Section 17.1 - Gordon State College

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Transcript Section 17.1 - Gordon State College

Section 17.1
Vector Fields
VECTOR FIELDS
Definition: Let D be a set in  (a plane region). A
2
vector field on  is a (vector-valued) function F that
assigns to each point (x, y) in D a two-dimensional vector
F(x, y).
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Since F(x, y) is a two-dimensional vector, we can write
it in terms of its component functions P and Q as
follows:
F( x, y )  P( x, y ) i  Q( x, y ) j  P( x, y ), Q( x, y )
F  Pi  Q j
The functions P and Q are sometimes called scalar
functions.
EXAMPLES
Sketch the following vector fields.
1. F(x, y) = −yi + xj
2. F(x, y) = 3xi+ yj
VECTOR FIELDS (CONCLUDED)
Definition: Let E be a set in  . A vector field on 
is a function F that assigns to each point (x, y, z) in E a
three-dimensional vector F(x, y, z).
3
3
We can express F in terms of its component functions P,
Q, and R as
F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k
Three dimensional vector fields can be sketched in space.
See Figures 9 through 12 on page 1094.
PHYSICAL EXAMPLES OF
VECTOR FIELDS
• Velocity fields describe the motions of systems of
particles in the plane or space.
• Gravitational fields are described by Newton’s Law
of Gravitation, which states that the force of attraction
exerted on a particle of mass m located at x = x, y, z
by a particle of mass M located at (0, 0, 0) is given by
 m MG
F( x ) 
x
3
|x|
where G is the gravitational constant.
PHYSICAL EXAMPLES OF
VECTOR FIELDS (CONTINUED)
• Electric force fields are defined by Coulomb’s Law,
which states that the force exerted on a particle with
electric charge q located at (x, y, z) by a particle of
charge Q located at (0, 0, 0) is given by
F( x ) 
 qQ
x
3
x
where x = xi + yj + zk and ε is a constant that depends
on the units for |x|, q, and Q.
INVERSE SQUARE FIELDS
Let x(t) = x(t)i + y(t)j + z(t)k be the position
vector. The vector field F is an inverse square
field if
k
F(x )  3 x
|x|
where k is a real number.
Gravitational fields and electric force fields are
two physical examples of inverse square fields.
GRADIENTS AND
VECTOR FIELDS
Recall that the gradient of a function f (x, y, z) is
a vector given by
f ( x, y, z)  f x ( x, y, z)i  f y ( x, y, z) j  f z ( x, y, z)k
Thus, the gradient is an example of a vector field
and is called a gradient vector field.
NOTE: There is an analogous gradient field for
two dimensions.
CONSERVATIVE VECTOR
FIELDS
Definition: A vector field F is called a
conservative vector field if it is the gradient of
some scalar function f , that is if there exists a
differentiable function f such that F  f . The
function f is called the potential function for
F.
EXAMPLES
1. Show that F(x, y) = 2xi + yj is conservative.
2. Show that any inverse square field is
conservative.