- Metric Spaces
- Convergence of sequences in metric spaces
- Cauchy sequences
- completeness
- contraction principle

- Topological spaces
- continuous maps
- Hausdorff spaces
- compactness
- connectedness

- The real numbers
- The real numbers as a complete ordered field
- closed bounded subsets are compact
- intermediate value theorem
- maxima and minima for continuous functions on a compact set

- Sequences and series of complex numbers
- standard tests for convergence and divergence of series
- absolute convergence and rearrangements

- Differentiation
- Differentiation of a function in one real variable
- Mean Value Theorem
- L'Hopital's Rule
- Taylor's Theorem with error estimates

- Riemann integration of functions in one real variable
- Definition
- Riemann integrable functions
- integration and anti-differentiation

- Sequences and series of functions
- power series and radii of convergence
- uniform convergence of sequences of functions
- uniform convergence and integration
- integration and differentiation of power series

- Differential Calculus for functions from n-space to reals and
reals to n-space
- Parametrized curves
- tangent vectors
- velocity
- acceleration
- partial derivatives
- directional derivatives
- the gradient
- the chain rule
- Taylor's theorem
- local maxima and minima
- level surfaces of functions
- tangent planes to surfaces in 3-space
- Lagrange multipliers

- Differential Calculus for functions from n-space to m-space
- notion of derivative
- affine function with best approximates a differentiable function at a point
- chain rule
- inverse function theorem
- implicit function theorem

- Integral Calculus in several variables
- The integral, path and surface integrals
- Green's theorem in the plane
- the divergence theorem in 3-space
- the change of variables formula

## References:

The first nine chapters of Principles of Mathematical Analysis, 3rd edition, by Walter Rudin.

Other useful references:- Wendell Fleming:
*Functions of Several Variables* - Kennan T. Smith:
*Primer of Modern Analysis* - Harold Edwards:
*Advance Calculus of Several Variables* - Marsden and Hoffman:
*Elementary Classical Analysis*