Lecture 1: Introduction and readiness check
brief review on calculus for single variable functions
Lecture 2: Functions of several variables
examples of multivariable functions, graph of a function of two variables
Lecture 3: Partial derivatives
definition of of partial derivatives, geometric interpretation of partial derivatives
Lecture 4: Tangent plane and chain rule
differentiable functions, tangent planes, chain rule
Lecture 5: Higher-order partial derivatives
definition of higher-order partial derivatives, equality of mixed partials
Lecture 6: Taylor's theorem and extreme values
Taylor's theorem, Hessian matrix, second derivative test, the least-squares technique
Lecture 7: Implicit function theorem
implicit function theorem for 1)f(x; y) = c, 2)f(x; y; z) = c, and
3) three variables two equations
Lecture 8: Lagrange multiplier
constrained optimaztion, the method of Lagrange multipliers
Lecture 9: Double integrals
double integrals over a rectangle, double integrals over a general domain
Lecture 10: Itereated integral
calculation of double integrals using iterated integrals, geometric interpretation of
double integrals
Lecture 11: Change of variables
polar coordinate systems, various coordinate systems and Jacobians
Lecture 12: Improper integral
improper integrals in one variable, improper integral in two variables, Gamma function,
Gaussian integral
Lecture 13: Line integrals
definition of line integrals, vector field, Green's theorem