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April 15 Mathematics 206a Mr. Haines 2004 Multivariable Calculus Final Examination
I. (5) If a = 3 i - 2 j - k and L is the line with parametric equation x = (1, 2, 3) + t a for all t ∈ℜ, give the equation of any line through the point (1, 2, 3) that is perpendicular L
II.(10) The curvature of a curve C with parametrization c : ℜ → ℜ^3 is the number
3 '()
c t
c t × c t κ =.
A. If C is any straight line, such as one of the lines in exercise I, calculate the curvature of C.
B. If C is the circle of radius 3 centered at the point (0, 0, 4) and lying in the plane z = 4, calculate the curvature of C.
III. (5) Suppose f (^ x , y )= xy^2 + x^2 y +^5 x and a = (1, 1). What is the value of the directional
derivative of f at a in the direction parallel to the line y = 3x.
IV. (5) If f ( x , y )= x^4 + xy + y^4 , give the Hessian for f at (1, 0).
VII. (10) If M is the part of the surface whose equation is z = x 2 – y^2 that lies inside the cylinder whose equation is x^2 + y^2 = 1, give a parametrization for ∂ M. M looks like a potato chip.
VIII. (15) Let F be a vector field given by F (x, y, z) = (ye z, xe z, xye z).
Let C be the boundary of the square in the xy-plane with vertices (0, 0, 0), (1, 0, 0), (1, 1, 0), (0, 1, 0), (0, 0, 0) oriented in that order and let D be the diagonal of that square connecting (0, 0, 0) to (1, 1, 0).
A. Find a potential function for F.
B. Evaluate ∫ •
C
F d x.
C. Evaluate ∫ •
D
F d x.
IX. (15) Let S = {(^ x , y , z )| x^2 + y^2 + z^2 ≤^4 }be the solid ball of radius 2 centered at the origin.
Recall that the volume of a ball or radius r is (^4 /^3 )π r 3 and that ∂ S ={(^ x , y , z )| x^2 + y^2 + z^2 =^4 }is the spherical boundary of S.
A. If (a, b, c) is a point on the surface ∂ S , give the equation of the tangent plane to ∂ S at the point (a, b, c). Your answer will contain two a's, two b's, and two c's because the tangent plane depends on the point (a, b, c).
B. Calculate n , the unit normal vector to the surface ∂ S at any point (x, y (^) , z). Your answer will contain x's, y's, and z's because the direction of the unit normal changes depending on where (x, y, z) lies on ∂ S.
C. Find a vector field F = ( F 1 , F 2 , F 3 ) on ℜ^3 with the property that F • n = x 2 + y + z.
X. (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) Stokes's Theorem says ∫∫ ∫
∂
M M
curl F n d σ F dx for suitable M and F. Evaluate
M
curl F n d σ by first converting it to a line integral using Stokes's Theorem
and calculating the resulting line integral.
