Topology in \mathbbRn: Open sets, closed sets, compact sets, Heine-Borel theorem, path connectedness in \mathbbRn.
Differential Calculus: Directional derivatives and its drawbacks, total derivative, comparison with differentiability on \mathbbR, chain rule and its applications, Ck functions, mixed derivatives, Taylors theorem, smooth functions with compact supports, inverse function theorem, implicit function theorem and the rank theorem, examples, maxima and minima, critical point of the Hessian, constrained extrema and Lagranges multipliers, examples.
Integral Calculus: Line integrals, behaviour of line integral under a change of parameter, independence of path, conditions for a vector field to be a gradient, concept of potential and its construction on convex sets, multiple Riemann integrals, Fubinis theorem, change of variables, integration by parts.
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