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Sec 17.1 Vector Fields

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Sec 17.1Vector Fields

DEFINITIONS:

1.

2.

Examples:Sketch the following vector fields.

1.F(x, y) = −yi + xj

2.F(x, y) = 3xi+ yj

Example:

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

Sec 17.2Line Integrals

Definition Let C be a smooth plane curve defined by:

If f is defined on C , then the line integral of f along Cis

Notes:

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.

Line Integrals of Vector Fields

Definition

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

Sec 17.3 The Fundamental Theorem for Line Integrals

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

Note:

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

path.

Theorem:

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

that

Theorem:

is a conservative vector field, where P and Qhave continuous first-order partial derivatives on a domain D, then throughout D we have

Theorem:

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

throughout D.

Then F is conservative.

Sec 17.4 Green’s Theorem

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

An application of Green’s Theorem:

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: