By A. F. Bermant
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Additional info for A Course of Mathematical Analysis, Part II
0). Hence (see Sec. hand side of equation (*) is the differential dz: dz = (f~
In fact, the general form of equation between x, y and z is f(x, y, z) = o. On substituting any given values x = xo' y = Yo for x and y, we obtain an equation in z: f(x o, Yo' z) = 0, from which the value (or values) of z can be found corresponding to x· Xo and y = yo' Hence z is given as a definite function of x and y by the equation f(x, y, z) = O. Such a function is said to be implicit. Definition. A function defined by an equation between x, y and z and not solved with respect to z is said to be an implicit function z of the two independent variables x and y.
Lim f(P') -; f(Po) = lim f(P) - f(Po) . P' ..... p. P Po P-+P. P Po In particular, Hthe curve Lis a level line of the function z = f(P), we have f(P) = f(Po) when P belongs to L, so that azjos = o. Thus the derivative of a function z = f (P) with respect to one of its level lines is equal to zero. This can be taken as characterizing a level line as a line of constant values of the function. 42 COURSE OF MATHEMATICAL ANALYSIS III. We shall take some examples of directional differentiation. . Example 1.