Practice problems are linked to here.

If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this page.

QuickTime free download.

	Allegany College of Maryland has a Calculus III course online. Unit 5 includes topics on vector analysis. Click on Unit 5 at the top of the page when you get there.
	Here are some terrific class notes for Calculus III from Paul Dawkins at Lamar University.
	Online Calculus III from the Springfield Technical Community College distance learning project is another place you might find help.
	Hotmath You can look at free solutions to problems in exercise sets from a wide variety of mathematics textbooks including Calculus by Larson, Hostetler, Edwards, eighth edition. The last time I looked they only had solutions through chapter 12.
	Here is a nice applet from Duke University relating to line integrals and work done by a vector field.
	Examples
	Worksheet
	Lake Tahoe CC Examples
	DPGraph Animations: Little Bang Plus One (Click on Animate and then Continuously Rotate for a nice effect),Little Bang 5, Little Bang with Source

Vector Fields

A vector field is described by a vector valued function that assigns to each point in its domain a unique vector. The gradient would be an example of a vector valued function that describes a vector field. The first function F below would describe vectors in the plane and the second vectors in space. Vector Field Applet

The picture at the right represents a portion of a vector field (in the plane) corresponding to the vector valued function given below.

A vector field F is said to be conservative if there exists a differentiable function f such that the gradient of f is F. The function f would be a potential function for the vector field F. Not all vector fields are conservative.

You will see why in class (see Theorem 15.1).

Here is an example of how you can construct a conservative vector field. Begin with a function f. I will use the one given below.

Now suppose we were starting with F and had not just seen where it came from. If we wanted to determine whether or not F is conservative we would have to check to see whether or not

Comparing (1) and (2) we see that if h(x) = 2x and g(y) = 0 we will have a function f that satisfies the conditions required for it to be a potential function for F. Here is a DPGraph Picture of the potential function f with x, y, and z all varying from -2pi to +2pi.

Piecewise Smooth Curves

See the definitions and at the beginning of Section 15.2.

Example

Find a piecewise smooth parameterization of the curve shown in the figure at the right with the piece from B(2,2,0) to C(2,4,4) following the path of a parabola in the plane x=2 with its vertex at B. The pieces from A(0,0,0) to B and C to A are linear. Click on the picture to see an animation. Click here to see a Maple worksheet generating a picture of the curve. The perspective in the picture was altered from the default perspective using the mouse and cursor.

We can easily represent the path from A to B by

The path from B to C projected onto the yz-coordinate plane would be modeled by the yz-equation

z = (y - 2)2.

In the parameterization from B to C we will have to be at B when t = 1. One option would be

In getting from C back to A we must be at C when t = 2. One option would be

Line Integrals (Integrating a function over a curve)

See the definition of a line integral in Section 15.2 of your text and Theorem 15.4.

Definition of Line Integral

If f is defined in a region containing a smooth curve C of finite length, then the line integral of f along C is given by

Theorem 15.4 Evaluation of a Line Integral as a Definite Integral

Let f be continuous in a region containing a smooth curve C.

Example

Evaluate the line integral of f(x,y,z) = 1 + x over the piecewise smooth curve C (made up of C1, C2, C3) demonstrated earlier.

Note that we could have parameterized each of C1, C2, C3 with t going from 0 to 1 as follows:

If the curve C represented a wire whose density at any point on the wire was x + 1 then the line integral just computed would have given the total mass of the wire.

Here is a Maple Worksheet investigating the line integral being indicated in the picture at the right. This is the line integral of the function f(x,y) given below over the path described by r(t).

Line Integrals of Vector Fields--Work

The work done relative to a force field F in moving an object along a space curve C from point A to point B wherer(t) is a parameterization of C and t = a at point A, t = b at point B, would be

Note that this is not meant to imply that the force vectors in the force field F would necessarily move the object from point A to point B (they might be pushing in the opposite direction). The formula above is simply giving the work doneby the force field. It might be some other force acting to overcome the force field that is moving the object. Thus the integral above could result in a negative value.

The formula is similar for work done in moving an object along a plane curve.

Example

Find the work done by the given force field F on an object moving along the indicated path. Click on the picture at the right to see an animation of the curve C.

Conservative Vector Fields and Independence of Path

Look at Theorem 15.5: Fundamental Theorem of Line Integrals. Note this statement from your text:

"The Fundamental Theorem of Line Integrals states that if the vector field F is conservative, then the line integral between any two points is simply the difference in the values of the potential function f at these points."

Example

Find the work done by the force field F on an object moving from P(0,0) to Q(2,4) three ways. Use two different paths and then use the fact that F is a conservative force field. Here is a DPGraph Picture of the potential function with one unit representing ten on the z-axis and here is the potential function along with the points (0,0,0) and (2,4,64).

Methods 1 and 2:

Maple Worksheet

HAPPY HOLIDAYS (If It Is Christmas Time)

A Triangle Grab the corners to change the shape. An Isosceles Triangle