Calculus: Analytic Geometry and Calculus, with Vectors

564 Partial derivatives

Looking forward to derivation of a formula (the chain formula) for w'(t)
we write

Thus

where

w(t + At) = u(x(t + At), y(t + At), z(t + At)).

w(t + At) = u(x(t) + Ax, Y(t) + AY, Z(t) + Oz),

Ax = x(t + At) - x(t), AY = Y(t + At) - y(t), As = z(t + At) - z(t).

Applying (11.222) then gives

w(t + At) - w(t) = [t{=(x(t), y(t), z(t)) + Ei][x(t + At) - x(t)]

+ [uv(x(t), y(t), z(t)) + e21[y(t + At) - Y(t)]

+ [tts(x(t), Y(t), z(t)) + e3][z(t -}- At) - z(t)],

where el, e2, e3 are quantities which approach zero as At approaches zero.

Dividing by At and taking limits as At approaches zero gives the chain

formula (11.232). The following theorem sets forth conditions under

which the chain formula is correct.

Theorem 11.23 (chain rule) If u is continuous and has continuous

partial derivatives us, u,,, u, over a region R in E3, and if

(11.231) w(t) = u(x(t), y(t), z(t)),

where x, y, z are differentiable functions oft over some interval T, and if the

point P(t) having coordinates x(t), y(t), z(t) traverses a curve C in R as t

increases over T, then w is differentiable over T and

(11.232) w'(t) = u,,(x(t), y(t), z(t))x'(t)

+ u,(x(t), y(t), z(t))y'(t)

+ u2(x(t), y(t), z(t))z'(t)

when t is in T.

In case [x'(t)]2 + [y'(t)]2 + [z'(t)]2 = 1 so thatP(t) moves along C with

unit speed, the number w'(t) is called the directional derivative of u in the

direction of the forward tangent to C at P. For this and other reasons,

some of which will appear later, the chain formula (11.232) is extremely

important. We temporarily suspend production of mathematics to

ponder consequences of the rude fact that it takes a long time to write

the formulas (11.231) and (11.232). We can wonder how much we can

abbreviate these formulas without abbreviating the life out of them and

without creating confusion that wastes more of our time than the abbre-

viations save. We can write (11.231) in the form

(11.233) u = u(x,y,z)

and carry in our minds the idea that the left side stands for the value of

a function of t and so also does the right side but, on the right side, u is