Math 2153 Final Exam

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Published on 31 Jan 2019
School
Ohio State University
Department
Mathematics
Course
MATH 2153
Professor
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 ti0tπ
(c) (2 points) Evaluate IC
F·dr
where F(x, y) = hxy2, x2yiand 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:
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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.