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Unformatted text preview: f . If such an f exists, ﬁnd it. (a) F ( x,y ) = (cos xyxy sin xy ) i( x 2 sin xy ) j (b) F ( x,y ) = ( x p x 2 y 2 + 1) i + ( y p x 2 y 2 + 1) j (c) F ( x,y ) = (2 x cos y + cos y ) i( x 2 sin y + x sin y ) j . 6. (10 Points) Section 8.4, Exercise 2. Let F = x 3 i + y 3 j + z 3 k . Evaluate the surface integral of F over the unit sphere. 7. (10 Points) Section 8.4, Exercise 14. Fix k vectors v 1 ,..., v k in space and numbers (“charges”) q 1 ,...,q k . Deﬁne φ ( x,y,z ) = k X i =1 q i 4 π k rv i k , where r = ( x,y,z ) . Show that for a closed surface S and e =∇ φ, ZZ S e · d S = Q, where Q = q 1 + ··· + q k is the total charge inside S . Assume that none of the charges are on S ....
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.
 Spring '08
 Ramakrishnan
 Math, Calculus, Linear Algebra, Algebra

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