Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 131Remark A developable sugace is one which can be obtained from a plane by bendingwithout stretching. Examples are cones, cylinders, and the surfaces of this problem. (SeeChapter 2 of the book by Struik listed at the end of the chapter.).: fLet F(x, y, z) be given in a domain D of space. To compute the partial derivative13Fla.x at a point (x, y, z) of this domain, one considers the ratio of the change A Fin the function F from (x, y, z) to (x + Ax, y, z) to the change Ax in x. Thus onlythe values of F along a line parallel to the x axis are considered. Similarly, aF/ayand aF/az involve a consideration of how F changes along parallels to the y and zaxes, respectively. It appears unnatural to restrict attention to these three directions.Accordingly, one defines the directional derivative of F in a given direction as thelimit of the ratioof the change in F to the distance As moved in the given direction, as As approaches0.Let the direction in question be given by a nonzero vector v. The directionalderivative of F in direction v at the point (x , y, z) is then denoted by V, F(x , y, z) or,more concisely, by V, F. A displacement from (x, y, z) in direction v correspondsto changes Ax, Ay, Az proportional to the components v, , v,, v,; that is,Ax = hv,, Ay = hv,, Az = hv,, (2.111)where h is a positive scalar. The displacement is thus simply the vector hv, and itsmagnitude As is h (vl. The directional derivative is now by definition the limit:V, F = limh+OF(x +hv,,y +hv,,z+hv,)-hlvlF(x, y,z)(2.112)In this definition we have allowed h to approach 0 through positive or negativevalues, just as in the definition of the derivative dyldx as the limit of AylAx as Axapproaches 0 through positive or negative values. One can think of the directionalderivative of F at (x , y, z) in direction v as the derivative of a function of one variableobtained by considering F as a function of a coordinate along an axis through (x , y , z)in the direction of v with the same unit of distance as along the x , y, and z axes.If F has a total differential at (x, y, z), then, as in Section 2.6,Thus