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December 12, Mathematics 206a Mr. Haines 2003 Multivariable Calculus Final Examination
(3) I. Let a = (1, 3, 7) , b = (0, 3, 6), and C be the straight line segment connecting a to b. Give a parametrization for C.
(9) II. Let x (t) = (t, t 2 , t 3 ) from t = 0 to t = 3 be the parametrization of a curve in ℜ 3.
A. Give an equation of the tangent line to this curve at the point where t = 2.
B. Give the cosine of the angle between x (t) and this tangent line at t = 2.
C. Set up but do not evaluate an integral whose value is the length of this curve.
(12) III. Give examples of:
A. Equations of two distinct parallel planes in ℜ^3.
B. Parametric equations of two distinct parallel lines in ℜ^4.
C. A non-constant vector field defined on ℜ^3 that is path independent.
D. A quadratic form.
(7) VI. Evaluate the line integral ∫ − +
C
xydx x^2 dy , where C is the boundary of the triangle cut
from the first quadrant by the lines x = 2, y = x, the x-axis, and the y-axis.
(7) VII. Calculate the value of the double integral ∫∫
R
dA , where R is the region whose
boundary is the circle parametrized by x (t) = (2 + 3cos t, 5+ 3 sin t) for 0 ≤ t ≤ 2 π.
(7) VIII. Set up but do not evaluate an iterated integral to compute the volume of the solid
below the surface x^2 + y^2 + z = 3 which lies above the region R which is the right triangle with vertices (0, 0), (0, 2), and (1, 0)
(7) IX. Suppose f ( x , y , z )= x^2 y^3 + xy − z − 3 y. Compute the line integral of ∇ f , the gradient
of f along the straight line path connecting (0, 0, 0) to (1, 1, 1).
(7) XII. Evaluate ∫∫ F • n dσ
∂ S
, where F = x i +z 2 j + y 3 k and S is the solid box determined by
the three coordinate planes, the plane x = 2, the plane y = 3, and the plane z = 4.
(7) XIII. State Stokes's Theorem.
