Multivariate Calculus - Bruce E. Shapiro

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