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Unformatted text preview: Vector Calculus 110.202 Final Exam Thursday May 6 2004, 9  12 AM, in 0 Bloomberg 272 if you are registered in an oddnumbered section, and '0 Remsen 101 if you are registered in an even—numbered section. MY NAME IS: I AM IN SECTION #: Please Show ALL WORK for which you hope to get credit. The
correct answer is in all cases worth exactly ZERO points: your mission
is to convince the grader that you arrived at your answer by a valid
chain of reasoning. Comments and hints as to What you think you are doing are helpful and welcome. Good luck!! I. [Four parts, worth 15 points each:] Notation: Let r = (m, y, 2) denote the radial vector from the origin in R3 to the
point (x,y, z)7 let
7‘ = II"! = x/(562 +112 + 22) be its length, and let B be the vector ﬁeld deﬁned on R3 — {0} (Le. Where 7‘ 7’: 0)
by
k — 3r‘2zr 13(937 y, z) = r3 7 Where k = (0, O7 1) is the standard unit vector pointing along the zaxis. [This
is what physicists call a dipole ﬁeld, directed along the z—axis: it is a model
(for example) of the Earth’s magnetic ﬁeld] 1. Calculate the divergence V ~18 . 2. Calculate the curl V X B . 3. Calculate the ﬂux f/SlBds of the vector ﬁeld 18% across the surfaces of a sphere S of radius R centered at the origin. 4. Calculate the gradient V45, Where (Mr) 2 zr‘3 . II. [Two parts, worth 20 points each:] Notation: Suppose e1 and eg are vectors of length one in R3, which are perpen—
dicular to each other: e1  e2 2 0. Then (7", 0) r——> r(r, 0) = 7"(el cost? + e2 sin 0) (for 0 g 0 g 27r, O S r g R) parametrizes a disk D of radius R centered at O in
the plane spanned by the vectors e1, e2. Note that this plane is perpendicular to
the unit vector e1 x e2, which I will call n. If we keep 1“ ﬁxed, say with constant
value R, then this parametrization restricts to deﬁne a parametrized circle C : r(0) r——> r(0) = R(e1 cos 0 + e2 sin 0) in the e1, eg—plane: this is the circle 0 = 8D which bounds the disk. Let V(r) = w X r be the velocity vector ﬁeld deﬁned by a rotating rigid body
With angular velocity given by the constant vector w = (1111, 1112, 1113). 1. Calculate the circulation per unit area of V around this disk: this is the ratio c(V,R)=W—;3/CVds. Then evaluate the limit limRho C(V7 R). 2. Calculate the similar ratio
~ 1
c(V,R)=W//D(VXV)ds, and evaluate its limit as R —> 0. ...
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 Fall '04
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 Calculus, Derivative, Angular Momentum, Vector Motors, Vector field

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