Homework #12

Do all integrals
entirely by hand showing all the steps, except where noted otherwise. 
You may, if you wish, check your calculations with Maple.

14.7	Integration over parametrized volumes.
	X. All the following by hand.
	   (a) Derive the jacobian for parametrization by cylindrical coordinates:
		x(r,theta,z) = r*cos(theta)
		y(r,theta,z) = r*sin(theta)
		z(r,theta,z) = z
	       Show all the steps of your calculation.
	   (b) Make a labelled sketch of the conical region enclosed by the surfaces 
			r = h - z,
			z = 0 .
	       h is a constant.
	   (c) Sketch the corresponding region in (r,theta,z) space.
	   (d) Determine the volume of the region (using cylindrical coordinates).
	   (e) Determine the location of the center of mass if the material filling
	       the region has uniform density. 

15.1	4. (a) Sketch a planar vector field by hand. Sample the
               field at, say, 12 or 16 points.
	   (b) Use Maple's "fieldplot" to create a more detailed picture of the field.
	
15.2	X. Compute the tangential and normal line integrals of the 
	   vector field F(x,y) = [x2,y2] along the straight line
	   segment that runs from the point [1,0] to the point [0,2].
	   (You have to devise a parametrization of the line segment. Make sure
	    the parametrization increases along the specified direction.)

15.4    Green's theorem
        2. Use Green's theorem to evaluate the circulation of a field
           around a square loop (without doing a line integral!).