Summary of lectures

Lecture 1

We introduced vectors in analytic way, defined addition and scalar multiplication of vectors, discussed their geometric significance, and listed their important properties.

Lecture 2

We defined the dot product of vectors, proved the Cauchy-Schwarz inequality, and learned how to define the angle between two vectors in R^n.

Lecture 3

We learned the orthogonal decomposition and the orthogonal projection of vectors. We also introduced lines and line segments in R^n.

Lecture 4

We learned that a vector-valued function of a real variable can be visualized as the trajectory of a moving particle (i.e., as a curve) and defined the concepts of limit, continuity, derivative, and integral for such functions. We saw that the derivative of such a function is a tangent vector to the curve or the velocity vector of the moving particle. We also discussed some examples of real-valued functions of two variables and visualized them by drawing their graphs. We saw that the level curves of such a function can help us in sketching its graph. For a real-valued function of three variables, we can think of its graph as made up of level surfaces. 

Lecture 5

We introduced the limit of a function of two variables, formulated a necessary condition for the existence of limit in the plane analogous to the concept of left and right-hand limits in the line, and learned how to check the (non) existence of limits using different paths. We also computed limits using the epsilon-delta definition.

Lecture 6

We listed some properties of limits such as the limit of a sum is equal to the sum of the limits etc. We defined continuous functions, discussed some of their properties such as the sum of continuous functions is continuous, the composition of continuous functions is continuous, etc., and saw how to apply them by means of examples. As a first step to generalizing the notion of derivatives to a function of several variables, we introduced partial derivatives and learned their geometric significance. We saw that there are functions that have all the partial derivatives but the function fails to be continuous. This suggests that partial derivatives are not a true generalization of derivatives.

Lecture 7

We observed that one cannot simply imitate the one variable definition of differentiability as a limit of a quotient to several variables because that leads to division by vectors. 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 made a formal definition of the differentiability of a function from R^n to R in terms of linear maps. We also learned to compute derivatives from the definition.

Lecture 8

We saw more examples of derivatives from the definition. We proved the uniqueness of the derivative. We proved that differentiability implies continuity. We also found a geometric interpretation of derivatives. We learned a sufficient condition for differentiability. We also learned a technique to compute the derivative in terms of directional derivatives provided we already know that the function is differentiable.

Lecture 9

We computed derivatives of some functions using directional derivatives. We introduced the Jacobian matrix and gradient. We learned to write derivatives using partial derivatives. We also computed the equation of the tangent plane to the graph of a function.

Lecture 10

We computed the Jacobian matrices of some functions from R^n to R^m. We learned the chain rule, expressed it using Leibniz notation, and discussed some examples.

Lecture 11

We discussed the sum, product, and quotient rules of differentiability. We defined tangent vectors, tangent lines, and tangent planes to a surface at a point. We computed the tangent planes to level surfaces and graphs.

Lecture 12

We introduced local extrema and saddle points of a function, defined second-order partial derivatives, and discussed the second-order Taylor approximation and second-derivative test.

Lecture 13

We discussed the implicit function theorem and the method of Lagrange multipliers.