Warm-Up
Notes
Paper airplanes.
Warm-Up
Notes
Which of the following statements about the gradient of a
differentiable function f (x, y ) is not true?
~ is a vector field.
1. ∇f
~ is orthogonal to the level curves of f .
2. ∇f
~ points in the direction of the greatest rate of change
3. ∇f
of the function (that is, the direction where the slope of
the tangent plane is greatest).
4. All of the above are true.
5. All of the above are lies.
6. I’m not sure what the correct answer is.
receiver channel: 42
session ID: mth275
Notes
Directional Derivatives
&
Chain Rules
sec 14.5, 14.6
12 March 2014
Today, We Will:
Notes
I
Use the geometric definition of the dot product to prove
properties of the gradient.
I
Define and practice computing the chain rule on paths
(chain rule in a single variable).
I
Define and practice computing the general chain rule
(chain rule in several variables).
Properties of the Gradient, Revisited
Notes
Suppose the first partial derivatives of f (x, y ) are continuous.
Then f is differentiable and:
~ is orthogonal to the level curves of f .
1. The gradient ∇f
2. The slope of the tangent plane is greatest in the direction
~ (the gradient points in the direction in which the
of ∇f
function is increasing most rapidly).
3. The value of the slope of the tangent plane in the
~ is k∇f
~ k (the magnitude of the gradient
direction of ∇f
gives the value of the greatest rate of increase of the
function at a point).
(These properties can be proved using directional derivatives.)
Properties of the Gradient 1 and 2.
The derivative Dû f is slope of the tangent plane above the line
spanned by û . Using the geometric definition of the dot product:
~ · û
Dû f = ∇f
~ ||||û || cos θ
= ||∇f
~ || cos θ
= ||∇f
Since −1 ≤ cos θ ≤ 1:
I
Dû f is maximized when cos θ = 1, which occurs when û
~ . (Property 2).
points in the direction of ∇f
I
~ || ≤ Dû f ≤ ||∇f
~ ||, and Dû f = ||∇f
~ || when û points in
−||∇f
~ (Property 3).
the direction of ∇f
Notes
Example: Directional Derivatives and the Gradient.
Notes
The directional derivative of f (x, y , z) at a point P is greatest
in the direction of ~v √
= ı̂ + ̂ − k̂ . In this direction, the value
of the derivative is 2 3.
I
I
~ at P?
What is ∇f
What is the directional derivative of f at P in the
direction of the vector ı̂ + ̂ ?
Clicker Question: Directional Derivatives and the
Gradient.
Notes
~ = 2 ı̂ − 1 ̂ , is
If the gradient of f (x, y ) at a point (a, b) is ∇f
there a direction û such that Dû f (a, b) = 7?
1. Yes (high confidence — definitely yes).
2. Yes (low confidence — maybe yes).
3. No (high confidence — definitely no).
4. No (low confidence — maybe no).
5. There is no way to tell; not enough information is given.
6. I’m not sure how to answer this question.
receiver channel: 42
session ID: mth275
A New Notation for Curves
Suppose a curve is parameterized by the vector-valued function
~r (t) = x(t) ı̂ + y (t) ̂ + z(t) k̂ .
The points on the curve (the terminal points of ~r (t)) will be
denoted:
c(t) = x(t), y (t), z(t) .
c(t) is sometimes called a path.
Notes
Derivatives of Paths
Notes
A path c(t) is differentiable if the coordinate functions x(t),
y (t), and z(t) are all differentiable functions of t.
The derivative of c(t) is defined to be the derivative of ~r (t):
dc
dx
dy
dz
= ~r 0 (t) =
ı̂ +
̂ +
k̂
dt
dt
dt
dt
Recall: The Chain Rule from Calc. 1
Notes
Chain rules apply to taking derivatives of compositions of
functions.
Recall from Calc. I:
d
df du
f (g (t)) =
dt
du dt
where u = g (t).
The Chain Rule Along Paths (One Variable)
Suppose c(t) = x(t), y (t), z(t) is a differentiable path in
the domain of a differentiable function f (x, y , z). Then:
d ~ · d c(t)
f c(t) = ∇f
dt
dt
~ · ~r 0 (t)
= ∇f
=
∂f dx
∂f dy
∂f dz
+
+
∂x dt
∂y dt
∂z dt
Notes
Example: Chain Rule on Paths
f (x, y ) = x 2 y 2
Notes
c(t) = (t, t 2 )
d f c(t) using the chain rule for paths.
dt
I
Compute
I
What is the rate of change of the function along the path
at t = 1?
I
At what time(s) t is
df
= 0? (Note that these are the
dt
times when ~r (t) is orthogonal to the gradient.)
0
I
An ant is skating on the xy -plane. She passes through the
df
point (1, 1) with velocity ~v =< 1, −2 >. What is
dt
from the perspective of the ant?
Discussion Question
Notes
What is the relationship between the chain rule:
df
~ · ~r 0 (t)
= ∇f
dt
and the directional derivative:
0
~ · ~r (t) ?
Dê~r 0 (t) f = ∇f
0
||~r (t)||
Notes