CHAPTER VI. PARTIAL DIFFERENTIATION. 132. Functions of Several Independent Variables. Our attention has hitherto been confined to methods for the differentiation of functions of a single independent variable. In the present chapter we propose to discuss the case in which several such variables occur. Such functions are common; for instance, the area of a triangle depends upon two variables, viz., the base and the altitude; while the volume of a rectangular box depends upon three, viz., its length, breadth, and depth; and it is plain that each of these variables may vary independently of the others. 133. Partial Differentiation. If a differentiation of a function of several independent variables be performed with regard to any one of them just as if the others were constants, it is said to be a partial differentiation. called partial differential coefficients with regard to x, y, The meanings of the differential coefficients thus formed are clear; for if we denote u by f(x, y, z) the operation It will throw additional light upon the subject of partial differentiation if we explain the geometrical meaning of the process for the case of two independent variables. Let PQRS be an elementary portion of the surface z=f(x, y) cut off by the four planes Y=y, Y=y+dy [Capital letters representing so that the co-ordinates of the corners P, Q, R, S are If PLMN be a plane through P, parallel to the plane of xy, and cutting the ordinates of P, Q, R, S in P, L, M, N respectively, we have considering y a constant is = Ltεx=0 LQ f(x+8x, y) − f(x, y) = Lt12 = Lt tan LPQ. (2) δα =tangent of the angle which the tangent at P to the curved section PQ (parallel to the plane xz) makes with a line drawn parallel to the axis of x. Əz Similarly which is obtained on the supposition Əy' that x is constant = Lt tan NPS, .(3) =tangent of the angle which the tangent at P to a section parallel to the plane of yz makes with a parallel to the axis of y. 136. If the tangent plane at P to the surface cut LQ, MR, NS in Q,' R', S' respectively, Also the section made on the tangent plane by the four bounding planes of the element is a parallelogram, and the height of its centre above the plane PLMN is given by MR' and also by (LQ'+NS'), which proves that MR' = LQ'+NS' 2 The expressions proved in (4), (5), and (6) are first approximations to the lengths LQ, NS, and MR respectively, and differ from those lengths by small quantities of higher order than PL and PN, and which are therefore negligible in the limit when dx and dy are taken very small. The investigation of the total values of LQ, NS, MR must be postponed until we have investigated the extension of Taylor's Theorem to functions of several variables. (Art. 156.) 137. Differentials. It is useful at this point to introduce a new notation, which will prove especially convenient from considerations of symmetry. I Let Dx, Dy, Dz be quantities either finite or infinitesimally small whose ratios to one another are the same as the limiting ratios of dx, dy, dz, when these latter are ultimately diminished indefinitely. We shall call the quantities thus defined the differentials of x, y, z. Also, as we shall be merely concerned with the ratios of these quantities, and any equation into which they may enter will be homogeneous in them, it is unnecessary to define them farther or to obtain absolute values for them. The student is warned again (see Art. 41) that the differential coefficient dy dx is to be considered as the result of performing the d operation represented by upon y, an operation de Ax The dy and dx of the symbol dy da scribed in Art. 39. cannot therefore be separated, and have separately no meaning, and hence have no connection with the differentials Dx and Dy as defined in the present article; but at the same time we have by definition Dy: Dx Limit of the ratio dy : Sx |