Practice Final Exam – Math 2153

1. Decide if the following statements are TRUE or FALSE and circle your answer. You do NOT need to justify

your answers.

(a) (1 point) If line integrals in the continuous vector ﬁeld F(x, y)are path independent then Fis a

conservative vector ﬁeld.

(b) (1 point) If Fis a conservative vector ﬁeld then line integrals in the continuous vector ﬁeld F(x, y)

are path independent.

(c) (1 point) If F(x, y)has continuous ﬁrst partial derivatives on the connected, simply connected

region Rand F(x, y)is irrotational then Fis conservative.

(d) (1 point) If F(x, y)has continuous ﬁrst partial derivatives on the connected, simply connected

region Rand F(x, y)is source-free then Fis conservative.

2. Give examples of the following. Be as explicit as possible. You do NOT need to justify your answers.

(a) (2 points) Give an example of a scalar function f(x, y)whose implicit domain is connected but

not simply connected.

(b) (2 points) Give an example of a non-constant conservative vector ﬁeld F(x, y, z)with domain R3.

(c) (2 points) Give an example of parametrized path in R2which is not a simple path.

(d) (2 points) Give an example of a non-constant source-free vector ﬁeld F(x, y, z)with domain R3.

3. Compute the following line integrals using any technique you like:

(a) (5 points) Evaluate ZC

F·dr

where F(x, y) = hxy, x −yiand Cis the straight line segment from the point (0,0) to

the point (2,1).

(b) (2 points) Evaluate ZC

F·dr

where F(x, y, z) = h6xyz, 3x2z, 3x2yiand Cis the path with parametrization

r(t) = ht, sin t, t sin ti0≤t≤π

(c) (2 points) Evaluate IC

F·dr

where F(x, y) = hxy2, x2−yiand Cis the closed square path with corners (0,0),

(0,2),(2,2) and (2,0) oriented clockwise.

4. Let

F(x, y, z) = hx2y, xyz, z2i

(a) (5 points) Compute

curl F

(b) (5 points) Compute

div F

(c) (5 points) Compute

div(curl F)

5. Compute the following surface integrals using any technique you like:

## Document Summary

Practice final exam math 2153: decide if the following statements are true or false and circle your answer. F dr where f(x, y) = hxy, x yi and c is the straight line segment from the point (0, 0) to the point (2, 1). (b) (2 points) evaluate. F dr where f(x, y, z) = h6xyz, 3x2z, 3x2yi and c is the path with parametrization (c) (2 points) evaluate r(t) = ht, sin t, t sin ti. F(x, y, z) = hx2y, xyz, z2i curl f div f div(curl f: compute the following surface integrals using any technique you like: (a) (5 points) evaluate. F n ds where f(x, y, z) = hz, z, zi and s is the upper half of the sphere or radius 2 with center (0, 0, 0) oriented inwards. (b) (2 points) evaluate. 0 xy 2 dy dx: (5 points) evaluate the double integral.