Advanced Calculus fi..

84 Advanced Calculus. Fifth EditionFigure 2.4Partial derivatives.are z,, f,, fl or, more explicitly, 7,(xl, yl), f, (sl. yl). fl (r,,.I). When subscriptsare used, there can be confusion with the symbol for components of a vector; hencewhen vectors and partial derivatives are being used rogerhe~ a notation s~lch asaz/a.r or a f 1a.r is preferable for partial derivatives.The function z = f (x, y) can be represented by a surface in space. The equatlony = yl then represents a plane cutting the surface in a curve. The partial derivativeaz/d.r at (.rl, yl) can then be interpreted as the slope of the tangent to the curve, thatis, as tan a, where a is the angle shown in Fig. 2.4. In this figure, : = 5 + x' - v',and the derivative is being computed at the point x = 1, v = 2. For y = 2. z =1 + r', so that the derivative along the curve is 2.r; for r = I, one finds f,(1, 2)to be 2.The partial derivative 5 is defined similarly; one now holds r constant.equal to xi, and differentiates f (.rl. v) with respect to y. One has thusThis can also be interpreted as the slope of the tangent to the curve in which theplane .r = s, cuts the surface z = f (x, y). One also writes f,(.r,, yl), f2(.rl, yl) forthis derivative.If the point (XI, yl) is now varied, one obtains (wherever the derivative exists) anew function of two variables: the function ,f,(.r, y). Similarly, the derivative a:/ayat a variable point (s, y) is a function f,(.r, y). For explicit functions r = f (.r, y),evaluation of these derivatives is carried out as in ordinary calculus, for one is alwaysdifferentiating a function of one variable, the other being treated as a cmstant. For7 7example, if ,: = .r- - y-, then