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What you really need to know from multi-variable calculus for economics is
- vectors and planes in R^n.
- the functions and their derivatives in R^n, chain rule, product rule, including a working knowledge of matrix notation of derivatives.
- constrained and unconstrained optimization, including problems subject to inequality constraints and the second order conditions.
- implicit function theorem in R^n, comparative statics.
Unfortunately, most of this is NOT taught in a multi-variable calculus course, even though the adcoms really want to see that course on your transcript. A typical sophomore calculus course is designed for engineers, and so you spend a while learning things like polar coordinates, greens and strokes theorem, line integrals, etc, usually just in R2 or R^3.
In addition, it is useful to have a good intuition to work with sequences, limits, convergence, continuity, closed and open sets, compactness, norms, Cartesian products of sets, etc in R^n (again, this stuff if taught at all is usually taught in a real analysis course).
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