Multivariate Calculus - Bruce E. Shapiro
Multivariate Calculus - Bruce E. Shapiro
Multivariate Calculus - Bruce E. Shapiro
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86 LECTURE 12. THE CHAIN RULE<br />
Substituting for x, y, and z,<br />
∂w<br />
∂s<br />
= ( 3(st) 2 + s + t ) t + 1 + st + 2(s + 4t)<br />
= 3s 2 t 3 + st + t 2 + st + 2s + 8t<br />
= 3s 2 t 3 + 2st + t 2 + 2s + 8t <br />
Generalization to Higher Dimensions<br />
If f is a function of multiple variables,<br />
f(u, v, w, x, y, z, ...) = f(u(t), v(t), w(t), x(t), ...)<br />
The figures are generalized in the obvious way; in between the node for f and the<br />
node for t we put nodes for each of u, v, w, x,... and so forth, and draw arrows as<br />
before. There are still only two derivatives in each path, but there are a whole lot<br />
more paths, and we have to add up the products over each path. The result is<br />
df<br />
dt = ∂f du<br />
∂u dt + ∂f dv<br />
∂v dt + ∂f dw<br />
∂w dt + ∂f dx<br />
∂x dt + ∂f dy<br />
∂y dt + ∂f dz<br />
∂z dt + · · ·<br />
Usually mathematicians use an indexed variable when the number of variables becomes<br />
large, and write a function of n-variables as<br />
f(x 1 , x 2 , x 3 , ..., x n )<br />
This is considered a function in n-dimensional space, sometimes called R n .<br />
chain rule for a function in R n is<br />
df<br />
n∑<br />
dt = ∂f dx i<br />
∂x i dt<br />
i=1<br />
The<br />
Relationship of Chain Rule to the Gradient Vector and<br />
the Directional Derivative<br />
If x, y, and z are functions of a parameter t, then the position vector<br />
r(t) = (x(t), y(t), z(t))<br />
traces out a curve in three dimensions. The rate of change of any function f (x,y,z)<br />
as a particle moves along this curve is df /dt. Using the chain rule, this derivative is<br />
df<br />
dt<br />
= ∂f dx<br />
∂x dt + ∂f dy<br />
∂y dt + ∂f dz<br />
∂z dt<br />
( ∂f<br />
=<br />
∂x , ∂f<br />
∂y , ∂f ) ( dx<br />
·<br />
∂z dt , dy<br />
dt , dz )<br />
dt<br />
= ∇f · d<br />
dt r(t)<br />
= v · ∇f<br />
Revised December 6, 2006. Math 250, Fall 2006