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### Sec 17.1 Vector Fields

### Example:

### Sec 17.2 Line Integrals

### Line Integrals of Vector Fields

### Sec 17.3 The Fundamental Theorem for Line Integrals

### Theorem:

### Theorem:

### Sec 17.4 Green’s Theorem

### An application of Green’s Theorem:

DEFINITIONS:

1.

2.

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

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.

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

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.

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

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.

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:

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