
# 14.1: Functions of Several Variables

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In single-variable calculus we were concerned with functions that map the real numbers $$\mathbb{R}$$ to $$\mathbb{R}$$, 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 $$f\colon\mathbb{R}\to\mathbb{R}^3$$, 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 $$f\colon \mathbb{R} ^n\to\mathbb{R}$$. We will deal primarily with $$n=2$$ and to a lesser extent $$n=3$$; in fact many of the techniques we discuss can be applied to larger values of $$n$$ as well.

A function $$f\colon \mathbb{R} ^2\to\mathbb{R}$$ maps a pair of values $$(x,y)$$ to a single real number. The three-dimensional coordinate system we have already used is a convenient way to visualize such functions: above each point $$(x,y)$$ in the $$x$$-$$y$$ plane we graph the point $$(x,y,z)$$, where of course $$z=f(x,y)$$.

Example $$\PageIndex{1}$$:

Consider $$f(x,y)=3x+4y-5$$. Writing this as $$z=3x+4y-5$$ and then $$3x+4y-z=5$$ we recognize the equation of a plane. In the form $$f(x,y)=3x+4y-5$$ the emphasis has shifted: we now think of $$x$$ and $$y$$ as independent variables and $$z$$ as a variable dependent on them, but the geometry is unchanged.

Example $$\PageIndex{2}$$:

We have seen that $$x^2+y^2+z^2=4$$ represents a sphere of radius 2. We cannot write this in the form $$f(x,y)$$, since for each $$x$$ and $$y$$ in the disk $$x^2+y^2 < 4$$ there are two corresponding points on the sphere. As with the equation of a circle, we can resolve this equation into two functions, $$f(x,y)=\sqrt{4-x^2-y^2}$$ and $$f(x,y)=-\sqrt{4-x^2-y^2}$$, representing the upper and lower hemispheres. Each of these is an example of a function with a restricted domain: only certain values of $$x$$ and $$y$$ make sense (namely, those for which $$x^2+y^2\le 4$$) and the graphs of these functions are limited to a small region of the plane.

Example $$\PageIndex{3}$$:

Consider $$f=\sqrt x+\sqrt y$$. This function is defined only when both $$x$$ and $$y$$ are non-negative. When $$y=0$$ we get $$f(x,y)=\sqrt x$$, the familiar square root function in the $$x$$-$$z$$ plane, and when $$x=0$$ we get the same curve in the $$y$$-$$z$$ plane. Generally speaking, we see that starting from $$f(0,0)=0$$ this function gets larger in every direction in roughly the same way that the square root function gets larger. For example, if we restrict attention to the line $$x=y$$, we get $$f(x,y)=2\sqrt x$$ and along the line $$y=2x$$ we have $$f(x,y)=\sqrt x+\sqrt{2x}=(1+\sqrt2)\sqrt x$$.

### Contributors

• Integrated by Justin Marshall.