Sec 17.1 Vector Fields. DEFINITIONS: 1. 2. Examples : Sketch the following vector fields. 1. F ( x , y ) = − y i + x j 2. F ( x , y ) = 3 x i + y j. Example :. If f is a scalar function of three variables, then its gradient,
Examples: Sketch the following vector fields.
1. F(x, y) = −yi + xj
2. F(x, y) = 3xi+ yj
If f is a scalar function of three variables, then its gradient,
is a vector field and is called a gradient vector field.
Definition: A vector field F is called a conservative vector fieldif there exists a differentiable function f such that
The function f is called the potential functionfor F.
Example: The vector field F(x,y) = 2xi + yj is conservative because
if , then
Definition Let C be a smooth plane curve defined by:
If f is defined on C , then the line integral of f along Cis
The value of the line integral does not depend on the parametrization of the curve, provided the curve is traversed exactly once as t increases from a to b.
The value depends not just on the endpoints of the curve but also on the path.
The value depends also on the direction (or orientation) of the curve.
Let F = P(x, y, z) i + Q(x, y, z) j + R(x, y, z) k be a continuous vector
field defined on a smooth curve C: r(t) = x(t)i+ y(t)j + z(t)k, a ≤ t ≤ b.
Then the line integral of Falong Cis
Theorem: Let C be a smooth curve defined by:
Let f be a differentiable function of two or three variables whose gradient vector is continuous on C . Then
This theorem says that we can evaluate the line integral of a conservative vector
field simply by knowing the value of the potential function f at the endpoints of C.
In other words, the line integrals of conservative vector fields are independent of
is independent of path in D if and only if
for every closed path C in D.
Theorem: Suppose F is a vector field that is continuous on an open
connected region D. If is independent of path in D, then F is a
conservative vector field in D; that is, there exists a function fsuch
is a conservative vector field, where P and Qhave continuous first-order partial derivatives on a domain D, then throughout D we have
Let F = Pi + Qj be a vector field on an open simply-connected region D. Suppose that P and Q have continuous first-order derivatives and
Then F is conservative.
Green’s Theorem: Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. If P and Q have continuous partial derivatives on an open region that contains D, then
If we choose P and Q such that
then in each case,
Hence, Green’s Theorem gives the following formulas for the area of D: