The n-dimensional Euclidean space, functions of several variable: Scalar product and the induced Euclidean norm on Rⁿ. Cauchy-Schwarz inequality. The basic properties of the norm. Examples: p-norms on Rⁿ. Topology of Rⁿ: open, closed, compact sets, interior, boundary. Cauchy sequences, completeness of Rⁿ. Borel-Lebesgue theorem for compact sets (possibly without proof). Limits and continuity of functions of several variable. The topological characterization of continuity, the level sets of continuous functions are open or closed. Convex sets, separation of convex sets and an exterior point, separation of two disjoint convex bodies (possibly without proof). 

Differentiation of functions of several variable: Differentiable functions. Partial derivatives, gradient, tangent hyperplane, Jacobi matrix, Jacobi determinant, chain rule. Continuously differentiable functions, higher order derivatives, Young theorem, Taylor formula (specifically of order two).  Multilinear mappings, positive and negative definite mapping. Convex functions and the second derivative. Local maximum, minimum and connection to the derivatives. Conditional maxima and minima, Lagrange multiplier. Banach fixpoint theorem, inverse function theorem, implicit function theorem. Rotation, divergence, nabla symbol, Laplace operator. The existence of scalar potential.  

Integration of functions of several variable: Definition and properties of the Jordan measure (without proofs). Integration of a continuous function of several variable on an n-dimensional domain. Integration by substitution: polar coordinates, spherical coordinates. Integration along a path, and along a surface.  Divergence theorem, Stokes theorem, Green theorems, (the proof of these theorems is only sketched). 

Function sequences and series: Pointwise convergence of a sequence or series of functions. Absolute convergence of a series of functions. Uniform and locally uniform convergence. The space of continuous functions with the sup-norm. Weierstrass criterion. Interchanging differentiation and the limit, integration and the limit. Term-by-term differentiability and integrability of a series of functions. Recall: properties of power series.   

Fourier series:  Fourier coefficients, Fourier series. The Fourier series of a twice continuously differentiable periodic function converges uniformly.