10) Integrate W =./x2 +y2+22 e tr+vrj in the region given by the set 11) Is the following vector field F conservative, compressible and rotational? F(asino) Fx, y 2Esnsn) 10) Integrate W =./x...
10) Integrate f(x, y) = sin (Vx2 + y2) over the region 0 < x2 + y2 = 16
10. For this problem, use the vector field Fx, y, z) = (y2, 23, ry+22) (a) 3 points Show that F is conservative. (b) 8 points Find a potential function f(x, y, :) such that F = V. (c) 4 points Evaluate SF. dr where C is any smooth curve from (1,0,-2) to (4,6,3). (d) 2 points What is the value of JF dr where is the circle 12 + y2 = 36 in the ry-plane?
evaluate JJ. (< –Y) A. ) Integrate f(x, y, z) = x2 + y2 + 22 over the cylinder x2 + y2 < 2,-2 <2<3 (IL dx dy dz Feraluate
Consider the given vector field. F(x, y, z) = (9 / sqrt(x2 + y2 + z2)) (x i + y j + z k) Find the curl of the vector field. Then find Divergence
y? +1 e a vector field 12) Let F(x,y) = (10 + 15y3 + cos (In(xe*)))i + (-6- 15x3 – sin" (ev? V In on R2. Use Green's Theorem to compute (F. dr Where C is the negatively oriented boundary of the region bounded by x2 + y2 = 4 and x2 + y2 = 9, in the first quadrant only.
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the surface S be the part of the sphere x2 + y2 + x2 = 4 that lies above the plane 2=1 and be oriented downwards. (a) Find the divergence of F. (b) Compute the flux integral SS. F . ñ ds.
+ cos(y) is conservative by responding to the 2. Show that the vector field F(x,y) = (ye* + sin(y))i + ( following steps: a.) Determine both P(x,y) and Q(x,y) given F. b.) Demonstrate your answers in a.) satisfy Clairaut's theorem. c.) Partially integrate P with respect to r to obtain the potential S(= y) = P(x,y)da = (1.x) + C) where (a,b) is the anti-derivative of P(x,y) with respect to r and C(y) is a function of y such that...
Triple Integration Problems. 1. Integrate zdV JJ w where ll' is enclosed by the planes z = 0 and cylinders x2 + y2 4 and x2 + y,: 9 = x+9+ 3 and by the 2. Integrate where E is bounded by the zu-plane and the hemispheres z/9-2y2 and z = V/10-22-27 Change the order of integration and evaluate x3 sin(уз)dydx. 0 Jr2 1. Integrate zdV JJ w where ll' is enclosed by the planes z = 0 and cylinders...
3. Consider the vector field F(x,y) = (27x D = {(1,y): 0 < rº + y2 <2}. +ya) defined on the region D where a) Directly compute SF. Tds using the definition of the line integral, where C is the unit circle oriented counterclockwise. b). Use Theorem 3.3 (Test for Conservative Vector Fields) from the text to determine if F is conservative. Is your answer consistent with part a)? If not, what is the source of the discrepancy?
2. a) Find a potential of the vector field f(x, y) = (a2 +2xy - y2, a2 - 2ry - y2) b) Show that the vector field (e" (sin ry + ycos xy) +2x - 2z, xe" cos ry2y, 1 - 2x) is conservative.