Test4_StudyGuide.pdf

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MA 242-001: Calculus III, NCSU, Summer II 2015

Test 4 Study Guide

Test 4 covers sections 13.1 - 13.5, 10.5, 12.6.

Treat the sample tests and practice problems as additional review, not as a complete representa-tion of the questions you’ll be given on the exam- you are responsible for all material we covered (i.e., all learning objectives below).

13.1 - Vector Fields

• Know the definition of a vector field on R2 and R3.

• Define a conservative vector field Fand its potential function.

Supplemental Homework Problems: 1-7 odd, 11-17 odd, 21-25 odd, 29, 31Extra Practice Problems: 6, 14, 18, 24, 26, 32

13.2 - Line Integrals

• Define a line integral along a path C .

• Evaluate the line integral of f (x,y,z ) along C = r(t).

• Parameterize curves such as circles and line segments.

• For curve C given by r(t), a ≤ t ≤ b,

C

f (x,y,z ) ds =

ba

f (r(t))r dt.

• Evaluate line integrals of vector fields F. For curve C given by r(t), a ≤ t ≤ b, C

F · dr =

ba

F(r(t)) · r(t) dt

.

• If F = P i + Q j + R k, then C

F · dr =

C

P dx + Q dy + R dz

.

• Find the work done moving a particle along a path through a vector field.

Supplemental Homework Problems: 1-17 odd, 16, 19, 21, 22, 33, 35, 36, 39-41Extra Practice Problems: 2, 4, 6, 8, 14, 20, 34


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MA 242-001, NCSU, Summer II 2015 - Test 4 Study Guide page 2

13.3 - The Fundamental Theorem for Line Integrals

Theorem 1. Let C be a smooth curve given by r(t), a ≤ t ≤ b. Let f be a differentiable function of two or three variables where ∇f is continuous on C . Then

C

∇f · dr = f (r(b)) − f (r(a))

.

Theorem 2. Let F be a vector field that is continuous on an open connected region

D. If

C

F · ds is independent of path, then F is conservative.

Theorem 3. If F is a conservative vector field, then P y = Qx.

Theorem 4. Let F = P i + Q j on an open, simply connected set D. If P y = Qx, then F is conservative.

• Identify open, connected and simply connected sets

• Test if a two-dimensional vector field is conservative.

• Find the potential function f of conservative vector field F.

Supplemental Homework Problems: 1-23 odd, 10, 14, 20, 31, 33, 34, 35Extra Practice Problems: 4, 6, 8, 12, 18, 19*, 24, 25*, 28*, 32

13.4 - Green’s Theorem

Theorem 5. Let C be a positively-oriented, piecewise smooth, simple closed curve in the xy-plane bounding D. If P and Q have continuous partial derivatives on an open region containing D, then

C

P dx + Q dy =

D

(Qx − P y) dA

.

• State Green’s Theorem.

• Use Green’s Theorem to find the area of a region D bounded by C .

A(D) = C

−x dy = C

y dx = 1

2 C

x dy − y dx

.

Supplemental Homework Problems: 7, 13, 17


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MA 242-001, NCSU, Summer II 2015 - Test 4 Study Guide page 3

13.5 - Curl and Divergence of Vector Fields

• curl(F) = ∇× F = (Ry − Qz) i + (P z − Rx) j + (Qx − P y) k

• Find the curl of F.

• Test if a 3-dimensional vector field F is conservative.

• Find the potential function f of conservative vector field F.

• Determine whether a vector field F is the curl of another vector field.

Theorem 6. curl ( ∇f ) = 0

Theorem 7. If F is a vector field defined on R3 whose component functions have continuous partial derivatives and curl( F) = 0, then F is conservative.

Theorem 8. If F is a vector field whose component functions have continuous second partials,then div(curl( F)) = 0.

Supplemental Homework Problems: 1-7 odd, 13-17 odd*, 18Extra Practice Problems: 4, 6, 14, 16, 19*, 25*

10.5 - Parametric Surfaces

• Parameterize surfaces in cylindrical and spherical coordinates.

• Parameterize z = f (x, y), y = f (x, z ), or x = f (y, z ).

Supplemental Homework Problems: 1-5 odd, 24,13-25 oddExtra Practice Problems: 2, 4, 6, 20, 22, 26

12.6 - Surface Area

• Find the area of a parameterized surface.

• If parametric surface S is given by the equation r(u, v) = x(u,v) i + y(u,v) j + z(u,v) k,where (u,v) are over the domain D, then the surface area of S is

A(S ) =

D

ru × rv dA

where ru = xu i + yu j + zu k, rv = xv i + yv j + zv k.Supplemental Homework Problems: 1-11 odd, 6Extra Practice Problems: 2, 4, 7, 10

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Test4_StudyGuide.pdf