Math 227
March 2016
Name:
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Mathematics 227
Midterm 2
March 16, 2016
There are four questions worth a total of 40 marks.
No calculators or notes are allowed.
10 40
1. Find the surface area of the part of the paraboloid z = x2 +y 2 that is inside x2 +y 2 +z 2 = 6.
Math 227
10 40
March 2016
Name:
2. Evaluate, by direct computation,
RR
S
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~ · n̂ dS where F
~ = yı̂ı + 2x̂ + z k̂, S is the surface
F
x2
y2
+
+ 4z 2 = 1
16
4
and n̂ is the normal with positive z component.
1
1
√ ≤z≤
2
2 2
Math 227
10 40
March 2016
Name:
3. Let ~a be a constant vector and let ~r = (x, y, z).
~ obeying ∇
~ ×A
~ = ~r.
(a) Find, if possible, a vector field A
~ obeying ∇
~ ×A
~ = ~a × ~r.
(b) Find, if possible, a vector field A
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Math 227
10 40
March 2016
Name:
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4. Let S be the surface of a region V in IR3 . Suppose that S consists of a finite number
of faces Fi , 1 ≤ i ≤ n, each of which is contained in a plane. Such a surface is called a
~ i be a vector
polyhedron. A cube is an example of a polyhedron. For each face Fi , let G
which is normal to Fi , points out of V and has length equal to the area of Fi . Compute
Pn ~
i=1 Gi · k̂. To receive any credit, you must justify your answer.
Math 227
March 2016
Name:
This page just provides extra work space.
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