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14 Partial Differentiation In single-variable calculus we were concerned with functions that map the real numbers R to 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 : R 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 : R n 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 : R 2 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 14.1 Consider f ( x, y ) = 3 x + 4 y 5. Writing this as z = 3 x + 4 y 5 and then 3 x +4 y z = 5 we recognize the equation of a plane. In the form f ( x, y ) = 3 x +4 y 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 14.2 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 323 324 Chapter 14 Partial Differentiation this equation into two functions, f ( x, y ) = radicalbig 4 x 2 y 2 and f ( x, y ) = radicalbig 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 4) and the graphs of these functions are limited to a small region of the plane. EXAMPLE 14.3 Consider f = x + y . This function is defined only when both x and y are non-negative. When y = 0 we get f ( x, y ) = 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 x and along the line y = 2 x we have f ( x, y ) = x + 2 x = (1 + 2) x . 10.0 7.5 5.0 0 0.0 y 2.5 1 2.5 5.0 x 2 7.5 0.0 10.0 3 4 5 6 Figure 14.1 f ( x, y ) = x + y (JA) A computer program that plots such surfaces can be very useful, as it is often difficult to get a good idea of what they look like. Still, it is valuable to be able to visualize relatively simple surfaces without such aids. As in the previous example, it is often a good idea to examine the function on restricted subsets of the plane, especially lines. It can also be useful to identify those points ( x, y ) that share a common z -value.
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