Problem 1

The position function of a particle is given by . At what time is the speed minimum?

Problem 2

Write a formula for a two-dimensional vector field which has all vectors of length 7 and perpendicular to the position vector at that point.

Problem 3

Give an example of a vector field in 2-space with the properties that is perpendicular to at every point.

Problem 4

Consider the vector field . Show that is a gradient vector field by determining the function which satisfies .

Problem 5

For each of the following vector fields , decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function (that is, ) with . If it is not conservative, type N.

(a).

The field is conservative

(b).

The field is not conservative

(c).

The field is conservative

Problem 6

For each of the following vector fields , decide whether it is path independent (conservative) or not. If it is path independent, enter a potential function for it. If it is path dependent enter NONE.

(a).

The field is path independent

(b).

The field is path independent

(c).

The field is path dependent

(d).

The field is path independent

(e).

The field is path independent

(f).

The field is path independent

Problem 7

For each of the following, decide if the given vector field is a gradient of a function . If so, give find the function and enter it as your answer; if not, enter the word none for your answer and be sure that you are able to explain why this is the case.

(a).

There is no function whose gradient is this vector field.

(b).

There is no function whose gradient is this vector field.

(c).

There is no function whose gradient is this vector field.

(d).

Problem 8

Find the line integral with respect to arc length , where is the line segment in the -plane with endpoints and .

(a). Find a Vector Parametric Equation for the line Segment so that Points and Correspond to and , Respectively

(b). Using the Parametrization in part (a), the line Integral with respect to Arc Length is

(c). Evaluate the line Integral with respect to Arc Length in part (b)

Problem 9

If is the part of the circle in the first quadrant, find the line integral with respect to arc length.

Problem 10

Compute the total mass of a wire bent in a quarter circle with parametric equations: and density function.

Problem 11

A wire in the shape of a helix for has a mass-density defined by g/cm, where , , and are measured in centimetres. Find the centre of mass of the wire. Enter your answer as a point including the parentheses.

Problem 12

Let and let be the line segment from to .

(a). Calculate and for the Parameterization for

(b). Evaluate

Problem 13

Calculate the integral of over the curve for .

Problem 14

Compute the line integral of the scalar function over the curve for .

Problem 15

Compute the line integral of the scalar function over the curve , for .