Math 275 Notes (Ultman)
Topic 3-6: Chain Rules for Multivariate
Functions
Textbook Sections: 14.5, 14.6
Homework: daily problem set #15
Prerequisites: In order to learn the new skills and ideas presented in this
topic, you must:
Know the chain rule for a function of a single variable (from Calc I).
Know the directional derivative for multivariate functions.
Be able to compute derivatives of parameterized curves (tangent/velocity
vectors).
Be able to compute partial derivatives and gradients of multivariate
funcitons.
Be able to compute dot products.
Learning Objectives & Big Ideas
You need to learn how to:
Apply the chain rule along paths to compute derivatives of multivariate
functions with respect to position along parameterized curve.
Apply the general chain rule to compute derivatives of multivariate functions with respect to general parameters.
Recognize .
The ideas from this topic that you should understand:
The chain rules for multivariate functions are similar to the chain rule
for functions of a single variable, in that they give a way to compute the
derivative of a composite function.
The chain rules for multivariate functions involve the dot product of the
gradient of the function, with an appropriate vector in the domain of the
function.
The chain rule along paths gives a way to compute the derivative of the
function with respect change of position along a curve. It is computed by
taking the dot product of the gradient of the function with the velocity
vector of the curve.
The general chain rule is used to compute derivatives of compositions
of a function of n variables with coordinate functions parametrized by
m variables. Mechanically, the general chain rule is similar to the chain
rule for paths, but all derivatives involved are partial derivatives.
The Big Picture
As with functions of a single variable, the chain rules for multivariate functions
gives a way of computing the derivative of composite functions.
Multivariate chain rules are computed using the gradient of the function and
a dot product.
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More Details
◦ The chain rule for paths gives a way to compute the ordinary derivative
df /dt of the composition f (c(t)). It measures the instantaneous rate
of change of a function with respect to a parameterized curve in the
domain.
◦ The chain rule for paths is computed by taking the dot product of the gra~ ·~
dient of the function with the velocity vector of the curve: df /dt = ∇f
v.
Writing the velocity as speed times the unit vector in the direction of
the velocity: ~
v = k~
v kv̂ , we see:
df
~ ·~
= ∇f
v
dt
~ · k~
= ∇f
v k~
v
~ ·~
= k~
v k ∇f
v
= k~
v kDv̂ f
So, the chain rule along paths is the directional derivative (rate of change
of the function in a given direction) multiplied by the speed of travel in
that direction.
◦ When applicable, the units of the chain rule are the units of the function
values over the units of the parameter — eg: if the function gives the
temperature in degrees celsius and the units of the parameter is time
measured in seconds, then the units of the chain rule will be ◦ C/s).
◦ Using Leibnitz notation and carry through the dot product, the similarities between the various versions of the chain rule become apparent:
single variable (Calc I):
df dx
df
=
dt
dx dt
multivariate, single parameter:
df
∂f dx ∂f dy
=
+
+ ···
dt
∂x dt
∂y dt
multivariate, general:
∂f
∂f ∂x ∂f ∂y
=
+
+ ···
∂t
∂x ∂t
∂y ∂t
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