Math 241 Fall 2023 Week 14 Merit Worksheet - Jie Yeo’s section
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Z
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1. Recall that given a vector field F : R → R , we define flux of F across a curve C as
(F · n) ds.
C
(a) What does n represent in this equation?
(b) What does flux measure in words?
2. Recall from lecture that for a surface S parameterized by r : D → S, we have
ZZ
ZZ
(F · n) dA =
S
Z
Use this to compute
F · (ru × rv ) dudv.
D
(F · n) dA where F = xi + yj + z 2 k and S is the sphere with radius 1 and
S
center at the origin. Feel free to check in with your TA about your expressions for (ru × rv ) and
F · (ru × rv ) as well as your final answer.
3. Let D be a region in R3 , F a vector field, and n the outward pointing normal from the surface ∂D.
(a) What does the divergence theorem tell us in this situation?
(b) Use your answer from part (a) to verify your answer to Q2.
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4. Fill in the blanks:
Green’s Theorem gives a relationship between a
curve C and a
integral around a
integral over the plane region D bounded by C. The
equation for Green’s Theorem is
.
The Divergence Theorem gives a relationship between a
of a region D in
Theorem is
,
and a
integral over the boundary
integral over D. The equation for the Divergence
.
ZZ
(F · n) dA where F(x, y, z) = h3x, xy, 2xzi and E is
5. Use the Divergence Theorem to evaluate
∂E
the cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 1.
6. Use the Divergence Theorem to calculate the flux of F(x, y, z) = 3xy 2 i + xez j + z 3 k over the surface
S bounded by the cylinder y 2 + z 2 = 1 and the planes x = −1 and x = 2.
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7. Let S be the cylinder x2 + y 2 = 4.
(a) Find a parametrization r(u, v) = hx(u, v), y(u, v), z(u, v)i for S. (Don’t forget to specify bounds
for u and v!)
(b) Calculate the normal vector ru × rv associated to your parametrization.
(c) Determine whether ru × rv points into the cylinder, or out of the cylinder. How can you modify
your parametrization to ensure that the normal vector ru × rv points in the opposite direction?
8. Let D be the unit square in the xy-plane. (We are still working in three dimensions.)
(a) Give a parametrization r(u, v) = hx(u, v), y(u, v), z(u, v)i with ru × rv pointing “up”.
(b) Give a parametrization r(u, v) = hx(u, v), y(u, v), z(u, v)i with ru × rv pointing “down”.
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9. Let R be the part of the cylinder x2 + y 2 = 4 that sits between z = −2 and z = 2.
(a) Make a sketch of R.
(b) In your sketch, identify the boundary ∂R of R.
(c) Suppose we give R the “outward” orientation. Describe the induced orientation on ∂R.
(d) In general, how do you determine the orientation of the boundary of an oriented surface?
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