Kenneth Kuttler

610 CHAPTER 30. STOKES AND GREEN’S THEOREMS

24. The cylinder x2 + y2 = 4 is intersected with the plane x+ 3y+ 2z = 1. This yieldsa closed curve C. Orient this curve in the clockwise direction when viewed from apoint high on the z axis. Let F = (y,z+ y,x). Find

∫C F ·dR.

25. Let F =(xz,z2 (y+ sinx) ,z3y

). Find the surface integral

∫S curl(F ) ·ndA where S

is the surface z = 4−(x2 + y2

), z ≥ 0.

26. Let F =(xz,(y3 + x

),z3y

). Find the surface integral

∫S curl(F ) ·ndA where S is the

surface z = 16−(x2 + y2

), z ≥ 0.

27. The cylinder z = y2 intersects the surface z = 8− x2 − 4y2 in a curve C which isoriented in the counter clockwise direction when viewed high on the z axis. Find∫

C F ·dR if F =(

2 ,xy,xz)

28. Tell which open sets are simply connected. The inside of a car radiator, A donut.,The solid part of a cannon ball which contains a void on the interior. The inside of adonut which has had a large bite taken out of it, All of R3 except the z axis, All ofR3except the xy plane.

29. Let P be a polygon with vertices (x1,y1) ,(x2,y2) , · · · ,(xn,yn) ,(x1,y1) encounteredas you move over the boundary of the polygon which is assumed a simple closedcurve in the counter clockwise direction. Using Problem 13, find a nice formula forthe area of the polygon in terms of the vertices.

30. Here is a picture of two regions in the plane, U1 and U2. Suppose Green’s theoremholds for each of these regions. Explain why Green’s theorem must also hold for theregion which lies between them if the boundary is oriented as shown in the picture.

U2U1

31. Here is a picture of a surface which has two bounding curves oriented as shown.Explain why Stoke’s theorem will hold for such a surface and sketch a region in theplane which could serve as a parameter domain for this surface.

Theory of Linear Ordinary Differential Equations

32. The following is a short list of Laplace transforms. f (t) denotes the function andF (s) the Laplace transform. f ∗g(t), the convolution, is given by

f ∗g(t) =∫ t

0f (t −u)g(u)du

Verify each of these formulas.

6102425.26.27,28.29.30.31.32.CHAPTER 30. STOKES AND GREEN’S THEOREMSThe cylinder x* + y? = 4 is intersected with the plane x + 3y + 2z = 1. This yieldsa closed curve C. Orient this curve in the clockwise direction when viewed from apoint high on the z axis. Let F = (y,z+y,x). Find [. F-dR.Let F = (xz,z*(y+sinx) ,zy). Find the surface integral f,curl(F’)-ndA where Sis the surface z = 4 — (x? +y’), z>0.Let F = (xz, (y3 +x) ,z3y). Find the surface integral f,curl (F")-ndA where S is thesurface z = 16— (x*+y"),z>0.The cylinder z = y* intersects the surface z = 8 — x* — 4y” in a curve C which isoriented in the counter clockwise direction when viewed high on the z axis. Find[-F-dRit F= ($2922),Tell which open sets are simply connected. The inside of a car radiator, A donut.,The solid part of a cannon ball which contains a void on the interior. The inside of adonut which has had a large bite taken out of it, All of R3 except the z axis, All ofRexcept the xy plane.Let P be a polygon with vertices (x1,¥1),(%2,92) °°, (%n, Yn), (41,91) encounteredas you move over the boundary of the polygon which is assumed a simple closedcurve in the counter clockwise direction. Using Problem 13, find a nice formula forthe area of the polygon in terms of the vertices.Here is a picture of two regions in the plane, U; and U2. Suppose Green’s theoremholds for each of these regions. Explain why Green’s theorem must also hold for theregion which lies between them if the boundary is oriented as shown in the picture.U2Here is a picture of a surface which has two bounding curves oriented as shown.Explain why Stoke’s theorem will hold for such a surface and sketch a region in theplane which could serve as a parameter domain for this surface.Theory of Linear Ordinary Differential EquationsThe following is a short list of Laplace transforms. f(t) denotes the function andF (s) the Laplace transform. f * g(t), the convolution, is given byreai)= [fe -wswduVerify each of these formulas.