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Transcript of Vector Field
7/31/2019 Vector Field
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Vector FieldsMATH 311, Calculus III
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan Vector Fields
7/31/2019 Vector Field
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Background
We have already discussed vector-valued functions which
are functions of the form r : R→ V 2 (or r : R→ V 3). They are
useful for describing a path in 2- or 3-dimensional space.
Vector field functions are functions of the form F : R2 → V 2(or F : R3 → V 3) which are useful for describing flows of fluids
in 2- or 3-dimensional space.
J. Robert Buchanan Vector Fields
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Definition
Definition
A vector field in the plane is a function F(x , y ) mapping points
in R2 into the set of two-dimensional vectors V 2. We write
F(x , y ) = f 1(x , y ), f 2(x , y ) = f 1(x , y )i + f 2(x , y )j,
for scalar functions f 1(x , y ) and f 2(x , y ). In space, a vectorfield is a function F(x , y , z ) mapping points in R3 into the set of
three-dimensional vectors V 3. We write
F(x , y , z ) = f 1(x , y , z ), f 2(x , y , z ), f 3(x , y , z )
= f 1(x , y , z )i + f 2(x , y , z )j + f 3(x , y , z )k,
for scalar functions f 1(x , y , z ), f 2(x , y , z ), and f 3(x , y , z ).
J. Robert Buchanan Vector Fields
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Example (1 of 2)
Consider the vector field function F(x , y ) = −y , x and plot
F(1, 0), F(0, 1), and F(1, 1).
J. Robert Buchanan Vector Fields
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Example (2 of 2)
1.0 0.5 0.0 0.5 1.0
0.0
0.5
1.0
1.5
2.0
x
y
J. Robert Buchanan Vector Fields
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Graph of a Vector Field
The graph of a vector field F(x , y ) is a two-dimensional graph
with vectors F(x , y ) plotted with initial points at (x , y ).
J. Robert Buchanan Vector Fields
G
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Graph of a Vector Field
The graph of a vector field F(x , y ) is a two-dimensional graph
with vectors F(x , y ) plotted with initial points at (x , y ).
Example
Consider the graph of the vector field function F(x , y ) = −y , x .
J. Robert Buchanan Vector Fields
E l
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Example
F(x , y ) = −y , x
1.0 0.5 0.0 0.5 1.0
1.0
0.5
0.0
0.5
1.0
x
y
J. Robert Buchanan Vector Fields
E l
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Example
F(x , y ) = sin x , cos y
2 Π
3 Π
2
Π
Π
2
0 Π
2
Π 3 Π
2
2 Π
2 Π
3 Π
2
Π
Π
2
0
Π
2
Π
3 Π
2
2 Π
x
y
J. Robert Buchanan Vector Fields
M t hi E i
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Matching Exercise
Match the following functions to the graphs of the vector fields.
F(x , y ) = y , x
G(x , y ) = 2x − 3y , 2x + 3y H(x , y ) = sin x , sin y
K(x , y ) = ln(1 + x 2 + y 2), x
J. Robert Buchanan Vector Fields
V t Fi ld i 3 Di i
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Vector Fields in 3 Dimensions
F(x , y , z ) = 0, 0, z
1
0
1
x
1
0
1
y
1
0
1
J. Robert Buchanan Vector Fields
Flow Lines
7/31/2019 Vector Field
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Flow Lines
We may think of the vector field F(x , y ) = f 1(x , y ), f 2(x , y ) as
indicating the direction of movement of an object at each pointin space.
If the object is initially at (x 0, y 0) its future position is given by
the solution to the differential equation:
x (t ) = f 1(x (t ), y (t ))
y (t ) = f 2(x (t ), y (t ))
(x (0), y (0)) = (x 0, y 0)
The parametric plot of the solution generates a flow line
through (x 0, y 0).
J. Robert Buchanan Vector Fields
Example
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Example
Consider the vector field F(x , y ) = x i − y j. Sketch the vector
field and several flow lines with different initial conditions.
J. Robert Buchanan Vector Fields
Example
7/31/2019 Vector Field
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Example
Consider the vector field F(x , y ) = x i − y j. Sketch the vector
field and several flow lines with different initial conditions.
1.0 0.5 0.0 0.5 1.0
1.0
0.5
0.0
0.5
1.0
x
y
J. Robert Buchanan Vector Fields
More Flow Lines
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More Flow Lines
1.0 0.5 0.0 0.5 1.0
1.0
0.5
0.0
0.5
1.0
x
y
J. Robert Buchanan Vector Fields
Example
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Example
Consider the vector field F(x , y ) = e −x , 2x . Sketch the vector
field and several flow lines with different initial conditions.
1.0 0.5 0.0 0.5 1.0
1.0
0.5
0.0
0.5
1.0
x
y
J. Robert Buchanan Vector Fields
Gradient and Potential
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Gradient and Potential
Definition
For any scalar function f , the vector field F = ∇f is called the
gradient field of function f . We call f a potential function forF. Whenever F = ∇f , for some scalar function f , we refer to F
as a conservative vector field.
J. Robert Buchanan Vector Fields
Example (1 of 4)
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Example (1 of 4)
Find the gradient field of f (x , y , z ) = x cos
y z
.
J. Robert Buchanan Vector Fields
Example (1 of 4)
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Example (1 of 4)
Find the gradient field of f (x , y , z ) = x cos
y z
.
∇f (x , y , z ) =
cosy
z
,−x
z siny
z
,xy
z 2 siny
z
J. Robert Buchanan Vector Fields
Example (2 of 4)
7/31/2019 Vector Field
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Example (2 of 4)
Find the gradient field of
f (x , y , z ) =mMG
x
2
+ y
2
+ z
2
J. Robert Buchanan Vector Fields
Example (2 of 4)
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Example (2 of 4)
Find the gradient field of
f (x , y , z ) =mMG
x
2
+ y
2
+ z
2
∇f (x , y , z ) = −mMG
(x 2 + y 2 + z 2)3/2x , y , z
J. Robert Buchanan Vector Fields
Example (3 of 4)
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Example (3 of 4)
Find a potential for the vector field
F(x , y ) = 2x cos y − y cos x ,−x 2 sin y − sin x
J. Robert Buchanan Vector Fields
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Example (3 of 4)
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p ( )
Find a potential for the vector field
F(x , y ) = 2x cos y − y cos x ,−x 2 sin y − sin x
If ∇f (x , y ) = F(x , y ) then
∂ f
∂ x = 2x cos y − y cos x
f (x , y ) = x 2 cos y − y sin x + h (y )
where h (y ) is an arbitrary function of y only.
∂ f
∂ y = −x 2 sin y − sin x = −x 2 sin y − sin x + h (y )
h
(y ) = 0Thus h (y ) = C , a constant and
f (x , y ) = x 2 cos y − y sin x + C .
J. Robert Buchanan Vector Fields
Example (4 of 4)
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p ( )
Find a potential for the vector field
F(x , y ) = ye x + sin y , e x + x cos y + y 2
J. Robert Buchanan Vector Fields
Example (4 of 4)
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p ( )
Find a potential for the vector field
F(x , y ) = ye x + sin y , e x + x cos y + y 2
f (x , y ) = ye x + x sin y +1
3y 3 + C
J. Robert Buchanan Vector Fields
Homework
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Read Section 14.1.
Exercises: 1–59 odd
J. Robert Buchanan Vector Fields