Problem 1

Consider the curve given by

(a). Find the Unit Tangent Vector as a Function of

(b). Find the Curvature as a Function of

(c). Determine the Principal Normal Vector at the point

Problem 2

A particle of mass has position and velocity at time . It moves under a force .

(a). Determine the Position of the Particle Depending on

(b). At what time after time Does the Particle Cross the Plane for the First Time?

(c). What is the Velocity of the Particle when it Crosses the Plane in part (b)?

Problem 3

On the following page, the vector field is plotted. In the following questions, give the answer that is best supported by the plot.

1. The Derivative at the point Labelled is (a) Positive, (b) Negative, (c) Zero, (d) there is not Enough Information to Tell
2. The Derivative at the point Labelled is (a) Positive, (b) Negative, (c) Zero, (d) there is not Enough Information to Tell
3. The Derivative at the point Labelled is (a) Positive, (b) Negative, (c) Zero, (d) there is not Enough Information to Tell
4. The Derivative at the point Labelled is (a) Positive, (b) Negative, (c) Zero, (d) there is not Enough Information to Tell
5. The curl of at the point labelled is (a) in the direction of (b) in the direction of (c) zero (d) there is not enough information to tell.
6. The work done by the vector field on a particle travelling from point to point along the curve is (a) positive (b) negative (c) zero (d) there is not enough information to tell.
7. The work done by the vector field on a particle travelling from point to point along the curve is (a) positive (b) negative (c) zero (d) there is not enough information to tell.
8. The vector field is (a) the gradient of some function (b) the curl of some vector field (c) not conservative (d) divergence free.

Problem 4

A physicist studies a vector field . From experiments, it is known that is of the form where and are some real numbers. From theoretical considerations, it is known that is conservative.

(a). Determine and

(b). Find a Potential such that

(c). Valuate the line Integral where is the Curve Defined by ,

(d). Evaluate the line Integral where is the Same Curve as in part (c)

Problem 5

In the following, we use the notation , , and is some number

(a). Find the Value for Which

(b). Find the Value for Which

(c). Find the Value of for Which

Problem 6

Suppose the surface is the part of the sphere that lies inside the cylinder and for which Which of the following are parameterizations of ?

• [ ] A. , ,
• [x] B. ,
• [ ] C. , ,
• [x] D. , ,
• [x] E. , ,

Problem 7

Evaluate the flux integral where and is the part of the paraboloid that lies above the triangle , . is oriented so that its unit normal has a negative -component.

Problem 8

Let be the oriented curve consisting of the 5 line segments which form the paths from to , from to , from to , from to , and from to . Let Evaluate the integral .

Problem 9

Let be the part of the paraboloid contained in the cone and oriented in the upward direction. Let . Evaluate the flux integral .

Let be the circle of radius in the plane and be the region bounded by and .

Problem 10

Which of the following statements are true and which are false?

• [ ] A. The curve defined by is the same as the curve defined by .
• [x] B. The curve defined by is the same as the curve defined by .
• [x] C. If a smooth curve is parameterized by ) where is arc length, then its tangent vector satisfies .
• [ ] D. If defines a smooth curve in space that has constant curvature , then is part of a circle with radius .
• [x] E. If the speed of a moving object is constant, then its acceleration is orthogonal to its velocity.
• [ ] F. The vector field is conservative.
• [ ] G. Suppose the vector field is defined on an open domain and its components have continuous partial derivatives. If , then is conservative.
• [ ] H. The region is simply connected.
• [x] I. The region is simply connected.
• [x] J. If is a vector field whose components have continuous partial derivatives, then when is the boundary of a solid region in