Vector Field

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Vector FieldsMATH 311, Calculus III

J. Robert Buchanan

Department of Mathematics

Fall 2011

J. Robert Buchanan Vector Fields

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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.


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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 ).


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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).


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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


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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 ).


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 .


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


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


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


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


Flow Lines

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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).


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.


Example

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Example

Consider the vector field F(x , y ) = x i − y j. Sketch the vector


1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

x

y


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


Example

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Example

Consider the vector field F(x , y ) = e −x , 2x . Sketch the vector


1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

x

y


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.


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

.


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


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


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


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


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Example (3 of 4)



p ( )


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 .


Example (4 of 4)

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p ( )


F(x , y ) = ye x + sin y , e x + x cos y + y 2


Example (4 of 4)

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p ( )


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


Homework

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Read Section 14.1.

Exercises: 1–59 odd


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Vector Field