22 the cylinder x 2 4 y 2 4 is intersected with the

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22. The cylinder x 2 + 4 y 2 = 4 is intersected with the plane x + 3 y + 2 z = 1 . This yields a closed curve, C. Orient this curve in the counter clockwise direction when viewed from a point high on the z axis. Let F = ( y, z + y, x 2 ) . Find R C F · d R . 23. The cylinder x 2 + y 2 = 4 is intersected with the plane x + 3 y + 2 z = 1 . This yields a closed curve, C. Orient this curve in the clockwise direction when viewed from a point high on the z axis. Let F = ( y, z + y, x ). Find R C F · d R . 24. Let F = ( xz, z 2 ( y + sin x ) , z 3 y ) . Find the surface integral, R S curl ( F ) · n dA where S is the surface, z = 4 - ( x 2 + y 2 ) , z 0 . 25. Let F = ( xz, ( y 3 + x ) , z 3 y ) . Find the surface integral, R S curl ( F ) · n dA where S is the surface, z = 16 - ( x 2 + y 2 ) , z 0 .
410 STOKES AND GREEN’S THEOREMS 26. The cylinder z = y 2 intersects the surface z = 8 - x 2 - 4 y 2 in a curve, C which is oriented in the counter clockwise direction when viewed high on the z axis. Find R C F · d R if F = z 2 2 , xy, xz · . Hint: This is not too hard if you show you can use Stokes theorem on a domain in the xy plane. 27. Suppose solutions have been found to 18.17, 18.16, and 18.12. Then define E and B using 18.14 and 18.13. Verify Maxwell’s equations hold for E and B . 28. Suppose now you have found solutions to 18.17 and 18.16, ψ 1 and A 1 . Then go show again that if φ satisfies 18.10 and ψ ψ 1 + 1 c ∂φ ∂t , while A A 1 + φ, then 18.12 holds for A and ψ. 29. Why consider Maxwell’s equations? Why not just consider 18.17, 18.16, and 18.12? 30. Tell which open sets are simply connected. (a) The inside of a car radiator. (b) A donut. (c) The solid part of a cannon ball which contains a void on the interior. (d) The inside of a donut which has had a large bite taken out of it. (e) All of R 3 except the z axis. (f) All of R 3 except the xy plane. 31. Let P be a polygon with vertices ( x 1 , y 1 ) , ( x 2 , y 2 ) , · · · , ( x n , y n ) , ( x 1 , y 1 ) encoun- tered as you move over the boundary of the polygon in the counter clockwise direction. Using Problem 13, find a nice formula for the area of the polygon in terms of the vertices.
The Mathematical Theory Of Determinants * A.1 The Function sgn n It is easiest to give a different definition of the determinant which is clearly well defined and then prove the earlier one in terms of Laplace expansion. Let ( i 1 , · · · , i n ) be an ordered list of numbers from { 1 , · · · , n } . This means the order is important so (1 , 2 , 3) and (2 , 1 , 3) are different. There will be some repetition between this section and the earlier section on determinants. The main purpose is to give all the missing proofs. Two books which give a good introduction to determinants are Apostol  and Rudin . A recent book which also has a good introduction is Baker . The following Lemma will be essential in the definition of the determinant. Lemma A.1.1 There exists a unique function, sgn n which maps each list of numbers from { 1 , · · · , n } to one of the three numbers, 0 , 1 , or - 1 which also has the following properties.
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