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Truneh Calculus 252

Truneh Calculus 252 - 1 Given F(x v z = xzi...

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Unformatted text preview: 1. Given F(x, v, z) = xzi + ye"j— _vzk , Find Curl(F), and diV(F). 2. Evaluate J-(x+ yz)ds if C is a line segment from (0,0,0) to (6,2,2). C 3. Show that the line integral is independent of path, and find its value. (4.0.1) [(210) + 5)dx + (x2 — 4z)dy — 4de (0.0.1) 4. Use Green’s theorem to evaluate the line integral J-ez"*"‘dx+ e‘ldy ; C is the triangle with vertices (0,0), (1.0), and (19])- C 5. Express the surface integral as an iterated double integral by using a projection of S on the xz-plane .( Don’t evaluate the integral). J‘J‘nyZZBdS ; S is the first octant portion of the plane 3x+4y+2z=l 2. S 6. Evaluate [(xz + yz)dx+ 2xydy , where C is part of a unit circle in the C first quadrant oriented counterclockwise. 7. Use divergence theorem to evaluate CHF- n (15 where 5 _) F = x 2 i + y 2j + z 2 k and S is the surface of the cube bounded by the coordinate planes and the planes x = 4, y = 4, and z = 4. a —> a 8. Given F = x i + y j + z k , and S is the first octant portion of the plane 2x + 3y + z = 6, and n is a unit upper normal to S, Find ”(F Zde S 9. Find the surface area of S, where is part of the paraboloid z = x2 + y2 cut off by the plane z = 9. a 10. Use Stoke’s theorem to evaluate ”(V X F) n dS, where F = 2y i — z j + 3 k and S _> S is the upper part of the paraboloid z = 4 — x 2 — y 2 cut by the plane 2 = 3 and n the upper unit normal to S. ...
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