Download MA 227-DW Test 4, Fall 2008: Vector Calculus Problems and more Exams Advanced Calculus in PDF only on Docsity!
Name:
- Part I
There are 6 problems in Part 1, each worth 4 points. Place your answer on the line to the right of the question. Only your answer on the answer line will be graded.
(1) Compute div F when F(x, y, z) = 〈cos(xz), eyz^ , x + y〉.
(2) Find the curl of the vector field F(x, y, z) = 〈 3 xz, 0 , − 5 x^2 〉.
(3) Compute ∇f when f = x^2 + y + z.
(4) Find a parametrization for the cylinder x^2 + y^2 = 1.
(5) Find a function f such that ∇f = 〈 2 xy + 1, x^2 〉.
(6) Evaluate the surface integral
S 2 dS^ when^ S^ is a disc with radius 1.
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- Part II
There are 3 problems in Part 2, each worth 12 points. On Part 2 problems partial credit is awarded where appropriate. Your solution must include enough detail to justify any conclusions you reach in answering the question.
(1) Let C be the circle with radius 1 centered at the origin and oriented counterclockwise. Evaluate (^) ∫
C
ydx + xdy
by two methods: directly as a line integral and using Green’s Theorem.
(3) Find the surface area of that part of the cone z = 1 −
x^2 + y^2 that lies above the x-y-plane.
