Homework8

# Homework8 - f . If such an f exists, ﬁnd it. (a) F ( x,y...

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1 Mathematics 1c: Homework Set 8 Due: Tuesday, June 1 at 10am. 1. (10 Points) Section 8.1, Exercises 3c and 3d . Verify Green’s theorem for the disk D with center (0 , 0) and radius R and P ( x,y ) = xy = Q ( x,y ) and the same disk for P = 2 y,Q = x. . 2. (10 Points) Section 8.2, Exercise 3. Verify Stokes’ theorem for z = p 1 - x 2 - y 2 , the upper hemisphere, with z 0 , and the radial vector ﬁeld F ( x,y,z ) = x i + y j + z k . 3. (10 Points) Section 8.2, Exercise 16. For a surface S and a ﬁxed vector v , prove that 2 ZZ S v · n dS = Z ∂S ( v × r ) · d S , where r ( x,y,z ) = ( x,y,z ) . 4. (15 Points) Section 8.2, Exercise 23. Let F = x 2 i + (2 xy + x ) j + z k . Let C be the circle x 2 + y 2 = 1 in the plane z = 0 oriented counterclockwise and S the disk x 2 + y 2 1 oriented with the normal vector k . Determine: (a) The integral of F over S . (b) The circulation of F around C . (c) Find the integral of ∇× F over S . Verify Stokes’ theorem directly in this case . 5. (15 Points) Section 8.3, Exercise 14. Determine which of the following vector ﬁelds F in the plane is the gradient of a scalar function
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Unformatted text preview: f . If such an f exists, ﬁnd it. (a) F ( x,y ) = (cos xy-xy 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 r-v 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.

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