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In single-variable calculus, we were concerned with functions that map the real numbers to real numbers, sometimes called "real functions of one variable'', meaning the "input'' is a single real number and the "output'' is likewise a single real number. In the last chapter we considered functions taking a real number to a vector, which may also be viewed as functions, that is, for each input value we get a position in space. Now we turn to functions of several variables, meaning several input variables, functions. We will deal primarily  $$\mathbb{R}^2$$ and  $$\mathbb{R}^3$$ spaces, however, many of the techniques we discuss can be applied to larger dimensions spaces as well.