How is multivariable calculus useful?

Answer 1

I am assuming you understand the distinction between single-variable calculus (calculus of one variable) and multivariable calculus (calculus of several variables).

Well, if you know the former, that is highly beneficial because the same techniques are used in the latter -- they are generalized to apply to calculus of n-variables. This is ultimately the goal of single-variable calculus.

Why?

Well, if you think about it, single-variable is not really applicable. Not many real world phenomena involve one variable. For example, in macroeconomics, GDP = Y is a function of many variables: Consumption (a function of net taxes and income), Investment (a function of real interest rates), Government Spending, and Net Exports. That is, Y=f(C(Y,T), I(r), G, NX). To perform many of the tools of calculus (e.g. finding how Y changes as G increases) to this function, one must know and apply multivariable calculus.