Lecture 1
We introduced vectors in analytic way---an n-dimensional vector is an element of the n-space R^n. We defined three algebraic operations on vectors, namely addition, scalar multiplication, and dot product, and listed their important properties. We also discussed their geometric significance in R^2. Finally, we stated the Cauchy-Schwarz inequality and as an application we defined the angle between two nonzero vectors in R^n in a purely algebraic fashion.
Lecture 2
We proved the Cauchy-Schwarz inequality and computed angle between vectors. We introduced the unit coordinate vectors in R^n. We defined orthogonal projection in R^n. We also defined lines in n-space and discussed significance in R^2.
Lecture 3
We defined planes in R^n by parametric equations. We briefly discussed the cross product in R^3 and use it to derive the normal form of the equation of a plane in R^3. We also derived the cartesian equation of a plane in R^3.
Lecture 4
We discussed several examples of functions of two variables and learnt how to draw their graphs in R^3 with the help of level curves. We also talked about level surfaces of a function of three variables. We introduced the limit of a function of two variables in an intuitive way, formulated a necessary condition for the existence of limit in the plane analogous to the concept of left and right hand limit in the line, and learnt how to check the (non) existence of limits using different paths.
Lecture 5
We defined limit of a function of several variables in a formal way, learnt how to compute limits from the definition, and listed some properties of limits.
Lecture 6
We defined continuous functions, discussed some of their properties, and saw how to apply them by means of examples. We wanted to defined the derivative of a function of several variables and observed that one cannot simply imitate the one variable definition to several variables. So we reformulated the single variable definition into one which generalises to several variables and saw how linear terms naturally pop up which contain information about derivatives. We defined linear maps from R^n to R and characterised them. Finally we made a formal definition of differentiability of a function form R^n to R in terms of linear maps.
Lecture 7
We learnt to compute the derivative of a function of two variables from the definition, discussed a geometric interpretation of the derivative, and saw that differentiability implies continuity.
Lecture 8
We introduced partial derivatives and the gradient vector, derived a sufficient condition for differentiability, and learnt how to compute derivatives using partial derivatives.
Lecture 9
We defined (parametric) curves in R^n and tangent vectors. We also learnt a version of the chain rule.
Lecture 10
We derived equations of tangent planes to the graph of z=f(x,y), and to the level surface F(x,y,z)=k.
Lecture 11
We discussed directional derivatives, local extremum, stationary points, and saddle points.
Lecture 12
We defined second-order partial derivatives, discussed equality of mixed partial derivatives, and derived the second order Taylor formula for C^2 functions.
Lecture 13
We defined positive definite matrix and characterised symmetric positive definite matrices in terms of their eigenvalues. We derived the second derivative test for local extremum.
Lecture 14
We defined linear maps from R^n to R^m and saw how to represent them using matrices. We defined derivative of functions from R^n to R^m, and learnt to compute Jacobian matrix. We also derived a chain rule for composition of vector fields.
Lecture 15
We discussed examples involving the chain rule. We found a sufficient condition for solving an implicit equation for one variable in terms of the others locally and derived a formula to compute partial derivative of the solution in terms of the given equation. This was the content of the implicit function theorem for functions form R^n to R.
Lecture 16
We took a geometric approach to find the extremum of a function of two variables with one constraint and developed a method called the Lagrange multiplier method for finding extremum with one constraint. We verified the validity of this method using the implicit function theorem and solved some practical problems.
Lecture 17
We discussed a procedure of computing area under the graph of a function of one variable and saw how it leads to the notion of the Riemann integral. We generalised this procedure to compute the volume under the graph of a function of two variables and defined double integrals. Using Fubini's theorem we learn to compute double integrals over rectangles and type I regions in terms of iterated integrals.
Lecture 18
I was away. Lecture was given by Venkata Krishna Kishore. He discussed Line integrals.
Lecture 19
I was away. Lecture was given by Venkata Krishna Kishore. He discussed Line integrals.
Lecture 20
We discussed the method of substitution for double integrals and some examples.
Lecture 21
We defined multiple integrals over regions in R^n and discussed some examples.
Lecture 22
We were interested in finding conditions on a vector field for its line integral to be independent of path. As a preparation, we proved the fundamental theorems for line integrals. As an application of the second fundamental theorem, we derived the principle of conservation of mechanical energy.
Lecture 23
With the help of the fundamental theorems, we proved that the line integral of a vector field on a domain is independent of path if and only if it is a gradient field. We learnt two methods of finding potentials for conservative vector fields. We derived a necessary condition for a vector field on planar domains to be conservative. We also discussed the Jordan curve theorem.
Lecture 24
We defined the positive and negative orientations of Jordan curves. We stated the Green's theorem and proved it in a special case namely when the region is of both type I and type II. Using the Green's theorem, we derived a necessary and sufficient condition for a two dimensional vector field to be conservative on simply connected domains.
Lecture 25
We defined parametric surfaces and saw some examples. We defined the fundamental vector product, discussed its significance, and learnt how it enables us to define the tangent plane. We formulated a definition of surface area in terms of double integral and computed it for the unit sphere.
Lecture 26
We defined surface integrals of scalar and vector fields and discussed their geometric interpretations.
Lecture 27
We learnt Stokes' theorem and saw how curl can be interpreted as microscopic circulation. We also discussed the divergence theorem. The course ends here.