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Unformatted text preview: 03/02/2004 14:40 FAX 510 642 9454 001 Math H53 Final Exam
December 16th, 2003 Dr. K. G. Hare 1: (5 pts) By use of Tangent Planes, estimate f(l.01,7r + 0.02) Where f($,y) =
552 sin(y) + cos(:c 2 a: (5 pts) Find a parametric representation for the intersect of $2 + 22 = 1 and
3:2 + y2 = 1 b: (5 PtS) Let
11va 1132 + y2 + e—m2fy2 Find all local minimums and maximums. 3: (5 pts) Let u = (11.1.1142) and v = <‘U1,’U2> be unit vectors in R2. Assume that f
has continuous partials of every order. Show, by means of Clairut’s Theorem, that
the directional derivatives feather: y) = fu,v(w: y) 4: (5 pts) Let f(:c,y, z) = 3:2. Find the maximum of f on the intersection of the
two surfaces 3:2 + y2 = 1 and x2 + 22 = 1. 5: (5 pts) Prove that the rational numbers Q have measure 0. 6: (5 pts) Find the volume of the region bounded by the function 2: = f (z,y) :
3 +352— aszsin(y:r) over D = {(33,y) :0 i :5 S 1,0 3 y S x} 7: (5 pts) Assume that e, f, g and h are continuous functions. Show that F(u, :15, y, z) =
(8(a), f (as), g(y), h(z)) is conservative. [Hint, consider the Fundamental Theorem of
Calculus] Find an example Where (f (y), is not conservative. 8: (5 pts) Let z : f (3:, y). By using the general equation for the surface area of a
parametric curve, show that the surface area of f over D is f/D 1+f.(m,y)2+.fy(as,y)2dA 9: (5 pts) Prove that the line integral fof  d?" : 0 for all closed curves 0 if and
only if F f  d7" is path independent for all curves P. (Do this directly, do not 11% 03/02/2004 14:40 FAX 510 642 9454 002 the fact that f is conservative.) 10: (5 pts) Let f(a:,y) = [i Show f0 f  CE?“ = O for all closed curves C, m24.3.2 a x2+y2
so long as C does not contain the point (0,0) in the interior. 11 a: (5 pts) State Greens Theorem b: (5 pts) Let D = {(m,y) : $2 + y2 g 1. Let F(:r,y) : (P($,y), 0) Prove Green’s
Theorem for this simpliﬁed case. 12 a: (5 pts) State Stokes Theorem. b: (5 pts) Use Stokes Theorem to evaluate ffS curlF  018 Where F($,y,z) :
($12,332,331) and S is the part of the paraboloid z = 4 — 3:2 — yz that lies above
the plane 2 = 3 , oriented upwards. 13 a: (5 pts) State the Divergence Theorem b: (5 pts) Give an example of the following
i: A disconnected unbounded 1—manifold.
ii: A clOSed bounded 2manifold. iii: A 3—manifold. 14: (5 pts) a: Convert (m,y) : (1,2) from Cartesian (rectangular) co—ordinates to polar co—
ordinates. b: Convert (r, 6) = (1, 3:) from polar co—ordinates to Cartesian co—ordinates. (3: Convert (:12, y, z) = (1, 1, 1) from Cartesian co—ordinates to cylindrical co—ordinates. (‘1: Convert (.38, y, z) = (1, 1, from Cartesian co—ordinates to spherical co—ordinates. e: Convert (r, «9, z) from cylindrical co—ordinates to spherical co—ordinates. ...
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 Spring '08
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