Test 1
Final Review
MA 432 S06
The nal will be just over chapters 7 and 8. So be able to do line and surface integrals of all types. Also know general facts about parametrizable surfaces such as how to nd an element of surface area, and how to nd the tangent plane to such a surface at a given point. Know the three main theorems in chapter 8 : Greens theorem, Stokes theorem, Divergence theorem, and be able to do integrals using these theorems. Finally be able to tell when a line integral is inpendent of path in an open set and how to nd gradient elds. The following problems from the end of chapters 7 and 8 are good problems to practice on. chapter 7 (page 514), 1, 3, 5, 7, 9, 11, 15, 17, 21, 23, 27. chapter 8 (page 605) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21.
Practice Test 1 25 pts 1. (a) Find 1 y = x , 1 x e.
C
xy 2 dx x2 dy , where C is the arc of the hyperbola,
(b) Find a scalar function f so that f = F, where 2 F = xexy (xy + 2)i + (x3 exy + 2yey )j. (c) Find
C
Fds, where F is as in (b) and C is the curve, x = t5 , y = t4 , 0 t 1.
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(ex + 3xy) dx + (ln(y 2 + 4) + 3x) dy where C is the triangle with (d) Find C vertices : (2, 0), (0, 1), and (0, 0) oriented counterclockwise. 25 pts 2. Find (y 2 + z) dS, where S is that part of the sphere x2 + y 2 + z 2 = 8 S which lies inside the cone z = x2 + y 2 . 25 pts 3. Find F n dS where S = {(x, y, x2 + y 2 ) : x2 + y 2 1 }, n is the unit S normal to S with positive k component and F(x, y, z) = (x, y, z). 25 pts 4. Find ( F) n dS where F = (x3 , y 3 + 3z 2 sin z, ey sin z + x4 ) and S S is that portion of the sphere x2 + y 2 + z 2 = 1 with z 1/2. 25 pts 5. Use the more general version of the divergence theorem to nd n dS where S is any smooth closed surface containing the origin, n is the outer unit normal to S, and = (x2 + y 2 + z 2 )1/2 .
S
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10 pts Extra Credit: Use the divergence theorem to show that if u is harmonic in an open set whose boundary is a smooth surface S, then u n dS = 0, where S n is the outer unit normal to S.
Answers Practice Test 1 1. (a) e, (b) f = x2 exy + ey 2. (16 7 2). 3 /2. 3. 4. 0. 5. 4. EC. Can do in class if asked.
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(c) 2 e - 1, (d) 1.
Practice Test 2 25 pts 1. Given the surface S = {(u2 + v, v, u3 ) : 0 < u < 2, 1 < v < 3}. (a) Find the tangent plane to this surface at u = 1, v = 2. (b) Find the surface area of S. 25 pts 2. Which of the following line integrals c u ds are independent of path? For those that are evaluate the line integral from (1, 0, 0) to (0, 0, 1). 2 (a) u = (ex , x + 3(y + 3)2 (z 1), (y + 3)3 ). 2 (b) u = (xex , 3(y + 3)2 (z 1), (y + 3)3 ). (c) u = (2xyz 3 + z, x2 z 3 , 3x2 yz 2 + x ). 25 pts 3. Use Greens theorem to nd F ds where C (a) C is the parabola y = 1 x2 and the line segment y = 0, 1 x 1 oriented counterclockwise and F = (x2 y, xy 2 ). (b) C is the rectangle with vertices (a, b) oriented counterclockwise and F = (x3 , xy 3 ). (c) C is any smooth closed curve and F = (f (x), g(y)) where f, g are continuously dierentiable. 25 pts 4. Find ( v) n dS, where v = (y + x2 )i + j + xyz 2 k and S is S 1 1 that portion of the ellipsoid x2 + 4 y 2 + 9 z 2 = 1, with z 0, while n is the normal to S with positive k component.
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25 pts 5. Find F ndS where S is the surface of the ice-cream cone, x2 + y 2 + S z 2 = 1, z 0, and (z + 1)2 = x2 + y 2 , z 0, while F = x...