 Good morning. So, in this lecture, we will study the phase curve and that will be our first lesson on vector analysis vector calculus. In this, I would advise that before going through the lecture, you should ensure that the basic notions of vector algebra are firmly in your mind, so that there is no difficulty in following through the discussion that we do in vector calculus. So, the text book, the first section of this chapter of the text book, recapitulate the basic notions of the vector algebraic relationship. We will not go through the details of that, but I advise that you go through those things before following this particular lecture. It should be clear the, what are the implications of dot and cross products in vector algebra and scalar and vector triple product because they will be very frequently used in this lesson and the next. Further, the differentiation rules of vector functions should be clear in following the analysis that we take up now. A few points are special which I would like to remind you. One is that when we try to interface the typical vector algebraic notions with the matrix algebra, then we find that the dot product in between two vectors is equivalent to this A transpose x, which is the inner product in vectors as understood in the linear algebra terminology. Next, quite often you will find that you come across expressions of this kind A dot x, B in which A, B and x all three are vectors. So, in that situation, if you take the expressions of A dot x as it is and put that in this place, you will find that the components of x, that is x 1, x 2, x 3, they get somewhere lost in the interior of this expression and sometimes it happens that you want to treat x as the right side of a linear algebra expression. In such situations, you can consider B to be kept on this side and it does not matter, because A dot x is a scalar and if you do that, if you keep B on this side and then you have an expression which looks like B A dot x and here in this A dot x, if you use this relationship that is A transpose x, then you will find that then after that you have all three linear algebraic notions B is a vector, column vector, A transpose is a row vector and then x is again a column vector and then using the associativity of matrix multiplications, you come to this relationship in which B A transpose multiplied together gives you a 3 by 3 matrix and x is a 3 by 1 vector and you find that you have got the vector x on the right side and that may be helpful in situations where you want to treat x as the unknown or variable vector with A and B as constant and known vectors. Another important interfacing between the vector algebra and matrix algebra is in the matter of cross product. Cross product is something which unlike the dot or inner product is a matter which is very special to three dimensional vectors. For general n dimensional vectors, you do not have anything called a cross product. Now, in the case of three dimensional vectors, which are sensible in geometry ordinary Euclidean spatial geometry, you have got the notion of a cross product between two vectors and in 3D space, it makes direct sense and the result is a vector. In 2D space also in certain circumstances, there is an interpretation which makes sense. So, for example, in the x y plane, if you have two vectors A and x, then their cross product is a vector which is actually perpendicular to the plane of A and x and that means perpendicular to the plane under consideration and that way, if you take the straight away expression from a cross x, then you get a scalar value. So, in the vector algebra sense, whatever you get as A cross x, if you want to put it in the linear algebra sense, then you can do it by this relationship. In the case of 2D vector, you get a perpendicular transpose x. Now, this particular notation A perpendicular is a vector which is A rotated by a right angle counter clockwise. So, if the vector A is A x A y, then the rotated vector rotated by 90 degree is this vector and inner product of this vector with x gives you the same result as you would get as A cross x. In the case of 3D vectors, if A and x are 3D vectors, then A cross x can be represented through this matrix vector multiplication, where A tilde is a 3 by 3 matrix which is a skew symmetric matrix with elements of A A, that is A x A y A z arranged in this manner. So, quite often in simplifying complicated expressions involving vector algebra notions into linear algebra formulation, these relationships will help you. Now, with this background of the basic notions of vector algebra rules, we proceed to the study of parametric curves or curves in their parametric representation. When we try to describe a curve in space, we can talk of several kinds of representation. This is called the explicit equation of a curve in which y and z coordinates are expressed as functions of x coordinates. As you know that for all curves, this may not be possible. That is for a general curve, it may not be possible. It will be possible only when for any given x, the values of y and z are unique. So, if a curve goes like this, then with a given value of x, there can be several values of y and z which will be on the curve and such a curve will not be expressible in this kind of a representation. This representation through two equations in x y z like f of x y z equal to 0 and g of x y z equal to 0 is another representation of a curve and this is called the implicit equation. In this case, the idea is to represent a curve as an intersection of two surfaces. f of x y z equal to 0 is the equation of one surface and g of x y z equal to 0 is the representation of another surface and the intersection of these two surfaces is a curve and that is the curve which is expressed through this relationship. Now, this is a representation which is actually unnecessarily complicated. For the description of a curve, the prior description of two surfaces is going to be much more complicated than the original issue. In comparison to these two notions, these two representations, you will find that the parametric representation is much more straightforward and completely general. In that, the position vector of a general point in a curve is represented as a vector function of a parameter. In this case, t and in terms of t, you can express all the three x y and z components as x of t, y of t, z of t and when you combine them together, you get this as the representation of the curve. So, as the parameter t changes, you find that the x y z coordinates of the point on the curve changes and accordingly, if you try to plot, you get the curve. You can represent this curve in this manner or in this manner. The two representations are equivalent. Now, if you represent a curve in this kind of a parametric representation, then a few notions appear immediately. If you differentiate this vector function of the scalar variables, then you get this derivative and this is called the tangent vector. So, if this is the curve at this point, suppose for a value of t, value of the parameter t, you get this point, say origin is here, then by the standard definitions of derivative, you get the derivative of this position vector at this point as a vector like this. This is the tangent vector to the curve at this point and this is r prime. Now, if you take the modulus, the magnitude of this vector, then what you get is called the speed of the curve, speed in what sense as if the parameter t is time, then you will find that r prime gives you the velocity of a particle moving along this curve with time t as the parameter. So, if you consider the parameter t to be time, then what you get as speed is indeed the physical speed that we know of. If you consider the unit vector along this direction, then you get the unit tangent as this vector divided by its magnitude. In this manner, if you find out the derivative, then you can use it to find the length of a curve from one point to another. Suppose, this is the point corresponding to the value of t as a and this is another point which corresponds to the value of t equal to b. Now, from here to here, if you want to find out the length of the curve, then what you can do? The speed at which it moves with respect to t at this point. So, you take that and multiply that with a little small change d t, then you get this segment. Similarly, you get lots of segments. So, the sum of these segments will give you the length of the curve as the length of each of the segment tends to 0 and there are infinite such segments, very large number of segments and each segment extremely small. That means, it is the sum of a large number of small parts that is integral. So, you get the length of the curve from t equal to a to t equal to b as this in which you have d r mod here, which means d r by d t and d t separately. So, d t is here and d r by d t modulus is this. So, this gives you the length of the curve. Now, if you keep the initial point for calculating the length as constant and the final point, if you do not specify as a particular value, but if you keep that as variable, then through that exercise, you can define a very important function, which is called the arc length function. So, if you keep b variable, then you get this arc length function, which is the integral from a to t. That means, starting from this point, for every value of t, you get an arc length, which is the length of the curve starting from this point to the current point. So, this is a function of t and this is called the arc length function in which you get d s is equal to the same mod of d r, which is this and from here, if you represent them as d x by d t whole square plus d y by d t whole square plus d z by d t whole square, then a term d t comes outside and here, what you are left with is simply d s by d t. That is this. Now, a curve r of t is called regular, if the tangent vector never vanishes. That is at every, if a curve is called regular, if at every point of it, there is a non 0 tangent vector. The direction is clearly representable from the derivative, from the first derivative and in such situations, you can effect a re parameterization of the same curve with respect to some other parameter say t star, which must be a strictly increasing function of the current parameter, which is t. Now, if you check the arc length function for a possible parameter, then first you note that arc length s of t is obviously, a monotonically increasing function. That is as t changes as t increases along the curve, you keep on getting the values of s, which is continuously increasing. So, arc length is a monotonically increasing function. Next for a regular curve, if r prime is not 0, then d s by d t will not be 0. And in that case, the function, arc length function s of t has an inverse. That is, its inverse can be expressed as t of s. And if we can do that, then wherever we had the representation of the function as r of t, we can revise it as r of t of s. That is, by virtue of the fact that the arc length function has an inverse. That is, from s of t, we can derive the inverse t of s. And if we can derive the inverse, then we can insert that inverse here. And then finally, this becomes a function of s. So, in this manner, we can define the curve as with s as the parameter, with the arc length itself as the parameter. And in fact, the arc length s gives you a natural parameter to describe the curve. If we parameterize a curve with the arc length as its parameter, then we get something interesting. What we get in that case is that, if we represent it as r s, then as we try to find out the derivative for the tangent, then we get r prime s. And this will be always of unit length. This will be always of unit length. And that is why a curve parameterized with arc length as the parameter is called a unit speed curve. Because for every small distance covered, you find that the length of the small segment is same as d s. So, this derivative will be always of unit length. So, the tangent that you will get will automatically have unit length. So, you do not have to really divide it with r prime norm. r prime norm is 1. So, in that case, that unit tangent you get directly as u of s as r prime s. Now, we come to the most important property of a curve, that is the curvature. After defining the tangent from the first derivative, you try to take the second derivative. And then you can see how this tangent changes, how this tangent turns. And now with the arc length parameterization you have got the unit tangent. The ordinary tangent turns out to be a unit tangent directly. And in that case, all that you need to see is that how this unit vector changes as you proceed along the curve. And that gives you the notion of curvature. In undergraduate calculus for planar curve, you might have studied this notion of curvature. That is, curvature is d psi by d s in which psi is the angle that the tangent makes with the x axis. So, here also you find the notion is same. That is, as a particle proceeds along the curve, then with respect to distance covered along the curve, how fast the direction changes, direction is represented by psi. The same thing is here, in which we say that with respect to s, how fast this tangent vector u is turning. So, we try to take the derivative of u with respect to s. So, d u by d s. And that gives you the measure of the curvature. And the way this curvature function is defined, it is going to be non-negative always. Because we are defining the curvature function with a norm, which will be never negative. So, you can say kappa of s, that is curvature function with respect to the arc length is norm of u prime or norm of r double prime. So, from here as you do that, you note that the unit tangent, if this is the u, u of s in which case, we will be defining the curve not with respect to t, but with respect to s. Then, if this is u s, then note that this is a unit vector. And with respect to the arc length, the rate of change of the unit vector will be perpendicular to this vector itself. Because for any small segment, here it is a unit vector, here also at the next point also it will be a unit vector. So, if you take this as u, then at the next point also it will be a unit vector, say u plus delta u. And then, this is the change delta u. And as s tends to 0, as this change becomes very small, then you will find that this change of u, that is delta u, turns out to be perpendicular to u. So, any unit vector you take and you consider its derivative with respect to arc length itself, then you find that the derivative is perpendicular to the unit vector. So, in this case, that perpendicular vector will be in this direction, the way it is defined, the vector like this will be always in the interior, that is in the direction in which the turn is taken by the curve. It could be 0, if around that place, around that location, temporarily the curve was going straight, then it would be s 0 vector. Otherwise, if there is a turn, then it will be in towards the interior of the turning. So, this vector is a normal, normal vector to the curve at this point. And as we know that its magnitude is defined as kappa. So, its direction, the unit vector along this direction, which is found by dividing u prime with kappa, that is called the unit principle normal. Unit, we can understand, because this is going to be a unit vector, because we are dividing u prime with its own magnitude. So, it will be a unit. Normal, we can see, because it is at right angles to the tangent vector. So, that is normal to the curve at this point. But, why do we call it unit principle normal? Why principle? Because, at this point, if this direction gives us the tangent, then perpendicular to this line, at this point, we can actually draw a plane. And every vector in that plane passing through this point is actually a normal. And this is the principle normal. Other vectors in this plane are also normal, but they are not the principle normal. This vector is called the principle normal, because it, this vector along with the tangent defines the plane, which can be taken as the plane of the curve at least locally, at least around this point. That is why the normal, which falls in the plane of the curve in this local neighborhood, that is the principle normal. If you take the cross product of u and p, if you take the cross product of u and p, then you get another normal out of the blackboard, out of the plane of the blackboard, and that is called the binormal. Now, this is easy for a representation with respect to arc length as the parameter. In the case of a general parameterization, in which we use the parameter t, not the arc length, for this curve, you will find that it will not be so easy to derive the expressions for the principle normal and the curvature function. So, for that, what you can do is, you can consider it in this manner, that r prime t has a magnitude, which is not unity in general. So, you can say that r prime is its magnitude, not necessarily 1 into the direction, unit vector along the tangent. Now, if you differentiate this for developing r double prime, then you will find that it will have two parts, one because of the variation of this and the second because of the variation of this. In the case of r of s, this part was missing because it was constant. So, if you consider a general parameterization like this, then you get r prime, which can be represented in this manner, and then if you formally differentiate it using the product rule, then you will get the second derivative in this manner, that is the derivative of r prime norm into u t plus r prime norm into d u by d t. And if you differentiate it like this, then you will find that in the derivative, this part d u by d t will be d u by d s, d u by d t will be d u by d s into d s by d t. Now, d s by d t is again this r prime norm itself and d u by d s is what we encountered earlier, that is kappa into p. So, when you insert those things, then you get this kappa into p and d s by d t gives you one r prime norm and another r prime norm is already sitting here. So, in the second term, you get this, first term has remained unchanged. Now, you find that this second derivative is not a vector in the direction of the unit principle normal, but it is a combination of u and p. The reason is that here the speed is not constant, speed is changing. So, you have this. So, because of the change of speed, you get this is acceleration actually. You get one component along the tangent, that is because of the change of speed, change of the magnitude of speed. And this is the representation of the part which is the result of the change in direction. So, now from here, if you want to find out the curvature, then what you need to do? From this second derivative, you need to subtract this part out. What remains? You examine that and from there, you get the direction vector p t and the curvature, because you have already r prime. So, when you consider r double prime minus this term, whatever is its component along the tangent, that part you subtract, then whatever is left from that, you get the direction and by inserting the values of r prime, you get from the magnitude, you get the curvature. Now, this is the situation. If you try to see it schematically, then at point a, this is the position vector r, r of t. That means at every point, r of t changes. And at this point, this is say r prime, r prime of t, unit vector size u, size 1, this is the unit vector u, u of t, this and then as you calculate the secondary derivative of r, it is not necessary that it is normal to the curve. It may be in some direction like this. Out of that, when you subtract away the tangential component, then what is left is in this direction and in this direction you get this vector. The unit vector along that direction is shown as p of t and whatever is kappa, that is the curvature. If you take 1 by kappa, then you get a length and in this direction, if you mark out that length, then you get a point denoted by c here in this figure and that is called the center of curvature, y center of curvature. Because locally around this point, in that case, you can represent the very close neighborhood of the curve by a circular arc, the center of which is at p. So, in that sense, this is the, this is called the center of curvature. And the plane that is defined by these two vectors u and p, that is the plane in which the curve resides in the immediate neighborhood of this point and that plane is called the oscillating plane. As we have already discussed that the two principle unit vectors u and p, if you take from the oscillating plane and define this cross product, then you get the third mutually perpendicular vector b like this perpendicular out of the plane of the blackboard. Then that vector is called the binormal. Why binormal? Because it is normal to the curve certainly and it is normal to both the important vectors that we have earlier defined u and p, that is why it is called the binormal. Now, you find that through derivatives, we have defined three unit vectors at this point which are mutually perpendicular u, p, b and u, p, b together define a right handed triad, the kind of three vectors that we typically use to represent vectors to get a frame of reference to describe 3D vectors. So, this particular right handed triad formed by that point as the origin, this particular point as the origin and these three vectors, these three unit vectors u, p, b as the three reference vectors, that frame of reference is called the serrate frenet frame. Now, in this frame, you can directly describe the oscillating plane which is the plane formed by u and p and then if you consider the plane formed by u and b, that is called the rectifying plane and the plane formed by p and b, the two normals that is called the normal plane because that entire plane formed by p and b is normal to the curve. After this, we get into the discussion of the next important property of a curve which will be there in the case of spatial curve, three dimensional curves. In the three dimensional curves, the tendency of the curve to twist out of this local plane, the oscillating plane to the tendency to twist out of this plane and become a spatial curve go out of this plane is measured by the quantity torsion. So, how would you describe it? You find that the plane, oscillating plane is the plane formed by u and p and the direction of the plane is given by the direction of it normal which is the binormal b in this case. Now, as the curve tends to come out of this plane, then at the next point, the oscillating plane will be somewhat different which means the binormal will be somewhat different. So, then the way the curve tends to twist out of the oscillating plane will be measured by the way this binormal changes that is from this unit vector, if it changes to this unit vector, then the manner in which this change takes place will give you a measure of the twisting tendency of the curve or torsion of the curve. That is why we measure torsion through the rate of change of b with respect to arc length. So, if you try to differentiate this to get the rate of change of b, then you get b prime as u prime cross p plus u cross p prime. Now, here u prime we already know as kappa of s into p. So, we insert that and then we find that the first term has p cross p which is 0. So, this goes out. What remains? You have u cross p prime. Now, we know u p b, now we ask what is p prime? Now, what is p prime? Till now we have not come across the expression for p prime, but one issue is very clear p is a unit vector and therefore, its derivative with respect to the arc length must be perpendicular to p itself. So, if there is a vector which is perpendicular to p, that means it will be in the plane of u and b. So, we can consider p prime as a linear combination of u and b. So, let us take it like this and try to put it here. So, this kind of an expression with a linear combination of u and b sigma u plus tau b we insert here. As we do that, we find u cross this. Now, the first term in this product will involve u cross u which is 0 and the second term will be tau into u cross b. Now, u cross b will be minus b because this is a right handed triad. So, you will get this. See, we have got b prime the rate of change of the binaural in terms of the known vector. So, it is a vector in the direction of the principal normal p and the magnitude tau that is we find here is called the torsion of the curve. And how do you find it? You just take inner product of this equation with p and then you get p dot b prime negative of that as tau. So, from here we get first the torsion of the curve, second the derivative of the binormal which is in this manner a vector in the direction of p itself. Now, you see that earlier we defined u prime and we got that as kappa into p when the parameterization is with respect to the arc length. Now, we have defined and got an expression for b prime. Now, out of the three vectors u p and b for two of them for u and b we know we have described how they change that is we have evaluated their rate of change u prime and b prime. Now, what about p prime? Here we assumed that p prime is this linear combination of u and b because we knew that it must be in the plane of u and b, but this sigma and tau we did not determine at that time tau turns out to be the torsion of the curve and that is how it is defined. Now, sigma is left if we can determine the value of sigma then we would have completed the description of p prime that is the rate of change of p that is how this principle normal changes. So, we can do that now completely because we have got now both u prime and b prime and from this triad we find that p is b cross u. So, using that we can try to determine p prime from this expression itself. So, if p is b cross u then its derivative will be b prime cross u plus b cross u prime and if we use the expression for b prime as we derived just now. That is b prime is minus tau p if we use that here then we find minus tau p cross u plus b cross u prime which is kappa p. Now, if we simplify this we find that p cross u is minus b. So, we get tau b and b cross p is minus u. So, we get from here minus kappa u. So, the sigma that we assumed here turns out to be minus kappa and with this now we have got p prime also. This complete set of three formulae that give the rate of change of the three unit vectors u p b in terms of those vectors themselves are called the serrate finite formulae. Now, with these formulae if you know the curvature and torsion as functions of the arc length then given the initial vectors u p b at the starting point of the curve you can determine the curve for all subsequent values of p. How do you do that? If you know not all subsequent values of t, but s the arc length function. If you know kappa and tau as functions of s then with the given value of u p and b at s equal to 0 you know the complete right hand side. That means you know the rate of change of these three vectors u p b. So, the initial trade is given and from these formulae you can find out the rate at which they change that is for a very small distance d s travelled along the curve you know how these will change the d u d p and d b you can work out. And with that with that knowledge you can effect that little changes in d in u p and b and proceed to the next point along the curve. As you know that next point along the curve and the u p b values u p b vectors at that point and again you know kappa and tau as function of s then at that new point you can again completely determine the right hand side and keep on determining the next point and the next trade u p b next frame or the next serrate frenet frame. And this way you can continue along the entire curve not only that if you have two different parameterizations of a curve starting with different points then also, but if you find that they are the same curve, but separated one curve here starting from this point and going like this and another curve starting from here going like this. If they are the same curve then with these two different starting points and different directions in which you proceed you can find out that these are actually the same curve moved and rotated by this distance and some rotation. So, this you can establish if you can find out what are the u p b vectors at this point and what are the corresponding u p b vectors at this point. And if you can give a compensatory motion to this compensatory rigid body motion a rotation and a translation then you find that the entire curve comes back at this location. So, that means that if the curvature and torsion kappa s and tau s are two given functions that is curvature and torsion as functions of arc length parameterization as long as these two functions are same for two curves the two curves will turn out to be exactly the same curve except that they may be rotated and displaced through some rotation and a displacement. So, that is why we can say that the serrat finite frame and the curvature and torsion functions give you an intrinsic representation of a curve intrinsic representation in the sense that they do not depend upon an external frame of reference. The u p b triad itself is its own most natural frame of reference in which to describe the curve and that frame keeps on moving along the curve as you change the parameter value that is the arc length. So, this result you can establish very easily that is the arc length parameterization of a curve is completely determined by its curvature function and torsion function except for a rigid body motion. Some of the exercises in the in the corresponding chapter of this of the text book you will find two different looking functions two different looking vector functions representing the same curve and through the reduction to the standard form through arc length parameterization you can establish the equivalence of the two different parametric representation for the same curve located at different locations. With this much discussion on curves we conclude this lesson with a little discussion on parametric surfaces. The way we describe curves by the help of a parameter we can also represent a surface by the help of two parameters say u and v. If you can represent the position vector of a point on the surface with the help of two parameters as a function of two parameters then the representation looks like this x of u v i plus y of u v j plus z of u v k. You can also represent it as a column vector with x y z functions of u and v being the members meaning the components in the vector. Now, for a function for a surface which is a two dimensional entity at every point you can find out two independent tangent vectors that is you can define a complete tangent plane and on that tangent plane you can define infinite tangent vectors two of them will be linearly independent in terms of which as linear combinations of which you can represent all the other tangent vectors. So, for example, suppose this is a surface then at every point of it you will find that you can describe a complete plane which is tangential to it. So, from this point you can determine several tangent vectors like this. So, in that plane all vectors in that plane which are touching the surface. For example, if you draw a cylinder like this then at this point you will find that you can find out a complete plane every point of it is a tangent. So, the line like this is also a tangent and in that plane whatever in the plane perpendicular to the board you can draw this kind of a tangent this kind of a tangent and so on. So, in that tangent plane two of the tangent are linearly independent as a linear combination of these two you can construct any other tangent in that same plane. Now, if you change y through a small amount u if you change u through a small amount the first parameter then you move along a particular tangent on the surface and that tangent is known as r u that is the derivative of r with respect to u that gives you one tangent vector and derivative of r with respect to v gives you another tangent vector. Now, if the parameterization is not degenerate at that particular point then you will find that r u and r v are linearly independent. There may be a point there may be some points where the two parametric tangent vectors turn out to be in the same direction and that will be a degenerate point for this particular parameterization. Now, considering a situation in which the point that we are considering is not degenerate you can find two linearly independent tangent vectors r u and r v by straightforward derivative. So, as you find two such vectors two such a tangents in the tangent plane then the cross product of these two tangent vectors will give you the normal to the surface like this which is normal to the entire tangent plane. So, that normal you can represent like this and quite often you are interested in the unit normal vector which you can find out like this. Now, the local shape of the surface around this point can be explored can be analyzed through the investigation of how this particular normal this unit normal changes as you move along the surface along this tangent or that tangent or any other tangent. So, if you have got say a surface like this at this point you have got this as r u and this is a tangent vector and this is another tangent vector this is r v cross product of them gives you this normal. Then at this point if you say that now along r u if I try to move on the surface then I get a point which is arbitrarily close to it close to this and a little in this in the direction of r u along the tangent plane then you move to this point which is very close to the original point here and what is the normal at this point. So, that normal you can work out similarly if you move along r v then you can get another normal and then you try to see the projection of these normal on the tangent plane again. So, that way for every move along a tangent vector you can get a new normal and you work out the rate of change of this unit normal unit normal n and through the those rates of changes you can work out the local shape of the surface around this point. So, how does you ask the question how does n vary over the surface. So, this gives you the information about the local shape local geometry of the surface. So, you can determine a curvature which is called normal curvature at every point along every direction. So, in the tangent plane if you take another direction another independent direction neither r u nor r v some any other direction arbitrary direction in the tangent plane which is a linear combination of r u and r v then you can cut the surface through a plane which includes n and this vector the normal and this vector. So, you can make a normal cut on the surface through the normal plane of n and this vector and that will define a planar curve and the curvature of that planar curve is called the normal curvature of the surface along this direction at this point. And then that is another candidate for a direction for the tangent and then if you can find try to if you try to find out those directions along which you can determine the normal curvature in such a manner that the curvature tensor that you get you see that curvature is actually a tensor because the moment you decide on a direction in the tangent plane finally, you get another vector in the tangent plane itself. So, from tangent plane to tangent plane that is a mapping. So, that way the curvature at every point is actually a tensor quantity and as you diagonalize the tensor quantity the curvature tensor you can find out those directions in which the entire curvature is contained within the normal plane. So, those directions are called principal directions and the corresponding normal curvatures are called principal curvatures. So, you can diagonalize this curvature tensor and find out two principal directions and the corresponding principal curvatures depending upon the signs of these principal curvatures you can determine you can classify the local shape as convex, one cave, saddle and so on. So, if both the principal curvatures turn out to be positive then you say that locally the surface is convex. If both of them are negative then locally the surface is concave. If one of the principal curvatures is positive and the other is negative then that point is termed as at that point the surface geometry is termed as a saddle point that is a saddle point. If one of the principal curvatures turns out to be 0 and the other is positive or negative then it is cylindrical. On the other hand if both the curvatures both the principal curvatures at a point turn out to be 0 then locally the surface the surface is a planar surface. So, this way you can work out the local geometry of a surface based on the way the normal varies in the immediate neighborhood. So, let us summarize the important points in this lesson. First point is that parametric equation is the general and most convenient representation of curves and surfaces in three dimensions. And second for curves for parametric curves arc length turn out to be the most natural parameter and the saddle turn out frame gives the most natural description of the curve which is intrinsic to the curve which does not depend on any external reference. Third important point is that the curvature and torsion are the only inherent properties of a curve. The tangents etcetera are themes or properties which are dependent on frame of reference also parameterization also and so on. Curvature and torsion as functions of the arc length parameter are the only intrinsic properties of a curve the rest of it is due to parameterization and frame of reference chosen. Finally for a surface patch for a surface patch the local shape can be understood through an analysis of the curvature tensor which is found by exploration of how the surface normal changes as we move in the immediate vicinity of the surface around a given point. Now, in the next lesson we continue the vector calculus topic and in this lecture we will see some of the differential operators on field functions. And in the next lecture we will continue into a discussion of the integral operations and integral theorems. First the differential operators a function of x y z which is a scalar function that is a mapping from three dimensional space to the real line is called a scalar point function or scalar field. Similarly, if you have a mapping from three dimensional space to a three dimensional space that is a vector field or a vector point function you can represent it with this in this manner. Now, on both of the scalar and vector fields you can apply an operator the differential operator first order differential operator which is denoted as del or nabla and the meaning of which is this. Note the particular way this particular this operator is written normally when we write a vector with components along i j k we write a x i plus a y j plus a z k, but here it is written as i into something plus j into something plus k into something the reason is that this something is not a quantity, but an operator. So, when you carry out the algebra of this del or nabla operator there may be situations where slight changes will mean different things. So, to save oneself from confusion one must remember these three things one is that del is a vector quantity second it signifies a differentiation and third it operates from the left side. You cannot multiply this thing to a quantity from the right side and expect that you will get a quantity back you always apply del from the left side. So, that is why two notations del dot v and v dot del mean two completely different things you see v is a vector which is v x i plus v y j plus v z k i v is a vector quantity this del is a vector operator it is also a vector. So, you can think of dot problem between them you can think of this and you can think of this, but both of them are meaningful expressions, but they mean two completely different things. Normal vectors for example, v a vector like that w another vector like that in that you know that v dot w and w dot v are same that is dot product is commutative, but not when one of them turns out to be an operator a differential operator like this del. What is this if you apply the del operator through a dot product then as you apply that on this on v and you work out all the dot product as usual then you will get something which will be del v x by del x plus del v y plus del y plus del v z by del z which is a quantity which is scalar quantity and this is this has a meaning this is called the divergence of v as we will shortly see. On the other hand this one will mean v x del by del x plus v y del by del y plus v z del by del z. Now, this is also scalar, but this is not a quantity this still is an operator waiting to operate on some function on this side. So, these two mean two completely different things because this del is an operator and it operates from the left side. So, when on the right of del there is nothing given. So, it remains still an operator after we put something on the right side of it then this operator will operate from the left side and produce a quantity. Now, another important operator which is a second order operator for that matter is this del square which is this. This is called the Laplace-Tien operator because when this operator is operated on a function phi then you get this equation which is the famous Laplace's equation and a solution of this kind of an equation that is a function phi which satisfies this equation is called a harmonic function which we will come across quite often in the course particularly in this chapter also. Now, when you apply the first order differential operator del on a field function scalar field or vector field you can develop three different notions of vector calculus. First is the gradient which is the result when you apply the del operator on a scalar function a potential function. So, that is the gradient quite often denoted as grad phi or simply del phi like this. If you apply that del operator term by term on phi then you get this and this is the gradient vector. This gradient has the same meaning as the gradient which we discussed in the previous lecture. That was in the in that case the dimension was free in this case the dimension is just free. In that in the other case in the linear algebra sense the dimension could be anything it was free here it is only for three dimensional vectors that is why three components i j k are there. So, now if phi of x y z is a scalar field then phi of x y z equal to constant will give the level surfaces or equipotential surfaces of that potential function and at any point the corresponding gradient vector that you can determine like this turns out to be normal to that or so on and to the level surfaces that is the gradient. Now in the scientific problems in engineering problems this is very important because if in a potential flow you have the potential described by a function phi then the negative of the gradient phi gives you the velocity vector. Now on a vector function when you try to apply the del operator then del itself being a vector on a vector function when you try to operate it you can operate in two manner one is by a dot product and the other is by a cross product. So, through the application of a dot product between the del operator and v you get this kind of a thing and which is called the divergence of the vector field or divergence of the vector point function. So, the vector field is given like this v x v y z are its components along x y z directions and the divergence is given by this expression. Now this also has a direct meaning in the context of fluid flow that is divergence of rho v where rho is the density and v is the velocity vector gives us the flow rate of mass per unit volume out of a given control volume. There are similar relationships between field vector and the flux in the case of electricity and magnetism. Now if rather than dot product between the operator del and the vector function v if you have a cross product then as a result you get a vector function because cross product between two vectors is a vector and that is called the curl and that is defined in this manner curl of v is like this cross product which you can evaluate like this and get the three i j a k components in this manner. So, that word curl literally means rotation turning and that is the precise meaning when you try to explore the situation in the case of a fluid flow. For example, if v equal to omega cross r represents the velocity field then the curl of the velocity vector gives twice the angular velocity that is twice the rotationality. So, that means if curl of the velocity vector is 0 that means omega is 0 and that means it is an irrotational flow. So, curl represents the rotationality in the flow. In electromagnetism the relationships between electric field and magnetic field is also established through the curl of the corresponding field. Beyond this we will consider a few further differential operators, composite operators and second order operators and integral operators in the next lecture. Thank you. .