Vector Fields
Line Integrals
FTC of line integrals
Green's Theorem
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.

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

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

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

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

(a)

(b)(c)

a

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

200

Find a potential function for the vector field

F(x, y) =< x2, y2>

(X3y3)/3 + C

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

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

 (2x)i - ((1/2)y)j

300

Evaluate the previous line integral. 

32.32 

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

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))

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