 Good afternoon. So, in the morning we tried to talk a little bit about some essential vector calculus that is required for doing a course on electromagnetism. But there were some questions raised in the last session and these were basically on what is the interpretation of 0 divergence. See the basically the idea of a divergence let me repeat it again. The as I told you that I am dealing with point functions a vector field which is defined at every point. Now if you if you now considered a certain volume now let us suppose that see I take the example of fluid because it is much easier to understand that supposing you have a closed volume certain amount of fluid is going in certain amount of fluid is coming out. Now you see the normally you would think that whatever is going in that amount will only be coming out that is what happens in a normal water reservoirs. So technically if the amount of flux that is going in that is inward flux is equal to the outward flux then of course this is the situation of 0 divergence that the amount going in is amount coming out. Now when you look at a picture what you will find is that the people decide that the length of an arrow is a representative of how much is the strength of the in our case the electric field. So if there is a bigger one bigger arrows it means that these are stronger and smaller ones means smaller. So it is possible let us say less amount is going in more is coming out. Now how is that possible? Now this is possible if there is a source here the it is something like this bad example but supposing you have a closed volume in and I am talking about water going in and water coming out. Then let us suppose inside that closed volume I also have a tap now in which case the amount of water going in could be less than the amount of water coming out because there is a source of the water inside that volume. Alternatively if there is a sink if there is a sink then what would happen is that of the amount of volume that is coming in the sink could eat away certain part. So therefore whatever is going in not all of it comes out a less amount comes out and so this situation where less is coming in more is coming out is a situation of divergence being positive. If the reverse is true and if I have a sink then the divergence is negative. The situation where the amount going in is equal to amount comes out is the one which corresponds to divergence equal to 0. Now you notice that our equation of continuity essentially takes care of that. So basically what we said is that divergence is equal to the rate of change of the density because after cancelling out the common volume factors and all that. So coming back to the what we are doing just before we ended so we have defined divergence. So I said that since divergence is a point function you have to consider small or infinitesimally small volume. Now you have the volume each volume is bounded by its own surfaces here I have given examples of cubes which of course have 6 surfaces but it could be any type of surface. The vector field is a point function. So because of that I am choosing these infinitesimal volumes. Now if I choose an infinitesimal volume and take its surfaces then do a surface integral then divide it by that infinitesimal volume. Now in the limit of that volume becoming 0 this quantity that is the surface integral divided by the volume of the element that is what I define as the divergence of F. Now notice it is a point function because they remember that divergence is very similar to a differentiation. So therefore this delta v you have to take it infinitesimally small so that the whole thing comes to a point. So that is what I defined as the divergence. So as I told you that also in relation to the previous question that basically divergence tells me how much does a vector spread out. Now this I had shown earlier I am doing it for completeness. So for example in this case you notice that the fields are all diverging out. On the other hand here if you look at the directions you notice that they are all converging in. So this is a situation of positive divergence this is a situation of negative divergence. A little while back I also explained to you what is meant by the 0 divergence and this theorem which I proved I will not repeat it. But basically what we pointed out is that this gives me the equation to continue. Now this was the example that I had to tell you this is related to the questions. So for example this is a field which is x square y times i plus x y square times j you calculate the divergence 4xy. Now both x and y are positive which happens in this quadrant or on this quadrant. You notice that the divergence is positive. This is both x and y are negative and here both x and y are positive. So here the divergence is positive and you can see it here that this is essentially fields spreading out. Now here is the example of a negative divergence. I am repeating that some people had asked me a question is there any meaning to this different colors that are there. No this is just a color scheme which is chosen by Mathematica there is no particular reason. The only thing that is there is that the length of the arrow is proportional to the strength of the field. So that is there and the direction showed by the arrow in some sense tells you what is the direction in which the field points. So these are pictures of the field and I am sort of telling you that the divergence in this example is positive in this example is negative. Whereas if you look at a situation like this xi minus yj its divergence can be easily shown to be equal to 0 and you notice there that if you take a volume here you notice that certain amount of arrows are going into that volume and certain amount of arrows are coming out of the volume. But the amount of things which are going in is equal to the amounts which are coming out as a result the it is a zero divergence situation. In this case you notice the arrows which are coming out are much bigger the arrows which are going in are much smaller. So as a result the net flux here is outward flux here is positive and I have a positive divergence. And here you notice that the arrows which are going in are much bigger than the arrows which are coming out. So this is an example of a negative divergence. I would advise you because this Mathematica or Maple these are you are all in engineering institutions this should be easy to be there just you know let your students plot this they will get a much better idea about what is happening there. There are two theorems connected with vectors which essentially becomes your you know I would not like to use the word but more or less biblical support. The one is known as the divergence theorem. Now divergence theorem basically comes out from the definition of divergence you notice that I define divergence as the limit of the surface integral divided by delta v that is the volume in the limit of the volume becoming 0. Now look at what it actually means that if this is your definition of the divergence then if you have an arbitrary surface it does not have to be infinitesimal anymore. So I have the an arbitrary surface I will try to work it out on the pad. So what we said is that the divergence of a vector field f is given by limit of delta v going to 0 on the numerator you have got f dot and ds which is the surface integral over that small volume the bounding surface divided by delta this is your definition of divergence a point at a function. Now suppose I am to calculate the surface integral of the field I am to calculate surface integral of the field over a any finite surface. Now what I do is this I split this into a large number of surfaces. So therefore I say that this is equivalent to that sum over all such surfaces delta s if you like and I have f dot n delta s. Now so this is nothing this is simply using the concept that integral is nothing but a summation. So what I have done is to split that total integral into a sum over smaller ones and now what I do is this I multiply and divided by the delta v here. So I multiply a delta v and put this f dot n ds integral here with a delta v there and then of course I need to take the limit of delta v going to 0 for this part. So notice that what I am trying to do is this I have essentially said that now this summation is over all such small integrals. So as a result in the limit of the small bits the infinitesimally small volumes this is nothing but an integration this summation becomes an integration and here you have the definition of a divergence. So therefore in that limit what I have got is that this quantity is your integrant which is your divergence. So divergence of the field f and let us say the this summation is written as integral d tau, tau I am using for the volume because I have been using here. So this tells me now notice I need something one more explanation that I will put a contour sign on that saying that this surface integral is actually over the bounding surface over the bounding surface of the volume over which the right hand side is taken. So in other words this is a connection between the surface integral of a vector field normal surface integral the normal component of the field ok surface integral of a vector field and it connects to the volume integral of a divergence. So let me repeat it again. So this is what I have I have your this is the surface integral f dot ds. Now this is actually ds essentially let me rewrite this. This is f dot n which is the outward normal ds this is what is meaning by ds vector and this is nothing but equal to the volume integral of the divergence of. So this is this is what it does the it connects. So this is the famous divergence theorem and this is very important there are two theorems which we needed for the electromagnetic theory course and this is very important that we do that properly alright. Now let me let me give you a an example I am asking I have give you a simple enough example suppose I have a constant field let me take that constant field to be v ok and so this is equal to some magnitude v and let me take it in the z direction. So vk now I want the surface integral of this field surface integral of this field over a hemisphere. Now notice this is a uniform field there is a uniform field in the z direction and let me make an attempt to draw a. So this is a hemispherical bowl of radius r and basically what I have is electric field or whatever field you have which are uniform uniform meaning is a constant field both in magnitude and in direction and I take the field to be in the z direction. Now what I am asking in this is how to calculate how to calculate the flux of the electric field or flux of this field v through the curved surface. So calculate flux through curved surface. Now this is an interesting problem and I will later on show you that this problem can be done in few seconds but let me do the more difficult way of doing it. Now notice that since I am looking only at the curved surface the normal to that surface has to be along the this direction that is a radial direction this is your normal direction. So let me represent this the angle between the field and the field direction is along the z direction and so this angle is my theta well that is the I could have drawn it here also. So this is my polar angle theta. Now in doing problems connected with geometry which is obviously spherical it is more comfortable to use a spherical polar coordinates. Now that is what I am going to actually do and basically what I do is the following that let me once again repeat the curve. So you have the field I am not drawing all the lines that I did earlier this is my vector field phi and this is the direction of the normal angle and this angle is theta. So I need to what is the area element. Now the area element on a surface ds. So if the radius of the hemisphere is r so it is r square remember r is constant because it is a hemisphere that is on the surface r square sin theta d theta and d phi. So this is this is my surface area on a in the spherical coordinate. So what I now do is I want to calculate how much is f dot n ds. Now notice that direction of n is nothing but the radial direction direction of n is the radial direction. So and if I substitute for f which is equal to v times the unit vectors in the z direction which is k. So I get the magnitude of v since the direction of n is the radial direction and the direction of the normal is the k direction. So f dot n simply gives me a cos theta factor. So therefore I have magnitude of v r square which is constant so it does not come into the integration integral 0 to 2 pi I do d phi that is the azimuthal angle because the integrand does not depend upon phi. So therefore I simply write it out and 0 to pi that is my polar angle there is a sin theta which came from here there is a cos theta which comes from the f dot n and now I have a d theta there. Now this is of course fairly trivial to do it because you have your sorry I made one small mistake I do not have a full sphere but I have a hemisphere. So this limit of this integral is not from 0 to pi but it is from 0 to 2 pi because see the notice is this that the theta goes from 0 to pi but because I am also doing an azimuth which is all over 2 pi. So for a hemisphere the theta goes from 0 to pi by 2 and since the integral differentiation of sin theta is cos theta. So this is a very trivial integration to do and if you do that you get v r square 2 pi came from there this you can immediately see will give me a factor of half because I can substitute sin theta equal to x or y or whatever and so I have got y dy and this from 0 to 1 so I get 1 by 2 there. So this is equal to v r square pi. Now this is a bit of a not very difficult but bit of a calculation. Now notice I will later on you all know already that this is much easier if you did not do it this way but you calculated how much is the flux going into this surface because after all the fields are constants so therefore there are fields going into the bottom cap also. Now since it is going in the amount of flux that is going in and the direction of the normal on the on the bottom cap is just opposite to the direction of the field. So therefore I simply need pi r square which is the area of the cap okay and of course v and because their directions are opposite I have a minus sign but you notice this that since it is a closed surface whatever number of lines are going in they must be coming out. So as a result the net flux will be equal to 0 and if the net flux is equal to 0 whatever is the flux through the base surface going in that amount of flux must be coming out through the curved surface and you notice going in there is a minus sign. So pi r square v okay there are questions I will take it up and coming out is v a pi r square v so therefore I can conclude from here since the net must be 0 it is equal to v pi r square. I have two queries one is from Coimbatore Institute of Engineering. I am Professor Smiles from the Coimbatore Institute of Engineering and Technology. One observation is you do not take the surface integral over any field vector it is a surface integral is normally taken over a flux density vector for example you do not take the surface integral over an electric field but you take the surface integral over a displacement density vector that gives a physical quantity otherwise you do not get it. Let me clarify that question see the thing is you realize I told you that I am giving you a short course on vector calculus okay so my fields here could be in principle anything okay I do not have to have a physical realization for my field I am saying that if you give me a point function which is also directed function I can define the a surface integral over that as long as it is a point function which is at the magnitude and a direction the point is this that what you are talking about is that I need a flux density type of thing I am not remember I have still not gone to electricity magnetism here I am still doing a basic vector calculus and in principle I can define a surface integral of any vector function as long as I have a surface defined and the I you give me a mathematical vector field what is the reason why you are saying I cannot define it mathematically it is okay but physically it is not correct well we will come to electric field later yeah the flux of the electric field will be defined after all right see the point is both D dot s will be defined as well as B dot s will be defined so it is not true that I can only define for certain types of field but yes there might be restriction in terms of physics we will come back to it at the moment I need mathematical fields okay thank you next silly guru I have two questions yes give some practical examples situation where as well as negative yeah and the second question yeah if you consider fluid motion yeah you wanted to know that give me a practical example of a situation where divergence is 0 of course you should all know it that the divergence of the magnetic field is 0 you see the thing is remember what I told you that the if I don't have a source or a sink in any closed region the the the divergence has to be equal to 0 and so far as we know there are no magnetic monopoles and as a result if you look at a magnetic field vector so divergence of the magnetic field vector is 0 that is in fact your first of the Maxwell's equation so here is an example is very much physical example of situation where the divergence is 0 what is your second question for incompressible fluid motion having velocity V divergence of the positive negative power 0 for incompressible fluid motion the incompressible fluid motion is fine but you see the I am trying to put in something else there I am saying that suppose there is a source in the volume see the thing is that if it is an incompressible fluid and there is no source or a sink in the region then of course whatever is going in what is that is must be coming out but supposing there is actually a source. I will at this moment I am not in a position to immediately cook up you see mathematical examples I can give you many but the divergence of magnetic field being equal to 0 immediately comes to my mind because it is a electromagnetism course but if you want to know any other physical example tomorrow the first thing in the morning I will take it up okay but immediately it does not come to my mind but I am sure there are many more physical examples. Thank you. Sure. Is there any relation between the divergence and electromagnetic wave which passes through the optical fiber? This question was from again Gangneh Institute of Technology yeah see the point actually is this that inside a fiber the I mean I will be talking about the fiber optics communication basically it is a guided mode and the divergence that I am talking about now is a mathematical quantity and it has some relationship with the physical way in which we understand divergence like for example you may say that our paths are diverging and things like that but at this moment I do not think there is an immediate connection between a fiber optic communication and a divergence because in fiber optic mode is a guided mode and so therefore there is no question of a divergence there but when I take up the optics and we do the fiber communication we will look at the question again. So aside this year I would like to put another question here aside this also I can add one thing yeah the electromagnetic wave completely controlled by the phenomenon called as the polarization is there any relation I mean correlation exist in between polarization and that the divergence. The polarization essentially tells you the way an electric vector is oriented as it is propagating or somewhere. So therefore I do not see the what is the connection of divergence with polarization now the answer is no. Thank you. Thank you. Thank you. Dronacharya college we are talking about this divergence say in somebody was asking about an example there exist a source another way we want the fluid is expanding or the fluid is contracting for example we take the for example the temperature we we study the flow a liquid underground water for example yes and there is a temperature source inside the underground. Yeah. Definitely there will be expansion of water and due to expansion the outflow will be greater than the inflow yeah that is why we can talk about the divergence that is the positive divergence because outflow will increase and inflow incompressible inflow if the temperature down. But you are absolutely right but this is not an example of a incompressible fluid then. In case of the incompressible fluid yeah 3j only example is that there should be source of the sink. Absolutely. You are you are right you are correct yes. The second case one was talking about the flux density yeah now we are defining the divergence theorem. Yeah. We are defining the divergence theorem. Yeah. And in the divergence theorem we are going to connect the surface integral to volume integral. Yes correct. We are going to connect the surface integral to volume integral. Right. So what is the surface integral of a vector. Yeah. And the electric field or magnetic field yeah we are connecting the surface integral to volume integral. Absolutely. As far as the flux density is concerned that is the definition of divergence in definition of divergence we take the flux density. Yeah. Or you need to volume. But we are here defining the gas divergence theorem and that leads the surface integral to volume integral. Absolutely. This is my submission. Thank you. So coming back to what we are talking about we discussed about the divergence theorem as I said this is a very important theorem which connects the divergence of a vector field the volume integral of a divergence of a vector field to the surface integral of the same field over the bounding surface. So that is that and the next because I need to complete the discussion you recall I defined the divergence of a field. Now one point I want to make sure that the divergence of a vector field is a scalar because you remember that what I had was integral f dot ds on the top divided by the volume. Now I define a curl the name curl came from the word circulation of a vector field all these nomenclatures comes from fluid mechanics. Now I define what is known as a curl of a vector by a very similar definition and there in case of a divergence theorem if you recall it was a relationship between a volume integral of a divergence with the surface integral of the field itself. Now I am going to do define a quantity which is known as the curl and there so let me define this curl as the ratio of a line integral to a surface. Remember divergence was ratio of a surface integral to a volume and here I come down one grade I say the ratio of the line integral of a vector field divided by deltas what do I mean by this. So basically what I am trying to say is this that supposing I have a surface this is just a some surface I am showing you. Now this surface that you have let me sort of do it here supposing there is something like a matka now what I am going to do is this that this is the rim of that matka now this surface I split into very large number of small surfaces I can do that now what I am trying to say is this that consider a small element of that surface. Now if one thing I want to tell you is this if you compare the proof of the divergence theorem with the proof that I am giving you now for the what is known as the Stokes theorem you will find there is a one to one similarity between our approach and the similarity comes because of this that there I had a volume which I split into large number of small volumes. Now I have a surface which I am splitting into large number of small surfaces now for example take a small surface like this now and which I am amplifying here. So this is a small surface element and I am saying now that make that surface small enough and just go around its contour like if you go around its contour remember that this is an infinitesimal surface and so I can define a surface integral so let me define this surface integral f dot dl I close it because it is a closed contour over the small contour and divide it by the area which is enclosed by this contour. Now this is my definition of the curl of the field so this is the curl so curl is when limit delta s going to 0 of the line integral of f over the bounding curve divided by delta s almost similar to the way we defined the divergence as the limit of delta v going to 0 of the surface integral divided by the volume. Now this picture on the right is a good way of understanding this see what I have done here is to have a bigger contour and I said that all right let us now split supposing this surface I have split into many such small elements and notice one thing that if I look at two adjoining surfaces the and I decide that my line integrals will be taken in a particular fashion let us say counter clockwise when I have looked at the top now in that case that so far at this area is concerned the line integral on this interface between this and that is travelled in this fashion on the other hand when this area is concerned when I take the line integral in the interface it is travelling in the reverse fashion. So, once I have it going like this another time I am having going like that so therefore the contribution to the line integral from the boundary will cancel away from two adjacent things and this would be true of all adjacent things excepting when I come to the edge now when I come to the edge there is nothing to cancel this parts so therefore what I have is that the total line integral is nothing but the sum of the line integrals due to each one of them separately now that is because a very large number of zeros have been added there. Now then it follows that if I split if I split my total line integral into large number of such small integrals then you can see immediately that integral line integral of f dot dl over this big curve then becomes the surface integral of the curve because of the definition of the surface integral itself because you take delta s to that side and do the summation like the way we did it. So, in this case what I have is that line integral of f dot dl the line integral of the field is equal to integral of the surface integral of the curve. So, basically what I tried to say is this that the I have two theorems which are important to look at and that is the one theorem is the divergence theorem the other one is the Stokes theorem and basically they connect one of them connects line integral to a surface integral the other one connects surface integral to a volume integral and I am not going to derive these things because this you are all familiar with that one can in the Cartesian coordinates calculate the curl of the field and this is given by this determinant and so, just to give you an idea of what it looked like on the right hand side I have got you can see why the name curl came in you look at the direction of the fields the essentially if the divergence told you the way a field was spreading out or spreading in at a particular point the curl of a vector tells you how is in a nearby point how is the direction of that vector changing is it curling around that is where the name came from really and in fact, I told you that most of these came from the discussion of water the fluid motion and supposing you have a paddle wheel inside a fluid and you can see exactly this is the way the what will make this paddle while turn and so, that is essentially the way the curl came into the picture. So, this essentially summarizes the thing so, we said now curl of a vector field is a vector unlike the divergence of a vector field which is a scalar quantity. Now, suppose this is a very important point suppose I express a vector field as a curl of another vector field that is I have a vector field f and supposing I can find a vector field a so, that the f is expressible as the curl of a in which case I will have the field to be divergence less and that is very simple because one can trivially prove that the divergence of a vector field which is expressible as a curl is identically equal to 0 this you can trivially prove by writing down the expression. Now, if a field is divergence less it is also called a solenoidal field it is also called a solenoidal field. Now, the question is this that there are fields whose curl is 0 now one can show that curl of a gradient is 0. So, notice divergence of a curl is 0, but curl of a gradient is 0. So, if you have a vector field whose curl is 0 that it is not curling as I showed it to you such a field is called irrotational you can see the name where did the irrotational come from. I will close the discussion of the vector field with a statement of a theorem which I will not prove which says that if you have a vector field you can express it always as a sum of two different vector fields one part whose curl is 0 another part whose which is divergence less one is solenoidal the another is irrotational. So, and this is the reason why a vector field can always be expressed like this. With this I more or less conclude our discussion on the vector calculus as I had said there are few questions coming up which I will take it up before going over to the electrostatics. KLE Institute of Technology, Hubli. Sir, I have no doubt regarding divergence of a vector field. So, you have explained if it is a inflow is equal to outflow then the divergence is 0. Yes. And if the inflow is greater than outflow the divergence is negative right. If the conditions are true for only for finite volume. No, no, no let me explain remember I told you that divergence is a point function what is meant by a point function because divergence is defined from point to point. So, at every point now what do I mean by at every point and a volume what it means is think of an infinitesimal volume located at a particular point. And then look at what is happening there because because the value of the divergence can vary from point to point. See it is for illustration purpose that one gives you that this is what it means there that supposing I talk about incompressible fluid. Now if I do not have any source or a sink then the amount of fluid that is going into a volume no matter how small that has to be equal to the amount of volume coming out because otherwise there will be an accumulation or you will have to create water or create this thing. So therefore, it is not a finite or a it is an infinitesimal volume with respect to which my definition of divergence is there. Okay sir. Thank you sir. Thank you. Next. SGS Institute of Practical Lodging. Okay sir. My question is from divergence if we put a positive point to charge inside a Gaussian surface or a closed surface. Yes. Then we will say that the divergence is positive. Yes. But what if the positive point to charge is outside of the Gaussian surface or closed surface what we say positive or a zero. You said that if you have a Gaussian surface and you put a charge inside then the flux that is coming out is of course it is positive or negative will depend upon the sign of the charge inside. But the point is that I have defined it mathematically. Now if you of course put a minus sign in front of it will become negative. See if you put a negative charge inside and calculate the flux because of the sign that is coming in with the source itself you will find a minus sign that is alright. But sir the last part of question is that if we put that point to charge outside of the Gaussian surface. Yes. Then the divergence over this surface will be positive or a zero. Zero. Because the intensity of these electric flux. Thank you for the question. I will explain that point. See this is the way we will actually show why in Gauss's law in the electrostatic Gauss's law it says that the flux is equal to charge enclosed divided by epsilon zero. If there is no charge enclosed the flux is actually rigorously equal to zero. Why it happens we will actually rigorously prove it. So wait for it. Thank you very much.