Advanced Calculus fi..

164 Advanced Calculus, Fifth Edition$49Figure 2.24Level curves of functionally dependent functions.point (xl, yl) of D, then on proceeding from this point in a direction normal to thelevel curve, f must either increase or decrease, since V f # 0; similarly, g musteither increase or decrease. Thus a sufficiently small neighborhood in the xy-planeis mapped by (2.164) on a curve in the uv-plane expressible either as u = $(v) or asv = $(u). Thus f (x, y) - $(g(x, y)) = 0 in the neighborhood. Along the normal,in terms of a suitable coordinate t, f and g are differentiable functions of t and at(x,, y,) ul(t) = a f /an = Vf . n # 0 in Eq. (2.1 17) and similarly vl(t) # 0, so thatdu/dv = ul(t)/v'(t) = +'(v). Hence $ is differentiable, as is F(u, v) = u - @(v)and f, g are functionally dependent in the neighborhood.The proof just given brings out the significance of the condition a(f, g)/a(x, y) E 0 for the mapping (2.164): This mapping maps D not onto a domainbut onto a curve or several curves. For the functions f = ex sin y, g = x +log sin ythe corresponding curve is given by part of the graph of log u - v = 0.The results obtained can be generalized to the case of three functions of threevariables or, in general, to n functions of n variables. Thus for three functions ofthree variables, functional dependenceis equivalent as previously to the condition 9This in turn is equivalent to the statement that the three vectors Of, Vg, Vh arecoplanar at each point, so that the three families of level surfaces have a commontangent direction at each point.One can also consider the case of n functions of m variables: fl (x,,..., x,,,), ...,fn(xl, ..., x,) and allow for several functional dependencies among them.Linear case. Here f, (x, , . . .) = a, ~ x,+ . . . + a,,x, for i = 1, . . . , n or, in vectornotation, f(x) = Ax, with A = (a,,) a constant n x m matrix. We restrict attentionto linear dependencies, of formZI fi + . . . + znfn 0,where zl, . . . , z, are constants, not all 0. (A general functional dependenceF(fi, . . . , f,) r 0 implies a linear one; see Problem 11.) Equation (2.165) is3"