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April 12 Mathematics 206a Mr. Haines 2005 Multivariable Calculus Final Examination
I. (10) Let M be parametrized by f ( , ) s t = ( s + t s , − t st , ).
A. Calculate a unit normal to M at the point (2, 0, 1).
B. Calculate the tangent plane to M at the point (2, 0, 1).
II. (5) Give an equation of the tangent plane to the graph of z = xy + x y^3 + x at the point
where x = y = 1.
III. (5) If f ( , x y ) = x^3^ + xy + y^5 , give the Hessian form for f at (1, 0).
V. (10) Let R be the triangle with vertices (0, 0), (1, 1), and (1, 2). Evaluate the double integral
∫∫ R^ (^ xy dA )
VI. (10) Suppose M is the part of the surface of the sphere of radius 5 centered at the origin that is cut off by the plane z = 4. M looks like a contact lens or a beanie or a cap.
A. Give a parametrization for ∂ M.
B. Give a parametrization for M.
VIII. (15) The Divergence Theorem (Gauss' Theorem) says ∫∫∫ ∫∫
∂
S S
div F dV F n d σ Explain
the meaning of each of these symbols, i.e. tell what they represent. You don’t need to give the conditions they satisfy, although that would be nice.
A. F
B. S
C. ∂ S
IX. (15) Let M be the portion of the surface of the sphere with radius 5 and center (0, 0, -2)
that is above the xy-plane and let F be a vector field given by F (x, y, z) = (y, -x, e xz).
A) What is the coordinate equation of this sphere (in terms of x, y, and z)?
B) What is the intersection of this sphere with the xy-plane?
C) Use Stokes's Theorem to calculate ∫∫ •
M
curl F n d σ by evaluating the line integral of
F around ∂ M , the answer to part B).
