spiked thing and therefore, this is also another example of a delta function. Delta function has many interesting properties, but I have just listed a few of them since I am not going to be using it right now I will just leave it in my notes. Now, one of the reasons why I introduced the concept of a delta function is to tell you what is the gradient what is the Laplacian operating on 1 over r I told you remember that 1 over r is my Coulomb potential. So, it is a very important potential that I have. So, let us look at that firstly supposing r is not equal to 0 remember 1 over r is not defined at r is equal to 0. So, supposing r is not equal to 0 the gradient because this is just a function of r.  So, therefore this gives you minus vector r by r cube right 1 over r square unit vector r the del square is del dot del of this and so that is del dot of this quantity. Now, this is the divergence of a vector multiplied with a scalar and this is a very standard formula you can calculate it trivially and you get that del square of 1 over r is equal to 0, but provided r is not equal to 0, but what happens if r equal to 0 is included. Now, for that I am claiming that del square of 1 over r is essentially a delta function accepting for a factor of minus 4 pi which comes with it. So, del square so look at this definition del square d cube r I said it is equal to del dot del 1 over r d cube r.   So, put all this now if you do that integration this integration does not vanish it becomes equal to minus 4 pi. So, I have worked it out here and so therefore, del square of 1 over r is a delta function, but for a factor of minus 4 pi that we need and that is because if r is not equal to 0 it is equal to 0 when the r is equal to 0 is included the volume integral turns up to equal to minus 4 pi. Now, this is a very important you know mathematical relationship which we will be requiring there. Now, why am I doing this I need it for proving the Helmholtz decomposition in the best possible shortest possible way there are many derivations, but most of them are long. So, therefore, this is the fastest. So, what I do is this supposing f is a vector field I define a vector v which is a function of vector r as an integral of f of r prime. Now, notice that this is a definite function depending upon the integration variable by r minus r prime d tau prime. Now, what happens if you take del square of v i i is the i th component. So, since f of r prime depends upon integration variable. So, it comes out because the del square is being taken with respect to r. So, it is del square 1 over r minus r prime see del square is being taken with respect to r because the left hand side v is a function of r only the vector r only. So, this del square of 1 over r minus r prime as we have just now seen is minus 4 pi delta r minus r prime. So, this gives me f of r. So, notice one thing the we have a given vector field f using this by applying the Laplacian operator. I would proved that they I can find a v defined through f corresponding to f I can find a v and this is my definition of v. So, let us let us proceed with that I need a an algebraic identity with which you are all very familiar which is del cross del cross of a vector field del cross del cross of v is given by gradient of divergence of v minus the Laplacian of v del square of v. Now, define remember I have defined v taking the expression for f. Now, what I do is using that v I define a scalar phi as minus del dot v and again using the same vector field I define a another vector a which is given by minus del cross of v. Now, that tells me del square v from this identity which is del of del dot of v minus del cross del cross v. Now, you can see immediately my del dot of v was minus phi. So, I have got it that minus del phi my del cross of v was minus a. So, therefore, this is del square v equal to minus gradient of phi plus del cross of v. Now, this is this is what we have, but del square v I just now proved is equal to f. So, you notice what we have done we have a vector field v f using that f I have constructed a v and showed that del square of v is equal to f. And now I have shown that del square of v is can be written as minus gradient of phi where phi and a are defined with respect to v itself plus del cross, but this is precisely what you want to prove that vector field f can be written as gradient of a scalar plus a curl of a vector and that is your Helmholtz decomposition. The other many people ask the question give us an example of vector decomposition. There are actually empty number of example, but let me try to give you some from our own subject namely electricity magnetism. So, you are all familiar with that the mean Maxwell's equation del dot of p is rho by epsilon 0 del cross of p is equal to 0. Now, notice that I can write e as equal to minus gradient of phi that is it does not have that it has only one component it is written as a quantity whose curl is equal to 0. And this plus 0 the the the solidarity part is 0. And this expression for phi is given by this the way we defined our v. So, this is an example of Helmholtz decomposition this is a very trivial decomposition because the second part is 0. Likewise, I can define corresponding to the magnetic field. So, this is I am going back to the same example I said phi is this 1 over 4 pi epsilon 0 rho by r minus r prime you can check that if you do del dot of p on this using the property of the delta function that we talked about it gives me rho by epsilon 0. So, that is there. Magneto statics provides you another example because del dot of b is equal to 0 del cross of b is equal to mu 0 j. And b is equal to del cross a once again this is an example where I have a solenoidal field, but I do not have a corresponding irrotational field. So, the expression for a is given by this. So, these are two examples from electrostatics and magnetostatics, but look at a little more general situation. I I claim that that I could define E the electric field for instance as equal to gradient of something V plus curl of some other field F where V is given by this expression what I do I have been given remember what I am trying to do I have been given an E. So, using E I construct this I know E. So, I say all right I have a scalar V of r which is del prime dot e r prime by 4 pi r minus r prime d tau prime. I have a vector F which is again constructing using E with this expression. Now, how do I know that this is correct? The way to know it is correct is the following I will give just one explain one I I am trying to say that E is given like this. So, what is supposing I calculated del dot of E? Now, this part is of course, identically 0. So, this is minus del square V is minus del square of this integration and once again the integration is with respect to r prime variable. So, therefore, it will go inside it will go inside giving me del square of 1 over r minus r prime which as you know is a delta function. So, therefore, this is gives me an identity and. So, therefore, this expression for V or similarly F that I have written down is correct. So, this is the way to understand Helmholtz decomposition. So, as I said I have spent more than an hour now in explaining the questions that came up and since I do not have really that much of time to start a new lecture what I will do is let me try to see what the new questions are coming in can you. Meghna the institute Meghna Chah Kolkata. Yeah, good morning Professor Ghavash. Good morning. Yeah, yesterday we talked about gradient of a scalar field and then we talked about curl of a vector field and we associated those things with some physical interpretation. And now we are talking about curl of a gradient of a scalar field is 0. And then we know divergence of curl of a vector field is also 0. So, what are the physical significance of those two mathematical forms? Number one. Number two, yesterday we talked about electrostatics Coulomb's law. So, which is essentially applied for charges at rest. So, can we apply Coulomb's law for charges in motion? If yes, why? If no, then why? So, let me let me talk about the first question first. You wanted to know to understand physically. You see the physically it should be obvious from the way you define the curl and the divergence because see we said that the meaning of the curl is how much the you know thing is rotating you know I mean it has a sense of rotation changing its direction. So, therefore, the if you take that physical interpretation of the curl right. So, the point is this that it is then understandable that such a field because it is curling out it is not diverging actually. So, I am just giving you a very hand waving argument on that. So, the and the same thing that comes out that see everything comes out from like for example, why it is called solenoidal that was the question which we tried to answer in the morning. The this is historical, the name does not have anything to do with solenoid, but it has got something to do with a tube because a solenoid has a tube like character. Now, if a vector field does not spread, if it does not spread the field lines can be confined in a tube and or in other words the field lines can be put inside a solenoid. So, the thing is this some of these mathematical things you can always prove rigorously mathematically, but at every stage of mathematics it may not always be possible to bring in a physical intuition, but but roughly this is what happens one of them is the field is diverging ok. The another one is the field is curling around and. So, therefore, there is it is a it is a divergence less field. So, they they I mean if you want a hand waving argument it is good enough, but it is good to sort of say that at every stage of mathematics you do not look for a hand waving argument. I mean I think it is best best to prove it rigorously, yes. Actually I am wondering I mean whenever we are taking gradient of a scalar quantity and it always gives us a curl 0. So, what really makes it so special that a vector which comes out from a scalar field always gives us curl 0, that was my. So, you know I mean I cannot probably answer it much better than what I did because what you are asking me essentially is this that take an arbitrary vector and then take a gradient and look at what the gradient I mean sorry take an arbitrary scalar then calculate a gradient and see what does that gradient look like pictorially and and then try to show that why such the curl of such a thing should be 0. I think it would the mapping of all this physically is going to be troublesome I mean because you know I can always think of one scalar, but then you will think of another scalar it is probably not never a good idea to try to understand every step of mathematics physically I mean I do not personally think. Now, the next question was connected with the Coulomb's law the Coulomb's law is valid whenever there are charges ok whether the I will be trying to tell you if not today maybe tomorrow whether the charges are at rest or at motion the thing that we will try to point out is this the electric field and the magnetic field are different manifestation of the same situation. So, we call say for instance supposing you have a charge at rest in your lab now you would say ok Coulomb's law is valid it is electrostatic from Coulomb's law is valid now supposing the charge is still the same, but you are moving now with respect to you the charge would appear to be in motion then you would say there is there is effectively a current if there is a current you would say there is also a magnetic field right. So, your friend who is sitting down there in the lab will say there is an electrostatic field and no magnetic field because the charges are at rest whereas you will say that I have a charge. So, therefore, there is an electric field I agree, but my charge is in motion. So, therefore, there is a magnetic field also the point is that I mean you can rigorously establish this if you go over to relativity, but I will try to give you some example of how they this thing that electric field and the magnetic field are also related to your frame of reference. Thank you. Yes, Silliguri go ahead. Good morning. Good morning. I just was just wondering then Helmholtz theorem you stated that it is not valid for just any vector field. I guess there are two conditions first one is it has to be twice differentiable and the second one I think it has goes faster than 1 by r I was just wondering why is the second condition needed. Okay, I mean I knew that somebody will come up with that question see my problem is that I have to mathematically justify it and it is sort of taking a lot of time. So, what I could do is if you mail me that question I can send an answer because I really cannot spend that much of time on mathematical niceties, but yes the second condition that the field must fall faster than 1 over r square is correct you are absolutely right and we need that condition in fact then you might ask me the question in which case how is it being used for electrostatics where the Coulomb's law is does not go faster than 1 over r square, but then that is a very limiting case. So, I will not be able to spend that much of time on mathematics here you know I mean we have already spent almost three two and a half three hours on mathematics today. So, if you send me a mail my mail address must be there on the model I will try to answer that So, yesterday when we concluded we had said that the there are two things which are going to be useful for us one is the Coulomb's law and of course we know what Coulomb's law is and we said that the if there is a test charge q at a point e then the it experiences a force q times e and the second session point is that if you have large number of charges the effect due to them add up that I called as vectorially add up and I call that as the principle of superposition. Now it is important that your students understand the that principle of superposition is a very important concept because it is not clear that when there are two effects which are acting at the same place simultaneously right it is a matter of simply adding them up sometimes you know the effects multiply, but so superposition principle says that if there are different forces the forces simply add up. So, that is a very important point so the we our entire basis of electrostatics electromagnetic theory is based on the you know validity of the superposition principle. So, that if I have a charge q i at the position r i then the field at the point p which is has a position vector r is given by 1 over 4 pi epsilon 0 sum over i q i r minus r i by r minus r i cube. So, this is my superposition principle and Coulomb's law. Now having talked about Coulomb's law let me jump to we discussed yesterday the definition of electric flux and Gauss's law. So, let me make one statement very clear Gauss's law is completely equivalent to the Coulomb's law, but the reason why you use Gauss's law is it is much simpler to deal with in some cases and particularly in cases which exhibits symmetry the Gauss's law is much better used over Coulomb's law. In fact, later on when you do magneto statics you will realize the role of Coulomb's law is taken by Barrett Savart's law there and there also we have a situation corresponding to the magnetic Gauss's law and once again or even there are things like Ampere's law even there such laws which depend upon symmetry are much easier to deal with. So, basically what it says is this remember how we define the flux as the surface integral of the electric field. Now here is for example a surface and there is this electric field coming in and so the there are incoming from one side outgoing from the other side. Now let us try to understand this Gauss's law a little bit because yesterday actually there was a question also which says why is it that if I have a charge inside a volume I get a flux, but if the charge is outside I do not get a flux. Now in order to understand that we need another mathematical concept and that that is the concept of a solid angle. Now what I told you yesterday is the solid angle is a straightforward generalization from the way we define an ordinary angle. Now remember in an ordinary angle what we do is to say that supposing I have two radial lines from a point at equal distances I draw an arc. Now this arc is of course perpendicular to the two radial lines and so therefore my definition of the angle that it makes is given by the length of the arc divided by the radial line and as a result the angle is a dimensionless quantity it is a different matter that we made some units like degrees radians and things like that, but it does not have any dimension. Now the situation with respect to solid angle is identical. So here what we do is instead of an arc we take a surface supposing I have this surface ds and at any point imagine that from that point we start drawing tangents all over on to that surface. So this will make a cone at the point b. Now this quantity which it makes that is obviously feels like an angle, but you notice that it is not a linear angle it has a conic angle if you like. Now how does one define remember our definition of angle theta was the length of the arc divided by the radial. Here I define exactly the same way I say it is the area the perpendicular area like there also we said it is the perpendicular one. Perpendicular area divided by r square because this has a dimension of r square this also has a dimension of r square. So therefore, this becomes and since if the radial line is vector r and the normal to the this surface remember that these are all valid for infinitesimal surfaces is n and if this angle makes an angle alpha then this is given by ds cos alpha by r square. So, now let us look at why did we suddenly talk about Gauss's law and solid angle. Now look at this point here supposing there is a volume there if I have a volume there then I am looking at what is the flux through this surface. Now so I have a q here and this is a enclosing thing remember the my definition of Coulomb's law. So, it will be 1 over 4 pi epsilon 0 q divided by r square is my thing there. So, therefore, since the way we defined our solid angle. So, solid angle was essentially the normal component of this thing divided by r square. So, therefore, what I did is to say the flux through this infinitesimal area is given by q times d omega by 4 pi epsilon 0 because remember that how d omega was defined it was this area the perpendicular area divided by r square. So, r square is there in the Coulomb's law and ds which will come in here perpendicular that is there in the definition of flux. So, therefore, the flux from this area element is written as d phi which is equal to q times d omega divided by 4 pi epsilon 0. Now if this is the situation then if you are looking at the total flux from all over then you notice that my solid angle is going to be have to be integrated and the total solid angle around is 4 pi. So, therefore, if I take how much is the flux. So, I will simply get this this will give me 4 pi I will get q by epsilon 0. So, now let us look at the same situation when this charge is outside because this was the question why do not you get a why do you say in Gauss's law that the flux from a surface is given by a closed surface is given by the charge enclosed divided by epsilon 0. So, look at this here now if you want to look at this point P and draw a tangential line this is going to intersect this closed surface not at one point, but at two points simultaneously. So, this is one area which it intersects this is another area which it intersects. Now the point is this that there is a slight difference between the two areas. So, far as this area is concerned the angle between the radial line and the normal is acute angle. So, far as this is concerned the angle between this area and the radial line is an octagon I mean actually it is the normal direction is outside above the. So, there is a opposite direction. So, therefore, since the both of them irrespective of the size of the surface that they enclose since they will give me the same solid angle. So, therefore, one of them gives me a positive angle another of them gives me a negative one. So, therefore, the total solid angle that is substituted is equal to 0 and so therefore, if the charge is outside it does not subtend any angle sorry about the any flux. So, that was of course, my the statement of the Gauss's law. So, we have our first Maxwell's equation here that is saying that the flux is equal to enclosed charge divided by epsilon. This is of course, what we learnt in school, but let us try to cast it in differential language. So, what we do is this remember we are talking about closed surfaces. So, flux is integral e dot d s integrated over the closed surface that is the definition of the this side. On the right hand side let me come back to this next, but on the right hand side which is there by the last term I have a 1 over epsilon 0 times enclosed charge. Now, how much is the enclosed charge now obviously, it is given by the integral of charge density inside that volume integrated over the volume. So, now, but we did what is called a divergence theorem this is a surface integral e dot d s. So, this must be the same as divergence of e volume integral. So, therefore, the Gauss's law also says that the volume integral of the divergence is 1 over epsilon 0 times the integral of rho d v which is nothing, but the charge. Now, if you look at this relationship, if you look at this relationship then you realize that the this is this must be true of any volume. Now, a relationship like this which is true of irrespective of the size of the volume mean that the two integrants must be the same because you know you can always satisfied for a particular volume, but if you say this is true of arbitrary volumes then the two integrants must be the same which tells me that the divergence of the electric field is equal to rho by epsilon 0. So, this is the first of the Maxwell's equation that we have derived that is basically equal to the Gauss's law electrostatic Gauss's law. So, I made a statement that Gauss's law is completely equivalent to the Coulomb's law. So, let me let me try to get that. So, to do that let me let me try to get this del dot p from Coulomb's law.  So, this is the this is equal to 1 over 4 pi epsilon 0. Now, what I do is this I take this gradient take this divergence del dot inside, but remember that the integration variables are r prime whereas, I am differentiating or this divergence is being with respect to the r variable.  And I need a minus sign now because what I now do is this I say all right this is this was divergence with respect to r you notice this quantity is entirely a function of r minus r prime. So, what I could do is I could make this divergence with respect to the prime variable and since this is a function of r minus r prime I put a minus sign there.  So, I write it as del prime of 1 over r minus r prime. Now, notice what I have done I am just doing some mathematical tricks the gradient of 1 over r minus r prime if it is with respect to minus r prime variable is plus 1 over r minus r prime cube into 1 minus r prime because the variable there is a minus sign that will come in successive differentiation. But with this there is a del prime dot del prime.  But this we have shown is given by minus 4 pi epsilon sorry minus 4 pi times a delta function. So, if you use the delta function to do this integration you find this is equal to rho r divided by epsilon 0. So, this is the mathematical derivation look at that here is Don Boskoff. Hello. Yes. What I want to say is in spite of the ease of applying the Goss law I believe it has got limited scope compared to the Coulomb's law here is because the Goss law can be applied only in case there is some kind of a symmetry. In case there is no symmetry obviously we have to fall back on the Coulomb's law only because the Goss law in the most general case it will only give you the total flux not the electric field. You are absolutely right I made that statement earlier also I said that Goss's law or for that matter Ampere's law later when we do it they have the limitation that they can be effectively used only in situations where there are sufficient symmetry. But remember that Coulomb's law is not very easy to use in general cases. So, in general cases both of them are equally bad. But in some high synthesis the Goss's law gives you an advantage over Coulomb's law they are completely identical in some situations you should use Goss's law because of the symmetry. I have I completely agree with you that the Goss's law can be used in very limited situation but mathematically it is as rigorous as Coulomb's law. So, do not be under the impression that see all that the difference that is there is I may not be able to calculate it using Goss's law whereas Coulomb's law at the worst case you can go to a computer and do a work. I agree with you you are right thank you which is that. Maulana Azad NIT yes. Hello very good morning sir. My question is related to the concept of potential that what is the... Related to concept now? ... concept of potential whether it is just a mathematical concept which is used for the sake of mathematical ease. Potential are you talking about? Yeah. Is that a potential you are talking about? Yes sir. Okay, so the point is this that is it just mathematical ease? Answer is only partly yes. Firstly you realize that it is much more difficult to deal with vector fields than scalar ones okay the potential is a scalar field the advantage of a scalar field is you can since superposition principle is valid the you can go on adding things up without having to worry about what supposing you are working with electric field you will need to worry about the directions of the electric field how to add up vectorially but what I can do is when there are multiple situations I can just add the scalar fields just like that okay and then after I have added all the effects I take a final gradient to find out the net electric field. Now I have avoided the entire problem of having to do a vector addition so it is not just a triviality now the question is there also a meaning to that the answer is yes in fact it is similar to you know many of these concepts even the name potential came from for example the fluid dynamics the point is this that you see when we are talking about fluid flow the level at or the height at which a fluid is located is also important because you make a statement that the fluid has the natural tendency to go flow from a higher potential or a higher level to a lower level now I actually borrow the same thing here also that the potential has a meaning because it is essentially telling me what is the level what is its electric level and supposing you feel put a charge there will it simply which direction will it go okay will it climb up will water climb up the answer is no. So these are also extremely intuitive concepts the the third thing is the potential though it is not equal to potential energy there is a intimate relationship between potential and the potential so therefore 1 it is a scalar function 2 it sort of tells me what is the natural tendency of the charged particles in an electric field which way they will move so I it is much easier instead of calculating the force to tell me the look if I know the potential I know it goes from a higher potential to a lower potential right and the third thing is its direct association with potential energy thank you. So let me give you a few examples of as I told you that we are not you know I mean strictly following a tutorial but we will be having some look at this interesting problem it is a question of I have a small charge charge q and a and a cube and this charge q is located at one of the corners my question is that what is the flux through one of these surfaces remember some of our friends pointed out the Gauss's law is useful only when there is sufficient symmetry at this moment it does not look like there is a symmetry because it is in one corner of a cube but you see if you are smart enough you can create a symmetry how do you create a symmetry what you do is you pack up this I have shown you here that put one more cube another cube below another cube here. So make it into four different cubes such that this charge q which was at one corner of the cube comes to the center of this four cube situation. Now I asked the following question supposing you had a charge which was at the center of a cube now then what is the flux that is going out from any one of the surface now you would say that look this is so symmetrical that from this bigger size cube supposing there is a charge at the center of a cube since a cube has six faces and with respect to the central position there is no special position of any of the faces there are the six faces. So q by epsilon 0 was the is the total flux through the whole surface. So q by 6 epsilon 0 should be the flux through any one of these surfaces fine but then this surface that I have got is four times as big as my original surface. So therefore my net through the shaded side here is just q by 24 epsilon 0 you notice a simple idea you try to use Coulomb's law you will be nowhere but beautiful symmetry of this problem which we have constructed. In the second session I will continue with application of Gauss's law and since we seem to be running out of time for the current session and you all we you all as well as myself we need to go out for tea we have just couple of minutes maybe one more quick question if there is a clarification you can take it up RBS engineering college can you quickly if there is a quick comment or a question you can tell me because otherwise we will be answering them all together yes go ahead good morning sir good morning sir my question is that apart from magnet magnet is there is any example where net divergence is zero well yesterday I told you velocity field for the there are lots of situations where the things could be zero magnetic field I talk about because this is subject of our course here but there are velocity fields which you can have where the divergence can be zero because the question is that if in through a particular volume the amount of fluid coming in is equal to amount of filling going down then of course the divergence could be zero because the fluid is incompressible so the row the density does not change there there I mean if you want to look up other examples from various fields we I am sure we can look it up because it is a very standard field and but we stick to because our magnetic field because of absence of magnetic monopole is that but remember even with electric situation if I have a dipole for example right I can create a situation where my situation would be zero but the you I will remember this question particularly if I mean fluid flow is one of them which I often remember if there are other examples which I can find out I will answer it next time thank you very close okay we have a 15 minutes break now for T and we'll come back and continue with quantum mechanics session


just needed me the first time I had hold hands like wow this lady is gifted and she is really anointed Pami-rams, welcome to the show again again I was so happy to have you again new introduction come up Did you know like I am just before I met they are those and are seeing you for the very first time they have interacted with the song now they are asking who is that just introduce yourself Pamirams is just a very humble lady who loves Jesus Christ. I love saying that all the time. I love serving Him because loving Jesus Christ is not just by our words. He wants us to love Him with our heart. So as weak as I am, I have this desire to dedicate my life to Christ just to serve at His feet. That is Pamirams. And when did you dedicate your life to Christ? Well, I dedicated my life long ago. I think this thing I always say that I was born in a very Christian family. So I began to know Christ since I was very a child. But of course I made that personal decision myself to accept Him as my Lord and Savior. So since then the journey has not been easy, but of course by the grace of God we are here. And you know people, you've mentioned something really interesting. People are born and then they grow up in church, they assume they are born again. Oh no, that is wrong. They assume they are born again. They don't make that decision. The fact that you are born in a Christian family doesn't make you a born again. You know you need to have a personal encounter with Christ. By yourself. Okofu niki tu personal.  Now kubali kuwacha dambi you say me I want to serve you. I want to work with you. Guide me. You know I was once like that. The salvation of your father will not save you. The salvation of your mum will not take you to heaven. So you need to work on yourself. Thank you for clarifying that part. Thank you so much Pamirams. You were here last in November.  And you know people went back to church. Thank God. And how have you been since then? What have been up to doing other projects as well? Yes, I've been well by the grace of God that's why I'm standing here. I really thank God. I've done a lot of new music. And I'm just wishing for those who forgot about me just to rush back and check on what the Lord has in store for them. It hasn't been easy but by the grace of God as I always say we live by grace that's why I'm here. Because God gave His grace it was sufficient for all of us. Yes. And you know I saw something on your YouTube channel. The live ministration. Yes. And I was hoping you would pursue that when you are releasing other music but you did not go that line. That's true. I didn't go that line but I'm still working on it. You know going live requires some nice budgets. So sometimes it's good also when you have supporters here will maybe bring their cameras will give you the place the location where you'll do your life or support one thing or the other. So just bringing everything together for yourself is not easy. So that's why you'll see maybe you go once in a while and then you also focus in the new normal things of life that you have to fix here and there. And then you'll go back again. But we're actually fixing on that. So very soon maybe we'll do another live session. Apakaribu nitu. On your YouTube channel. Please. And to nafini liya notification. Once you upload it or once you post something new we are notified about it. Pamirams is our YouTube channel. I know Adina Jewa. So after Kalu you did Majira. Yes. And I want to know about this Majira so. Wow. Because it's my... Okay all your songs are my favorite. She keep requesting Majira. Did you explain for me Majira? Jesus. I want to find out why did you write this song. Was it something that you are going through at that point and what exactly were you going through? Yes. What made you write this song? Majira is a song sung by faith. Because as I was really going to record Majira I was just recovering from a hearing loss. I had lost my ears completely. I couldn't hear. And how did you lose it? I think due to stress. That's why I always emphasize to follow what the Bible is telling us. You know the Bible ask us this question. By worrying do you add anything in your life? We say no we don't add anything by worrying. Because worrying what it does it only decreases us. So the best thing to do is just to trust in God. So I think I was a little bit stressed up here and there or depressed or something I don't really understand. But then it affected my sense of hearing and that's how I lost my sense of hearing. But I thank God for His grace. So Majira comes at that point where I was really broken and down and the Lord was speaking to me that I am rafa yohila. So hold there and I've been holding on to that word that He said is rafa my hila and I've been seeing the restoration of God. So you may be out there struggling with something. It may not be my hearing loss like mine. It may be something else. It may be depression. It may be I don't know stress. It may be another loss, kind of loss. I want you to know that Jehovah Rafa is there. I want you to know that Jehovah Nisi is available. I want you to know that our hila is still available. So trust in the Lord and you shall see His hand. I'm interested to ask the way you say it about Majira and looking at your songs. So many songs, music out there on YouTube na maybe zingi na maybe we jaya upload. We have the kind of hekalu, Majira. There is the new muema, sababu. There is Dio. Dio, noto. Like which one song najunda samazote zinaku minisariya but one that you sang and you felt by the time you are releasing it you apart from just having a testimony it was ministering to you the more more than the rest of the music Wow, just as you said like all of them minister to me differently but if I have to choose only one and be sincere, I will choose Majira. Why? Because Majira came when I was really desperate my brother. I was desperate because as an artist when you lose this you become useless actually. You need to find something else to do. So I was really desperate and I was really crying I remember my manager forcing me you will sing this song and siskivi zooda I am like I can't even hear well it was just recovering God wants you to go and minister just like that so I went crying with my heart it was my heart was heavy when I was singing the song Majira so it really touches me in a very very personal and different way That's why you can see even in the video it was hard for me to smile even when I tell him to smile it's very hard because my heart was heavy I was like God, God, God yes, I know this is in the season but God, where are you I want to see in my season console me, comfort me, give me strength But no one can tell you you are having a time even in the video I love it personally I love the song I remember you shared with me the song even before you released it and I listened to it the whole day I listened to it I prayed to it I prayed to the song and we love the song Thank you so much I remember my director telling me look on the side and I'm just there and he's like look on the side and then I'll turn and he's like now here and then I can't see and he'll be telling me and then I'll turn it was just it wasn't easy but I really thank God that I'm here look at it the song is a success yes it's doing so well check it out on our YouTube channel that is family rams yes and now your new song Muema wow Muema now it's also just a testimony yeah why because with COVID and everything that is around people are becoming desperate people are being discouraging and all that so we don't have to really focus on the negativity because in everything that happens to us if you look at it very clearly and keenly you will see there's a dose of a goodness of God in it even when I lost my ears when I sit down I'm like there's still a dose of goodness in it God knows why because he has a purpose for every circumstances so this is just to encourage people forget about the negativity of life forget about what you lost yesterday that you can't recover forget about whatever that happened to you about your tears can you just focus on the goodness of God that you're still breathing today that is the greatest testimony ever so let's just focus on the goodness of the Lord because the Bible says the Lord is good to everybody because I've lost my sense of hearing that he has stopped being good his nature is goodness so regardless of whatever that is around let's just focus on the positivity and remember and enjoy the goodness of God in our lives I noticed something about this video Muema, you wanna is she your sister? in the video yes you have power you can't win that's my elder sister I know too we take on a kwa good job I want people to run to your YouTube channel that's why I high five people I want people to run to your YouTube channel and check out her sister our YouTube channel is Pamirams Pamirams and I just want to take this opportunity for those who love you for those who love DJ Knicks kindly for those who love this show and for those who are loving me for being part of this show today kindly just rush to my YouTube channel and support me I think you will hype it even better than us because you are the host here kindly just run to my YouTube channel and support me let people know that I was here let just the neighbor know that I was here yes subscribe to my YouTube channel Pamirams and of course leave me a comment or just watch all the songs watch all the shit when you're dropping the comments make sure you say from JC's circle we are here now I laugh when he comments do that team JC team JC went and subscribed to her YouTube channel I'm not even lying I'm just telling you the truth you're going to love all her songs all and not live out even one actually when I heard Muema it just spoke to me the message you're saying about how people are worrying how people are people might be uncertain of what next and the song just mean it said to me immediately and I just felt wa I think this is what even the whole world should know and be glad that you came here to drop this song here and we're just waiting for another another one and another one and just to adapt what you're saying you know the goodness of God just as you said in the song it's not measured nor calculated imagine God will do in the morning anaku wake up vizuri you wake up healthy by his masses in the long term he will provide for you lunch you eat sometimes we think it's by our strength the Lord has done something great for you to receive that food that you're eating in the evening you will do a miracle that we are healthy we will go to bed some of us will not even pray but still it will just cover us with his love so if you think you have nothing else to see the goodness of God anaku concentrate naio just at least look at my life that I am healthy and alive today for some of us I'm telling you the truth where we are supposed to be the symmetry or something six feet under for real but by the grace of God and for his goodness not that we are so rich than the others not that we are prettier than the others but his love for us was so special and particular is like no I still need you here please let's just focus on the goodness of God today you blessed a sister who is watching us this day and she is wondering how can I reach Pamirams your socials wow Pamirams even before you reach me because the most important thing here I'm telling you is to know the person who is doing good things to you the Bible says test and see that the Lord is good it's possible that you may be there na uja okoka ume test drug ume test depression ume test fears and tears na bitterness this is another chance that God is giving you to test him by his word you know I remember him speaking to the Samaritan woman imagine you na kunyua uta patabado kiyukesho but the water that I'm giving you is eternal and the water that the Lord is giving you today is this word I want you to believe in the word of God tukama okoka palia uja okoka I want you to test my God I've seen the goodness of God I always say I am the living testimony so I want you to test the goodness of the Lord ume poke aleo as your personal savior and it's very simple una wabi atu God I need you today we cannot let this anointing go just like that kindly just lead us with the word oh my God yes, lead us yes, I just want you wherever you are just receive Christ that's the best gift ever that has been given to the humanity so wherever you are just say Lord Jesus I come to you just as I am I acknowledge that I am a sinner so today I bring my life to you as I render myself to you I want you to lead me I want you to be my personal savior I receive you as my Lord and savior from today kindly remove my name from the book of death and write my name in the book of life I have made my decision today to be yours forever in Jesus name I can promise you if you test this God your life will never ever remain the same again I tango that you came back stronger to ambient social media handle Zango social media handle Zango on YouTube as I said Pammy Rams kindly show me some love from JC Sacco please show me some love and subscribe to my YouTube channel that's the greatest support that you can give to me on Facebook you find me as Pammy Rams music and on Instagram at Pammy Rams Twitter Pammy Rams I'm at the load bless you if you need maybe somebody to pray with or to pray for you kindly feel free to text me I always always respond to the messages okay watch a people slide pale on their DM on your DMs as they subscribe to your YouTube channel maybe there's a few people that we would like to appreciate and send your shoutouts this day wow yes I would like to appreciate my manager Pastodania Mangubu for always standing with me and always be that teacher of the word of God in my life I want to appreciate the person who brought me here today Alexis Kabambi is just behind the camera thank you for coming with me and of course I want to appreciate my video director Mike Milo and my family my children and all my fans out there for loving me without you know knowing who I am but you just loved me may God bless you and remember you may God bless you too amen thank you thank you for having me Karibu tell now you're so humble and so kind may God continue just pouring more and more so that to release your songs worship and praise amen amen may you never run dry amen I receive that amen dropping it for the very first time no it's not the first time first time here here with me with Pamirams present yes live in studio live in studio yes do the orders yes Moema the brand new song by Pamirams and you got to love this jam and it's going to be a blessing to you so come with us