 I start with this question, who is afraid of vector calculates, I start with this question who is afraid of vector calculates, the purpose of starting like this is this, if the audience says nobody is afraid of vector calculates, you can go for early lunch now, but I think we all have some subconscious, some moment of fear in the subconscious mind or somewhere about vector calculates, so perhaps we can go through this right, because nobody is, if everybody stands up and says we are not afraid, the course of the action will be different. This is the plan of my talk, now don't think I have by mistake added some other power point, it is actually technical talk on vector calculates and the learning objective is that at the end of it you will understand all these pictures, you will start liking vector calculates or if you already like it, you will like it more, that is the learning objective of next one hour. Have a look at these pictures and perhaps it is already triggering, it started triggering something in your mind on the topic, Professor Beshmou gave a very short but very effective introduction to the concept of gradient, we will start with that, before that the correlation vector, he told you what a vector is and said I will amplify on that, a vector as all our students know very well is something which has got magnitude and direction, so this is the aspect which was discussed, let us say something which has got a magnitude 10 elephants now small magnitude and a well defined direction, a tritule temple or Tirupati temple as you like, it is a east Nada of that one, so there is a vector, there is a direction and it has a magnitude or if you are not impressed, let me present you directly something with a magnitude and a direction, that is an ideal example of a vector quantity looks like, no we know that we have seen the technical aspect of it earlier, this is not a vector quantity, then what is the vector quantity is something has got a magnitude and direction in a well defined a clear manner does not qualify as a vector quantity, so what you should do is turn the vector around, turn the elephant around and look for its reaction, if you are bold enough to do vector calculations, so this is what a vector is, you define it in terms of certain transformations, if the components transform in this manner as specified with this, we know this and then only it is defined as a vector, so vector, scalars, tensors etcetera are defined in terms of the transformation properties, so what is a scalar then simple, scalar is which one which does not transform at all, they are all actually the younger brothers of tensors, the family called tensors, there is a problem with the tensors, I have been teaching for several years, in every one of the class B, C, H, D or beta called MSC, the moment I say the word tensor, students pick up tensor and tensor, there is no need, it is something to help us, the whole notation is something to help us to put our complicated mathematical expression in a simple manner, that is the purpose of introducing not only tensor, any mathematical concept, including that of vector calculus which we are going to see, is it too mathematical, is this is going to be too mathematical, there is a quote here which says, if you want to read the book of the universe, you must know its language which is mathematics, who said so, the father of experimental physics and the father of experimental physics, must be the grandfather of engineering, engineering came out of this right and this is nothing none other than, so this is the grandfather of engineering, who said, so engineering students, civic students all have to learn this language, so they want to do their job properly, that is what we are going to see, great grandfather of technology, the concept of gradient is also explained to us in limits, we will look at the three concepts of vector calculus, gradient, divergence, and curve, as some limiting process, an expression which describes a limiting process, each one. Let us start with defining a unit vector as shown here, it is the limit of dr by dr, we will see what it is as we proceed. Let us look at the concept, you want to climb down a mountain, there is an earthquake or something in a roadways to climb down, so how will you climb down, at any point where you are, there are different ways of climbing down, different direction through which you can climb down, so proportional to how much you move in the x in the plane, how much for example, there are different ladders, ladders can be kept like this, kept like this, like this, so how many steps you have to take to cover a large height, that is the question as is the evident term, this definition what is going to come. So, let us define some quantity called gradient as shown here, this quantity gradient start by defining it, let us call it a give it a symbol and then define this, so dot product, this dot product it is a radius, what it is, this is we will see, it is one way of looking at it is to call it by name and then start describing it, so this is a dot product and it gives the rate of change, spatial rate of change of the field in a particular direction, u is any direction as the rate of change in that direction is decided, now we have to ask just one question, when is it a maximum and then we get the concept of field, in other words I will explain all this, gradient is the greatest slope, if you want to descend of course from the mountain you want to descend, then you are talking about the negative gradient, you want to descend in the most efficient manner, that is what is meant by this dr by ds and the limit of that, so if you want to know the gradient, if you want to know d psi by ds how much change per step you get, then you have to take this product, this dot product, so that gives you this, where u is defined like this, this is what persuasion was mentioned earlier, it is important to realize that gradient of a scalar field is a vector field, it is related to the directional derivative and gradient is the maximum directional derivative, where is the concept maximum coming from, it is because it is a dot product, whether it is maximum and when it is minimum and all that, so the quantity, the scalar quantity psi which can be anything any scalar quantity, this quantity increases most rapidly along the direction of the gradient and it decreases most rapidly in the direction of the negative gradient, why is it important, often we have to make a judgment in which direction we have to run to escape a particular field, like temperature, temperature is a scalar quantity, so the, I have seen the movie where the villain will be, the hero will be captured in a place called absolute zero temperature liquid helium, that is not possible to have such a big chamber, but then he is supposed to, he want to run away and go to the normal temperature, so which direction he should run, he has to make a decision to overcome the temperature gradient and come to normal temperature, room temperature, so his neck top has to do this calculation and find out the direction of the gradient, it is the direction of the gradient which gives the direction in which the change is maximum, so which direction you will run depends upon whether you want to escape the change or not, for example force is a negative gradient, why is it the negative gradient of potential, force is defined as a negative gradient of a potential in the case of conservative force states, that the negative helps us to get the work energy theorem right, let us look at this maximum directional derivative in a little more detail, the definition is here, this quantity, this d psi by d s will be maximum when this quantity, the new quantity which you have defined is parallel to the direction which you are moving, so in other words, great sign has a direction and magnitude of the greatest rate of change of the scalar function, in the case of the height, it is very clear in which direction you have to go, so that you get a maximum asset, which way you should keep the capital added, so that you climb larger number of steps per equivalent number of steps along the x direction, along the plane, the other aspect of it is, as I said, it decreases most strongly in that, why I am talking about decrease, because often we want to escape the gradient, escape from the gradient, as I said the negative sign has to do with other choice of discussing what is lower and what is higher, the other aspect of this is the direction of the gradient is also perpendicular to an eq potential surface, so it is very easy to find an eq potential surface, look for the direction of the gradient and see, because on that d psi is 0, there is no change, eq potential means there is no change in the potential, flat surface geometry, if we can prove it easily, let us assume the vector which is lying on this eq potential surface and then calculate d psi by d s, which is equal to the gradient of psi, the dot product of psi and gradient of psi and v, so this has for a non-zero, if gradient is non-zero then it has to be perpendicular to v, that is the only way we can have this, it is 0, so that proves that the direction of the gradient is perpendicular to the eq potential surface, this has a physical significance which will be shortly. To repeat, the directional derivative d psi by d s has a maximum value when taken in a direction parallel to the gradient and it is zero when it is taken in a perpendicular direction, it is a directional derivative, derivative taken with the concept of a direction associated with it, this is a vector point function, gradient is a vector point function. Let us look at some simple examples to get the idea clearly in our mind, the function, the scalar function is given as t equal to 20 plus x plus y, so are the constants of the proper dimension, do not worry about the reasoning of it, let us just look at it mathematically first, this is a plot of a temperature, that is temperature is a function of x and y position, so this is what I have plotted here, x and y here, the brighter regions have larger values of temperature, the darker regions, colder regions are shown with bluer and the brighter ones are sorry, the brighter ones have larger temperature and the larger ones lower temperature, so find out the gradient of this, this is very simple and let us calculate the gradient of this, if you do that by the usual vector, method of vector calculus, you get this, what you get is field is this, so let us look at the function, this is the gradient is e x plus e y, 1 along e x and 1 along e y and that can be plotted like this, so that shows the direction, this shows the arrow shows the direction from pole to part, so this is what I have, I was trying to explain by considering and that is a vector field, so the gradient can be depicted by this picture here, this is one example, let us go to another example, little more complicated, 20 plus x plus y and here again this is sorry, this is not 20 plus x plus y, this is not correct, this is 100 plus x y, 100 plus x y that is plotted here and if you look at the gradient, the gradient is going to be in this direction, so similarly you can, now let us go to a cylindrical polar coordinates, this is a simple equation again, T is 15 plus rho cos theta, this rho and phi have the normal meaning in the case of plane polar coordinates, again the same thing, the concept is the same, we plot T and then we plot the gradient of it, here you can work out the gradient, if you will talk about how to get the expression for gradient etcetera later and you see that it is e x, what does it mean physically, that is why I showed this, so it is easier to see in the rectangular coordinate system, the answer of this, otherwise this is the answer, now you see that it is equal to e x, which means this the gradient is along that side, that is what it means, that the magnitude you need. So, this way you can calculate for a given function, you can visualize how the function behaves and how the gradient behaves and you can again this is clearly shown here by the shady that a direction of gradient is like this, let us understand gradient, we all know from school days onwards that water flows down hill, water flows down hill is actually a statement of gradient, we are talking about flows, so when we go to the college school we know that water flows down hill, in the college we learned whatever flows, everything flows down hill, whatever I mean by whatever I mean all that, I am talking of transport properties, transport properties means somebody is running away with something, what is the running away with liquid flow, the current flow, charge running away with charge running away with heat running away with momentum, also flow, running away with particles, all these are called a different transport phenomena, in this case you have liquid flow, current flow, heat flow, particle flow, all these have mathematically similar expressions governing this and in all these cases there will be a gradient, for example where does current flow from a higher potential to the lower potential and where does heat flow again from higher potential to the lower potential and the particles where do particle flow from higher concentration to the lower concentration, where does money flow from the pore to the bridge, so that means it is all branches of engineering and finance, gradient is can be used to describe all branches, each one is a branch of an engineering, any branch of an engineering discipline, so that is the importance of the concept of gradient and once you look at some of the, you may have looked at some of these equations you will realize what I am talking about, the similarity in expression, mathematical expression and the concept of gradient, some examples from the familiar electric field and potential, we have, we know this electrostatic field which is a nice conservative field, electrostatic field inside a conductor is 0, these are the things which we teach our students, what is the connection with gradient, if it is not the situation will not be electrostatic anymore, induced charges, induced fields etc., into cancel the original field that is the way it is arranged, any net charge that resides only on the surface not inside, what will happen if these are the common things which we teach students, the conductor is in equipotension, so all this is related to, this can be explained quantitatively using what we learned just now, it comes to the gradient, the field line just outside the conductor must be perpendicular, this is another important aspect of the electric, this is just to give a flavor of where it can be used, this concept we should learn right now can be used, you can understand it by this converting a field into potential or finding out the field corresponding to potential or vice versa, this problem for example gives an electrostatic field and we are finding out the potential from which it has come, it can be done, both of these are standard problems if it level problems which you are all familiar with, I am just bringing in how it is related to the gradient, then obtaining the potential from a given field, field from potential, potential from field, I guess that you will be familiar with this, only connect it with the topic, in the case of for example here in this case, we must know what it is, what is meant by the constant that is again a standard, constant here means it is independent of the variable involved, so this is to obtain the standard problems to get field from the potential or vice versa, then you can plot all this nicely depending upon which expression you take, you can plot it and show the gradient also. Let us go to divergence, divergence before talking about divergence, let us, divergence we will understand conceptually, we will also fight the expressions and we will also talk a little bit about the physical meaning of this, the divergence theorem which I am going to describe and not describe to get is a mathematical expression of the conservation principle, what it conserves, it is best illustrated by recognizing that in the absence of any souls or sink the density of matter in a well-defined volume can change if and only if a matter flows in or matter flows out, we will see that, we will elaborate on this. There are some very nice things which last mentioned, you can read that, the enchanting charms of the sublime science appeal only to those who have the courage to go deeply into it, let us say use the word fear in my title, I have had my results for a very long time but I do not know how to arrive at this, Mahatma Gandhi said not only the final goal but also the method to arrive at that also should be pure, that is why he was waiting for finding out a mathematically, the equivalent here is a logical way of describing it. When you talk about divergence, the idea that comes to mind immediately is that of flux, what are the difference between flux and divergence, why do we need two different concepts to describe if it is the same thing, flux crossing a surface is given by the surface integral, the surface say A F dot dA or I have used both F dot dA or F dot dS alternative, to define a flux, you need a vector and a surface, a vector field and a surface, so if students have a problem in remembering how to define a flux, this picture we can in the equivalent of that will be the bull fight not only in Spain, we have in Tamil Nadu also, so that shows that you must have a field and you must have a surface, it can be a closed surface, both the bull and you can be enclosed in the nice enclosure, there is extreme divergence, so you need to define flux in the vector and surface, that is what this integral tells, you could the surface can be of different kinds, then you have to define this, this is important, I have shown it as an area, you know the concept of the directed area, the direction of an area, it is an outward drawn normal, for a flat surface of course there is an ambiguity, there is no outward drawn normal, you can take element by element and find out the total flux which is an additive property, so these pictures show how the show a field and an area, a field and an area and how the flux can be defined is in calculus, so this ok I will be we will now define a force is a vector point function, but the problem the issue is flux is not a vector point function, first of all it is not a vector, it is not defined at a point, flux is defined only over a surface, not over a point at a point, this is a standard textbook derivation of what I am going to show the meaning of geometrical meaning of or a physical meaning of also the physical meaning of flux, this is standard textbook thing, so I am not going to describe it, the idea involved here is to find out how much flow is there across each surface, consider two points close by points and look at the flow through each one, so you can do it in a step by step manner, take each pair of faces and find out how much is going, then add all them all of them then you get an expression like this, the flux this is standard textbook thing, you can see it in textbook, the flux is given by this quantity, the bracketed quantity of integral of the bracketed quantity, so we have a volume integral here, now if you define this quantity as the divergence of x and write it in this fashion then you can define the flux in this manner, so we have the divergence theorem which connects the volume integral into a surface integral, it relates the surface integral and the volume integral, the surface and volume are related, it is the surface which goes around this particular volume which you are talking about, that is the connection between the v and s or v and a as I have used that encloses the particular surface, now let us look at the meaning of it, so what we have done so far is we have looked at the flow of the flux considering two surfaces, then we considered all surfaces and got an expression, you can believe me you can get an expression like this and then we called it in a, this we called as a divergence of x and then we have this equation, so this is a volume integral of this quantity and this is surface integral and this is where our divergence is, now let us try to define it, let us look at the physical meaning of what is exactly diverging, what is meant by divergence and what is the relation between flux and divergence, this is nicely shown by this limiting case, as I said we will we have already looked at gradient as a limiting case, now we look at divergence as a limiting case, the flux cannot be defined at a point, flux is defined for an area, suppose we were to shrink that area, it is like the famous example given by Richard Feynman, when he discusses the velocity as a quantity, definition of velocity by calculus, the famous example in which the processor is driving a car at 70 kilometers per hour and the police stops sequences, you are going at 70 kilometers per hour, then the processor says neither I am going for 70 kilometers, I am going only for 2 kilometers and I am not going for one hour, there is a lecture after 5 minutes, so what is your problem, when he says no no do not argue with me, if you go like this you will cover 70 kilometers, cannot you see if I go like this I will hit against that wall, if I go like this I will hit against that then we cover on the road and who told you I am going to go like this in future, no no if you go like this you will cover, he is repeating that again, oh I will cover 70 kilometers after one hour, come and arrest me after one hour, you cannot arrest anybody for something he is going to do after one hour, so you can when you drive you can keep it there, remember this, the fellow will get sufficiently confused and then you will escape the consequences, this man gets very annoyed, just do not cheat me, at this instant you are going at 70 kilometers per hour, at this instant I am here, at this instant we are all sitting comfortably here, we are not going anyway, so what is your problem, the answer cannot be given by an ordinary policeman, so what is the answer, the answer can only be given by a real teacher or a master teacher, where we involve the concept of limit, that is where we involve the concept of limit, consider a very small displacement and consider the time taken for the displacement and then take the ratio delta x by delta t, tending to 0, delta t tending to 0 and that is the limit of, so that is the concept here, we take this quantity, we take this integral which you have defined just now here and consider a small volume and define the limit, this volume integral, so we consider look at the limit of this and that limit can be defined as the derivative, so it is defined over the volume, the volume shrinks and at that point, so we are asking at that point what is the flux, it is something like that, so this is the concept as a limit and we have to remember that flux is not a scalar field even, it is not a local quantity, whereas divergence gives you the same information in the form of a field, it is a scalar field at that point, local quantity, that is what you get out of this, now we have Gauss's divergence theorem, let us look at it in little more detail, this connects the volume integral, it is as I said, so people do not like volume integrals, they like only surface integrals or vice versa, they can use this, often we use this trick to jump across the dimension, jumping across dimensions constitutionally, the surface integral statement is this, the surface integral of the normal component of the vector f taken over a closed surface is equal to the integral of the divergence of f taken over the volume enclosed by the surface, please note that these are the surface and volume are related to each other, it is not any arbitrary, any surface. So here the surface integral of a vector field is related to the volume integral of divergence of that vector field, what it means is an integration of the sources or sins of the vector field over a volume provides the next, the next outflows going through this, this like some gardening issue to give a very crude analogy, it is like a gardening issue, you have several fountains and then several holes through which water can drain off and then you are looking at finally how much water is leaking to the neighbors back here, front yard and that is what you are, the neighbor complains that there is a leakage of to their front yard, so what will you do? You go around and count how many fountains are there and how many holes are there through which water goes in and finally they do not match with each other, if there is only one fountain and no hole, then all the water goes to the neighbors, so that is like a stop taking in some sense, it is a crude example but it is a stop taking in some sense of what is happening in a garden, in a flow, in the case of water flow that is what I did. The better example will be the electrostatic field, the electrostatic field, you know the electrostatic field and what is the divergence of the electrostatic field also, so you can represent it like that, so what I said just now is this, integration of the faucets, the source or the sink will be equal to the flux of water flowing out from the surface and closing the volume, the surface integral of a normal component of the vector taken over a closed surface is equal to the integral of the divergence of the vector taken over the volume and closed by the surface, this is the same thing, same thing. So gradient was a mountain, when we talked about gradients, we talked about mountains and we talk about divergence, we are talking about fountains, this is the example in a better way, you see the divergence clearly says at you in this picture, this is a charge, positive charge sitting there, it is a fountain and if the negatives are sitting there, goes in the other direction. So this is the concept of Maxwell's equation, you can clearly see what Maxwell's equation, Maxwell's equation in electrostatic says, that is the physical meaning of, one way of understanding the physical meaning of, it is an example actually, this is an example, you can do this, you can actually solve this, this is the Gauss's theorem, the famous Gauss's theorem, Gauss's law which you can actually derive using this mention. It is interesting to see that the flux is independent of the radius, how is it possible? I give it, leave it as a homework or you think. An interesting homework, a special homework will be, what is the divergence of a uniform, constant, constant is the word normally used, but constant means actually in time, but if here the various, the differentiation is done on space, not on time, so uniform is the word, better word. By the, you can use the divergence theorem, let us play with the divergence theorem a little bit, let us calculate the E dot ds here and using the ideas we have developed just now, we can write it like this and then you get an Maxwell's equation as I was telling. So this is the meaning of the picture which I showed here, please try to correlate that equation and this picture, then you understand divergence, this picture, this equation, this is a differential form, this is the first Maxwell's equation which is actually cool. Visualizing this by examples, this is the question where the divergence is 0, this is an example of the visualization of a field where the divergence is 0 and here it is, it has some value, some numerical value too for this particular field, you can calculate it by the standard So here in this case, why is it 0? Conceptually, the influx solenoidal, the case of solenoidal, the divergence of a vector field is the extent to which the flux lines behave like being near a source or a sink, there is another way of looking at it physically. You can see there are sources and sinks in general, in general in your garden there are contents and holes and then the input, so we relate the sources, the sinks and the net flow, inflow, outflow, total flow, that is what we are looking at. If there are no sources and sinks, then whatever goes in must come on, so now we are talking about a principle of conservation, I bring back the definition of this, then if you apply this to electrostatics again, you have to consider J current and then you look at the divergence of J and then you arrive at the equation of continuity. These are simple steps we do in our electrical magnetetics, but it is in DC such as perhaps what you get is this equation. Now, let us analyze this equation, equation of continuity, this equation tells you about a vector J, right now we are we can talk about electrodynamics or even without electrodynamics. Okay, let us take some vector J, this divergence is negative of 0 by dt, that is what it says, this equation tells us, here is the meaning of the divergence, the divergence of this vector is equal to the change in the rate, the rate of accumulation or otherwise of a quantity, some quantity, let us understand it. This can be the equation of continuity which relates to the conservation of mass or conservation of charge. In the electrostatic case which we are recently talking about now, it is charge conservation or if you look at it as a flow of matter, then it is a conservation of mass, then the rho becomes a density of matter, here it is a charge density. Okay, so this equation tells a lot of things, it talks about the conservation of mass, conservation of matter or conservation of charge both in the context, appropriate context. This is the importance of the equation of continuity. Divergence theorem is actually a mathematical expression for the conservation principle, it implies, it implies the conservation principle. In the absence of creation of destruction of matter, no sources are seen, the density within the region of space can change only by having matter flow into or outside. We will put it in say it in different languages, this is what it says, just now we can understand this, we can look at this example. So one way of looking at this will be, I think I have it later, so I will come back to that. So let us summarize what we learned about divergence. We started with flux and then we wrote this integral and we defined divergence as a local quantity, we can look at the limiting, the case of limiting case of flux, this integral in particular, at a particular, by divided by the volume and we found the local quantity. And then we looked at the divergence of the continuity, equation of continuity. The one way to tell, to make our students understand this is like this. Suppose they are staying in the hostel and their father is sending them money. If they go around seeing all movies, buying all things and all that, if you keep diverging your money, then remember there is a negative accumulation of the bank balance of the father. This is their divergence of money and this is the bank balance of the father, it goes negative, d r by d t is a decrease in the bank balance. So that is what it is, it is a conservation of money in financial things. So even in a country, in a country there are some developmental projects, lot of people taking drive, so sources and things are there. So the finance minister has to actually learn divergence theorem and see how to manage all these corruptions and the seems and sources in the finances and some who manage it. So students should find it very difficult to understand the concept of divergence in spite of all that. But divergence is actually their strength. It comes to Curl. We start on our journey on Curl by looking at, looking at going back at gradient once again. So it is for us to fix our ideas in our mind. We define the quick cover recap of what we did for the directional derivative and gradient. This is what we did. We looked at the limiting case of d r by d s and then we said the direction in which it happens, the maximum happens is the direction of the gradient. The maximum change happens along the direction of the gradient. Now look at this. There are two equations of one is s equal to m a and other one is a gradient, negative gradient. This is a negative spatial gradient of a potential in the case of a conservative force. How do they reconcile these two? There are two equations here. The point is that the compatibility of the two expressions emerges if and only if the potential psi is defined in such a way that the work done by the force is given by this quantity, negative gradient. In displacing the object, work done by the force, force is given by this quantity, work done by that force in displacing the object on which the force acts, that is what it is. So this is the force and we are looking at the work done. If the work done has to be independent of the path, if work done has to be independent of the path then only both of them are reconciled. So in other words, we find that this works for a conservative force field. We can write this only for a conservative force. This we know this for electrostatic force or gravitation force, the work done is independent of path and in a closed loop the work done is 0. So this is possible only with the path. So the path dependence of this integral is completely equivalent to an alternative expression which can be used to define a conservative force and that is brought out by the definition of curve. What is this curve? For a vector field f, we can define curve. Curve is a vector point function. It is defined as, it is a vector first of all and it is defined at every point of the vector field such that for an orthonormal basis set of unit vectors, this we can write this expression. Of course, you are more familiar with the standard expression then it is fine. Now this is related to circulation. We talked about the directional derivative and then we came to the gradient as a limiting case. Then we talked about the flux and then we took the direction and we took a limit of that and then came to the divergence. Now we talk about circulation. Just like the flux is defined with a point and with a force and in area, circulation is defined with a closed path and circulation is only over a closed path and not at a point. So how do you describe the circulation at a point? Again in this period of Feynman's example, that is what we have here. So you take the curl A is defined like this. Curl A is defined like this. To understand that, take one of its outars that is its projection in one direction. We do not understand infinity. We do not understand something like divine or God. So we take it but it is better to understand some of its outars. So we take the outar of curl along this direction, along a direction and try to understand that and then we can extract stuff like that. So this particular component of the new thing which is curl, which I have not properly defined yet, that quantity, that dot product is equal to again a limit. What is the limit here? Here we have this line with the circle, circulation and per unit surface that will give you this. So that is a little tougher to understand compared to what we did earlier, flux and divergence. But if you understand that properly, it becomes easier for you to look at this. So here, this is circulation. So look at circulation and then take the limit at a point. The point at which you want to define it, what is the circulation? Circulation is not at a point. Circulation is across a curl, closed curl. But you can define it. If you want to define a point function which conveys the same idea, then you can define it at a point. That is what we are trying to do here. Then you can write it in different coordinate systems. I will come back to that if time permits. This is independent of this. If you define it this way, it is independent of any particular coordinate frame. The average circulation. So one physical meaning of curl will be the average circulation per unit area taken at that point where the elemental area becomes infinitesimally small. Just like define an example, we have now an area and we shrink the area and look at that. So this is the meaning of it. The average circulation per unit area taken at the point where the elemental area becomes reduces to almost a point. This is the spirit of this definition. The limit when the area becomes unitary. Let us try to understand it a little bit more. First of all, this is circulation and circulation is on the field as I said. So shrink this curve. First look at the circulation. The circulation depends upon the vector at all points on C on the curve which you are talking about. We are taking an integral just like the case of flux. Then shrink that curve in the limit the circulation vanishes. So, so does the area. So this is something like 0 by 0. But in the limit, it does not really vanish in the limit. In the limit, we can define the curve. The ratio is finite in the limit and this local quantity at that point is defined. So that is the meaning of I am repeating what I wrote earlier to emphasize the point that it is a limiting circulation per unit area. That is mathematically what the curve is. Curve means how much a vector curves around the point. That is the physical or geometrical significant surface. About the point how much the vector curves around. So you can, so right now you let us associate the curve with a rotation. Quickly I am going to tell you that it is not necessary all the time. So this is the picture you can have in your mind. Circulation and the limit how much the circles around at that point is not circling around at that point. That is the concept of the limit. There is a famous example given by Feynman. How to understand is the physical meaning of curve. He says imagine a flow of, let us imagine the flow of water and arbitrary flow of water. It need not be a very uniform flow. And then you consider a path across the flow. It is a tube across the flow. The tube is an imaginary tube. So water flows through the tube. I mean it is just you are putting a line, drawing a line there and water is flowing through that. Then he says abracadabra and all the water freezes except that in the tube. Is it clear? Water is flowing in your, you consider a loop or actually a tube, imaginary tube, water flows through the tube, whatever you define the tube also. But then you freeze all the water except for what goes into it. And then you look at whether it is a rotating, whether there is a motion inside. If so, when you bring it to a point and at that point there is a non-management curve. That is Feynman's experience of what a curl is. So curl is now related to some rotation, some level of rotation. I will again give one more example for that. So then there will be no rotation if it is irrotation. This is called irrotational if it is zero. Conservation force is irrotational, I think. So we can look at the connection between all this gradient. In this case gradient circulation and curl. Let's look at some points. The curl of the vector field at a point represents the net circulation of the field around that point. Circulation is that integral image. The magnitude of curl represents a maximum circulation at any point. The direction is given by the right hand rule is normal to the surface upon which the circulation is latest. The circulation can depend upon where this. So this direction is the normal to this circle which you consider on which it is greatest. Just like the example of a gradient. You can have different paths and you can have different, if you put your ladder in different manner, you will be climbing so much like that. And if it is zero it is called irrotational for obvious reasons. Leprostatic, gravitational or irrotational is where they call it zero. So what you have to remember is that the criterion that a force field is conservative is that its path integral over a close loop is zero. If this is zero in the region then there will not be any curliness or rotation associated with it. This is a repetition of what I said just now. Examples in real life. A tornado that winds rotates about the eye and the velocity field. Now we are talking about the velocity field. The vector involved is the velocity. Would have a non-zero curl at the eye and possibly at other places as I said. That is an example of the concept of curl. In a vector field that describes the linear velocities of each part of the rotating disc, the curl will have the same value on all parts of the disc. We will come back to this point. The velocities of cars on freeway were described by, if that were described by a vector field and the lanes at a different speed limits then the curl on borders between the lanes would be non-zero. I will explain this also in a simpler way. So let us go ahead. If you get a circulation on the other faces and add up this, just like our, okay, what is this? This is the textbook derivation of the equation which I am going to derive. The textbook derivation is like this. It takes along different directions, something like what we did for divergence and finally you can derive this Stokes theorem. This derivation can be found in any textbook on this. And this, now look at, here we have a line integral and here we have surface integral and again jumping across dimension. This is the Stokes theorem. This is how we write normally curl is remembered by students but remember that it is not a determination. If you do not remember it will be determined to your knowledge in physics and mathematics. It is just an easy way of understanding the whole thing or easy way of working out. This is a more decent way of writing curl. This is also good because we are going to give the secret how to remember these things. This is the field which you have seen earlier and here is a curl here, non-zero curl and in this case also there is a non-zero curl. The direction and magnitude of the curl can be found. Now look at this. So far we were always talking about rotation. The moment I said curl I was having this picture in all the slides for curl. So you should separate our rotation all the time when the curl is involved. No. The best example will be if you somebody is really very eager and very curious to know about this point. Do we need always rotation to have a curl to define a curl? Can we define curl when the field lines are parallel? You have to do a simple experiment. You have to go to the main gate of pi 18 and there are vehicles coming at different speeds. Let us assume that that stretch of the road is parallel and just lie down on the board on the road and see whether your head are rotated with respect to each other. You will soon find that when Carvits hits your leg at some speed and another one hits your head at a different speed, so you rotate. So in this case also you can have a curl. The field this is what is mathematically shown here v is like c y here. So there is a curl existing there. So in flow and river the boat gets tilted. So we can actually mathematically show this. Zero curl does not necessarily mean that the stream lines are non-circular or circulation is zero. That is what I want to see. Some examples. Look at this field. You can see this is from Bertley series book and you can see all this is given there. So I will just quickly go through this. In this particular field you can see the field line clearly. Yeah the reason is also given there. Dive of this whatever field is depicted here is zero and curl is also zero. Field lines are straight not curved but the curl is non-zero. Sorry curl is non-zero. I said curl is zero. Curl is non-zero though it is so the curl. This can represent something going very fast and something going very slow. So if you lie down like this you will rotate. The curl is non-zero though the field lines are straight. This is the next picture. In this case we have the divergence non-zero but curl is zero. So looking at the picture alone may be deceptive so we have to look at the argument by which it is. So it is there in the textbook so I will not spend much time on it. This is where the divergence is zero. Curl is also zero. The whole field lines look very confused. This is the case where divergence is zero. Curl is also zero. Why is it zero? That is interesting. You should go to the book and read why it is. Another example where the curl is not zero here but divergence is zero. The example is given this is again straight away from the book. You can go back and look at it without standing in between our lunch and this. Divergence is non-zero. Curl is also non-zero. This is another example of that. Now the question is how do we write it in different coordinate systems? Look at this expression. So this is curl of grade 5. Before that before I go to that what is the curl of a gradient? It is simple I don't know whether you call it arithmetic or calculus whatever it is. Your simple steps will show you that the curl of a gradient is zero. So that is an important step today. It is very easy to show. Now what is the connection between curl and speed? Let us define the velocity and angular velocity. The relation between angular velocity and velocity of a rotating system. Look at the curl of a gradient. Curl of a gradient can be shown to be written like this and then you see that it has a value 2 omega. So the curl of linear velocity gives you a measure of the angular velocity. That is why we say it is related to rotation. This is a surface. So this is actually another way of explaining this. Again from Berkeley it is how to explain this. You consider path C, circulation and then you cut it into smaller pieces. You consider several regions of this and you see that here it is in the opposite direction at this along this line. So the sides are like that. So that is why you can write it as a constant letter. Finally you can arrive at this expression. That is the connection between the line immigrants and the curve. You do this systematically across all this and then add them up. That is you add them. This a dot d l for this and then what you get is this. The component of the curl. That is another way of looking at this Stokes theorem. Remember Stokes theorem is to be written carefully and this is named after Stokes. Although the first known statement was by William Thompson, not Kelvin and it appears in a letter to Stokes in 1850. This is the form in which, the same thing which is written earlier. Now there is an important thing to look at when we look at the Stokes theorem. We are talking about a surface, we are always talking about the surface and the curve. Yes, in this case the surface and the curve. So what is the relation between this curve and this surface? Is it unique? Given a curve, can we say that only one surface is associated with it? No. You can consider a ticket balloon and then you can balloon or some object like this and you can see that there are several surfaces which can be associated with this particular, there are several surfaces which can be associated with this. It is not unique. The Stokes theorem relates the line integral of the sector about a closed curve to the surface integral of its curve over the, curl over the enclosed area that the close curve points. I was always telling that there is a connection in the earlier case. I was telling in divergence the volume and surface are related to each other. Similarly here the curve and the surface are related to each other. Any surface bounded by the closed curve will work. You can pinch the butterfly net and distort the shape balloon. You can put it in different ways. Any of the surface it will work on any of those surfaces. But does it work on all surfaces? We will see that shortly. So you know the, this is C trans, traversed one way. In this case it is traversed the opposite. So we follow the right-hand screw convention when you take the curve. Extract the divergence of a curve. This is again a small exercise and you can show that this is zero. Diversion of a curve is a standard, this again is standard textbook derivation which tells you an important result that divergence of a curve is zero. You can, once you understand the physical and geometrical significance it is easy to see from this example which I have shown here. The important points to summarize, the surface S is not unique for a given C. That is what we are saying just now. The same C can correspond to an infinite number of open surfaces. Then given the sense of C the direction of curve is specified by the right-hand curve. The surface need not to plan out. Though conveniently we derived it for a simple circular surface. The theorem is applicable only to orientable surfaces for which the normal at every point is uniquely defined. What is this orientable surface? All surfaces are not a neat surface. Some of the surfaces give us problems. For example, look at these surfaces. Okay? So well-behaved surface or ill-behaved surface. A cylinder open at both ends. A cylinder open at only one end. This is supposed to be something like topology where we talk about orientable surfaces. This is a very nice picture. You look at the beautiful earring. This is a very nice earring. And what is so special about this earring? It is not obvious but it is called mobius strip. This is what is called a mobius strip where you have a very peculiar surface. You can easily make it. You can, while you are teaching you can ask the students to make it in a minute. You only need a long strips of paper. You take a strip of paper. If you just join it, you will get a ring like that. That is an ordinary ring. So that you take the thing and then give a twist. Okay? Instead of connecting like this, you connect like this at one point to the last point. Paste it. And then ask the students to cut it. The whole class will be in for a surprise. So they haven't seen it in YouTube already. Okay? Then if they are sufficiently, if you think their surprise is not enough, you take another one, take a strip, twist it and fix it. And then cut it twice. Then everybody, even the toughest guy who looks intellectual, who is not impressed by anything, will be impressed by that exercise. That is something you can do in your class. So such surfaces, it is not. So this is just for fun. It is used for various ornaments and all that. This is already available in the market. This is what the surface is talking about. This is one twist at the last. Instead of pulling like this, one twist at the end. So the thing is, you can give it to students and ask one to paint green on the inside and red on the outside or vice versa. And it will never be done because inside and outside. So somebody said, how many sides, you know, so how many sides these polygons have, various polygons have, the students give an answer. So somebody asked, how many sides does the circle have? That mathematical answer is infinity sides, that polygon becoming a circle. But the practical answer is two sides. The inside and the outside. So this does not have an inside and outside. It has got, so Aishar has made nice paintings out of it. You can look for these kind of pictures. This is how it looks like when you make it. And you can cut it once or twice or something. When you are free after another vacation, after the exams, you can look at this Aishar's paintings and pictures, which will be very nice. A mathematician confided that a mobius band is one-sided. And you will get quite a laugh if you cut one-and-a-half or it stays in one piece when divided. This is what I was asking you to do. You can have a three-dimensional version of this. It is called a claims body. Okay. Now, what is it that even if you forget whatever I said so far, what is it that you should take home? The take home message is that left of calculus is PC interesting and it is our friend rather than our enemy. How do you make an enemy? How do you kill an enemy? You make him your friend. Then the enemy becomes your friend, okay? If he may not become your friend as an enemy, but at least you will be different. That is our hope. So the expression, now the question is to write the expressions for curl in different coordinate systems. How is it done? It is very difficult, very complicated. How can we do it without the help of going to those lower than tables or something? I mean wherever it is available. Let us look at what it is. The curl is written like this in the row 5 system in the cylindrical coordinate system, okay? So the easy way to understand is this. You take this way. You understand it this way. This is E row D by 0 of this. Then E 5 1 by row D by D 5 of this quantity. E 6 D by D 7 of this quantity. And once you do that, the only thing you have to remember is what is this E row D by 0? How it comes about? And E 5 1 by row D by D 5, how it comes about? That is clear from the way we do differential coordinate systems in the beginning. And then of course, Z is easy. Once you take that of this quantity, this quantity is easy to remember, standard, okay? So the difference comes from here. For example, this quantity, that is what makes a difference in this case. Once you look at it this way and if you know the basic ideas of this, you can derive it at any point. You do not have to remember this expression for curl. Some details. For curl, this is for, I think this is the same things I have shown here. So we can actually work out every step and show that it is equal to the whatever standard expression we have. So row 5 3, each one, you involve this and then then cylindrical polar. Cylindrical polar is what we are discussing. And yeah, so this is finally what you get. It is these things which make the difference. So pay attention to this. Once you know this, the rest, this can be easily remembered, the other one can be easily remembered. And finally, get the correct expression. Similarly for half theta 5. So what will you do in the case of half theta 5? Remember this. So we have to concentrate on that difference here. D by D R, 1 by R D by D theta, and 1 by R by D theta. If you remember that, there is this. So this part is standard. And what you get, you can go through the steps. If you do this correctly, then you will get. So that is my, so I hope you find it at least a little bit more interesting than what you were thinking earlier. And that was my learning object. Thank you.