Calculus: Analytic Geometry and Calculus, with Vectors

11.2 Increments, chain rule, and gradients 565

linked to t by the intermediate variables x, y, z which are functions of t.
We can then abbreviate (11.232) to

du au dx audy au dz

(11.234)

dt - ax dt + ay St + -Oz dt

It turns out that, in practice, the version (11.234) of the chain formula
is often very convenient. For example, if

u = x2 + y2 + z2, x = sin t, y = et, Z = t2,
then

du

= 2x cos t + 2ye° + 4zt

at

and it is, in fact, not always necessary or even desirable to express the
right side entirely in terms of t. The abbreviations are not always so
agreeable, however. In case the parameter t is x itself, we must employ
considerable fortitude to comprehend the sentence containing (11.233)
and the formula

(11.235)

du_au +audy+au dz

dx ax ay dx az dx

It is awkward and perhaps even undesirable to be required to think of u

depending upon x in two different ways. It is easier to let

and to write

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

dw_au dx au dy au A

dx axdx+aydx+azdx

and then observe that dx/dx = 1. While Theorem 11.23 is the fully
meaningful theorem to which we can refer when meanings of symbols
must be carefully set forth, we give a restatement of the theorem in
terms of the simpler curly dee notation.
Theorem 11.24 (chain rule, second version) If u is continuous and
has continuous partial derivatives au/ax, au/ay, au/az over a region R in
Ea, and if

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

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

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

over T, then is is differentiable over T and

du_au d x audy au dz

(11.242) dt ax YT+ay 77+az dt

It is not often that we have an opportunity to make an observation