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# hw7 - Homework 7 Math 118C Spring 2010 Due on Tuesday June...

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Homework 7 – Math 118C, Spring 2010 Due on Tuesday, June 1st, 2010 1. Verify Stokes’ theorem for ω = x dx + xy dy with D as the unit square with opposite vertices at (0 , 0) and (1 , 1). 2. Evaluate the integral of ω = ( x y 3 ) dx + x 3 dy around the circle x 2 + y 2 = 1 using Stokes’ theorem. 3. Find a 1-form ω for which = ( x 2 + y 2 ) dx dy , and use this to evaluate integraldisplay D ( x 2 + z 2 ) dx dy, where D is the region inside the square | x | + | y | = 4 and outside the circle x 2 + y 2 = 1. 4. Show that the volume of a suitably well-behaved region R in space is given by the formula v ( R ) = 1 3 integraldisplay ∂R ( x dy dz + y dz dx + z dx dy ) . 5. Verify the invariance relation ( ) T = d ( ω T ) when ω = x dy dz and T is the transformation x = u + v w , y = u 2 v 2 , and z = v + w 2 . 6. Let F = F 1 e 1 + F 2 e 2 + F 3 e 3 be a C 1 mapping of an open set E R 3 into R 3 . With this vector field we associate the 1-form λ F = F 1 dx + F 2 dy + F 3 dz, and the 2-form ω F = F 1 dy dz + F 2 dz dx + F 1 dx dy.

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hw7 - Homework 7 Math 118C Spring 2010 Due on Tuesday June...

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