Calculus Chapter 17 Review

Vector Fields

100

What is a Function F that assigns to each point (x, y, z) in E a three-dimensional vector F(x, y, z)?

A vector field.

200

Match the following vector field with the graph: F(x, y) = 1/2(i + j)

(a)

(b)(c)

a

300

Determine the gradient vector field of f(x, y) = x² -(1/4)y²

 (2x)i - ((1/2)y)j

Line Integrals

100

The definition of a line integral is the integral of f(x, y) * ds along the curve C. What does the ds equal?

sqrt((dx/dt)² + (dy/dt)²)*dt

200

Set up the line integral: F = y³ , C: x = t³, y = t, 0 < t< 2. 

The integral from 0 to 2 of (t³)(9t⁴ + 1)^1/2 * dt

300

Evaluate the previous line integral. 

32.32 

FTC of line integrals

100

If F is continuous on an open connected region, then the line integral of F * dr is _______ of _______ if and only if F is conservative. 

Independent, path

200

Find a potential function for the vector field

F(x, y) =< x², y²>

(X³y³)/3 + C

300

Evaluate the previous integral of F * dr over the given curve C. 

C is the arc of the parabola y = 2x² from (-1, 2) to (2, 8) 

171

Green's Theorem

100

Fill in the blank: 

Green's Theorem: 

Let  be a positively oriented, piecewise ______, _____ ______ curve in a plane, and let  be the region bounded by . If L and M are functions of  defined on an open region containing  and having continuous partial derivatives there, then

 

smooth, simple, closed

200

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 

The integral of xy² dx + 2x²y dy over the curve C, the triangle with vertices (0, 0), (2, 2), and (2, 4)

12

300

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 

The integral of cos(y) dx + x²sin(y) dy over the curve C, the rectangle with vertices (0, 0), (5, 0), and (5, 2) and (0, 2)

30(1 - cos(2))