Southwestern College Line Integrals & Greens Theorem Calculus Questions Test 4

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fuehr

Mathematics

Math 281

Southwestern College

MATH

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Calculus
Complete the following 6 calculus questions. Show all the work for each problem as well.

do not include any explanation, just show the work

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Due May 26, 2020 Math 281 - Test 4 Name__________________ Show sufficient calculations to indicate that you are not relying on a calculator... or the Internet. 1. Let F( x, y, z ) = e x sin yi + e x cos yj + xyzk . a. Find curl F. b. Find div F. 2. Let C be the curve defined by a. Find # 3z b. Find # C C 2 x = 3 cos t, y = 3 sin t, z = 4t, 0 ! t ! " . ds . $2 y dx + 2 x dy + ( x 2 + y 2 ) dz . Due May 26, 2020 Math 281 - Test 4 page 2 of 4 3. Determine whether or not the vector field F is conservative. If it is conservative, find a function f such that F = %f . If F is not conservative, explain why not. a. F( x, y) = 3e3 x sin y i + (3 + e3 x cos y) j b. F( x, y) = (e y + y) i + ( xe y $ x ) j 4. Use Green’s Theorem to find # (3 ye3 x + xy) dx + (e3 x + x 2 ) dy where C is the path of line C segments from (0, 0) to (2, –2) to (2, 2) back to (0, 0). Due May 26, 2020 Math 281 - Test 4 page 3 of 4 5. Let F( x, y, z ) = ( y + z )i + ( x + z ) j + ( x + y)k . [One can test that F is conservative. Assume this.] a. Given that F is conservative, find a function f such that %f = F . b. Evaluate the integral # F & dr where C is the path of line segments from (2, 1, 2) to (2, 1, 5) C to (2, 7, 5). Use a method of your choice. [The Fundamental Theorem of Line Integrals will be easiest if you trust your work in part a. Note that C is not a closed curve.] Due May 26, 2020 Math 281 - Test 4 page 4 of 4 6. You are asked to calculate the line integral ' ) F & dr where C is the ellipse produced by (C intersecting the plane x + z = 0 with the cylinder x 2 + y 2 = 1, and F( x, y, z ) = z, x, y . Because '' you see that curl F = 1, 1, 1 , you choose to use Stokes’ Theorem ' ) F & dr = )) curl F & dS and (C ((S instead to calculate the surface integral, where S is the portion of the plane paramaterized by r(u, v) = u cos v, u sin v, $ u cos v , 0 ! u ! 1, 0 ! v ! 2" To that end you do the following: a. Calculate ru (u, v) . b. Calculate rv (u, v). c. Calculate ru # rv . 2" '' curl F & dS = ' ) )) (0 ((S 1 ' ) 1,1,1 & $ru # rv % du dv . (0 d. Calculate [e. Optional: C is paramaterized by r(t ) = cos t, sin t, $ cos t , 0 ! t ! 2". Show ' ) F & dr calculated directly gives you the same answer. For modest extra credit.] (C
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