Download Document

The Gradient and Applications
ƒ This unit is based on Sections 9.5 and 9.6 , Chapter 9.
ƒ All assigned readings and exercises are from the textbook
ƒ Objectives:
Make certain that you can define, and use in context, the terms,
concepts and formulas listed below:
1. find the gradient vector at a given point of a function.
2. understand the physical interpretation of the gradient.
3. find a multi-variable function, given its gradient
4. find a unit vector in the direction in which the rate of change
is greatest and least, given a function and a point on the
function.
5. the rate of change of a function in the direction of a vector.
6. find normal vector and tangent vectors to a curve
7. write equations for the tangent line and the normal line.
8. find an equation for the tangent plane to a surface
ƒ Reading: Read Section 9.5, pages 474-482.
ƒ Exercises: Complete problems 9.5 and 9.6
ƒ Prerequisites: Before starting this Section you should . . .
9 familiar with the concept of partial differentiation
9 be familiar with vector functions
1
Directional Derivatives: definition
ƒ Consider the temperature T at
various points of a heated metal
plate.
ƒ Some contours for T are shown in
the diagram.
T=15
T=20
A
T=25
B
ƒ We are interested in how T changes from one point to another.
2
Directional Derivatives: definition
T=15
ƒ The rate of change of T in the direction
specified by AB is given by:
T=20
(20-15)/AB = 5/h
an examples of directional derivative
A
T=25
B
ƒ In general, for a given function T = T(x,y), the directional
derivative in the direction of a unit vector u = < cosθ, sinθ> is
T ( x + h cosθ , y + h sin θ ) − T ( x, y )
,
Du (T ) = lim
h
h→0
where
h = ( ∆x ) 2 + ( ∆y ) 2
3
Directional Derivatives: definition
A
the gradient vector at a point A
¾ magnitude = the largest directional derivative, and
¾ pointing in the direction in which this largest directional
derivative occurs, is known as the gradient vector.
4
The gradient vector: grad
ƒ A vector field, called the
gradient, written:
grad F or ∇F
can be associated with a scalar
field F.
ƒ At every point the direction of
the vector field (∇F) is
¾ orthogonal to the scalar field
contour (C) and
¾ in the direction of the
maximum rate of change of F.
∂ ˆ ∂ ˆ ∂
ˆ
∇=i + j +k
∂x
∂y
∂z
∇ called ‘del’
5
The gradient vector: grad
ƒ Gradient of a Function
Given : w = F ( x, y , z )
∂F
∂F
∂F
∴ grad F = ∇F ( x, y , z ) = i
+j
+k
∂x
∂y
∂z
Example:
Given : F = xy / z , Find ∇F
2
3
6
The gradient vector (Cont.)
ƒ Key points:
¾ ∇F direction is the normal vector to the surface
∇F • tˆ = 0, tˆ is a tangent unit vector to F
¾ |∇F| magnitude gives the rate of change (the slope) (rate of
change) of F, when moving along a certain direction.
¾ ∇F points in the direction of most rapid increase of F.
¾ -∇F points in the direction of most rapid decrease of F.
¾ F is a scalar field while ∇F is a vector field.
¾ ∇F is not constant in space.
T
7
The gradient vector (Cont.)
ƒ Generalization of Directional Derivative in u direction:
tˆ
F(x,y)=C
∇F
u
Du ( F ) = ∇F • uˆ , uˆ = unit vector
ƒ The maximum value of Du(F) is ||∇F|| and occurs when u and ∇F
are in the same directions.
ƒ The minimum value of Du(F) is -||∇F|| and occurs when u and ∇F
are in the opposite directions.
r
Example: F = xy /( x + y ); u =< 6,8 > Find Du (F) @ ( 2,−1).8
The gradient vector: Some Applications
ƒ Engineers use the gradient vector in many physical laws such as:
1. Electric Field (E) and Electric Potential (V):
r
E ( x, y , z ) = −∇V ( x, y , z )
2. Heat Flow (H) and Temperature (T):
r
H ( x, y, z ) = k∇T ( x, y , z ), k = constant
3. Force Field (F) and Potential Energy (U):
r
F ( x, y , z ) = −∇U ( x, y , z )
Example
Given : T = 100 − 2 x − y,
Find the heat flow vector H
2
9
Insect Example!
ƒ Temperature Distribution:
T ( x, y ) = 5 + 2 x 2 + y 2
Coolest location : center
What direction the insect should go to cool off the fastet?
(Assuming the insect is
familiar with vector analysis
and knows the temperature
distribution!!)
d = −∇T ( x, y ) = −4 xi − 2 yj
= -16i- 4 j
10
Equation of Tangent Plane
ƒ A vector normal to the surface F(x,y,z) = c
at a point P (xo,yo,zo) is ∇F, and can be
denoted by no.
ƒ If ro is the position vector of the point P
relative to the origin, and r is the position
vector of any point on the tangent plane,
the vector equation of the tangent plane is:
r
r r
r
no • (r − ro ) = 0, no = ∇F ( ro ) at P
ƒ The equivalent scalar equation of the tangent plane is:
(x − xo)Fx (xo, yo, zo) + ( y − yo)Fy (xo, yo, zo) + (z − zo)Fz (xo, yo, zo) = 0
Example:
Surface : x 2 + y 2 + z 2 = 9, Point : ( −2,2,1)
11
Equation of Normal Line to a Surface
ƒ A vector normal to the surface F(x,y,z) = c
at a point P (xo,yo,zo) is no = ∇F.
ƒ If ro is the position vector of the point P,
and r is the position vector of any point on
the normal line, the vector equation of the
normal line to the surface is:
r
r
r r
r
no × (r − ro ) = 0, no = ∇F ( ro ) at P
ƒ The equivalent parametric equation of the normal line is:
x = xo + t Fx (xo, yo, zo),
y = yo + t Fy (xo, yo, zo),
z = zo + t Fz (xo, yo, zo)
Example:
Surface : x 2 + 2 y 2 + z 2 = 4, Point : (1,−1,1)
12