Set up the integral for the divergence theorem both ways. Then find the flux. F(x,y,z) = x2 i + xy j + z k E is the solid bounded by the paraboloid z = 4-x2-y2 and the xy plane.

Q: The electric field in the xy-plane due to a point charge at (0,0) is a gradient field with the…

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Q: Use the Divergence Theorem to evaluate [/F F. N ds and find the outward flux of F through the…

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Q: Let 2 Evaluate {(x,y) = = x 20, y ₂0 1 ≤ x² + y² = 4 polar coordinates and integrate by parts) So…

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Q: 8. Choose, refer and use one of the theorems of Vector Calculus to evaluate the line integral F- ds…

A: To find- Choose, refer and use one of the theorems of the Vector Calculus to evaluate the Line…

Q: Let f(x, y) = |x − 1| · yś. (a) Compute, if they exist, the partial derivatives fx and fy at the…

A: Partial derivative of a function f(x,y) at point (a,b) fxa,b=limh→0fa+h,b-fa,bh…

Q: 1. Aflock of birds is perched on a tree branch. The branch B can be modeled as a 3-D line segment…

A: We have to parametrize the branch B and evaluate the integral.

Q: (a) Compute directly I = S Ss F-dS, where S is the surface of the sphere x? +y? +z? = 4 and F =…

A: X²+y²+z²=4=2² Therefore radius of sphere is 2.

Q: . Evaluate the integral: ſf (V · n) da over any surface whose bounding curve is in the (x, y) plane,…

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Q: Use the divergence theorem to calculate the flux of F = (x - 8y)i + (y – 4z)j + (z - 8x)k out of the…

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Q: Evaluate fc F • dr, where F = (z, y2, x) and C is parametrized (in the positive direction) by r(t) =…

A: F=z,y2,x and C is the parametrized by r(t)=t+1,et,t2 for 0<t<2. We have to evaluate ∫CF·dr.

Q: Find the point P(z, y, z) at which the paraboloid has a tangent plane parallel to the plane (z, y,…

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Q: 2. Let C be the part of the curve z = y in the plane x = 1 from (1,2,4) to (1,0,0). Consider a cable…

A: 2) Given that C is the part of the curve  (a) Now we have to parameterize C with parameter t as…

Q: Find the flow of the velocity field F = (y-x) - (x² + y²) j along each of the following paths from…

A: A velocity field is a vector field that describes the velocity of a fluid or a gas at each point in…

Q: Find the flux of F = 3zi + 5y? i+ 3x k out of the closed cone y = Væ2 + z2, with 0 < y < 1. flux =

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Q: 1. Find a vector valued function that represents the curve of intersection of the semi-ellipsoid x²…

A: The equation of the semi-ellipsoid is The cylinder is

Q: 11, 31. Find the flux of F = (x²y, xy, y²z³) across the cube bounded by the planes x = y = ±1, z =…

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Q: Use the definition V = ( ) to prove the identity: curl (curl F) = grad (div F) – V²F Assume that…

A: To prove:

Q: Find the circulation of F= (xy)i+(yz')j-(2xyz)k around the curve C in the counterclockwise direction…

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Q: Parametrize the intersection of the surfaces y² − z² = x - 1, y² + z² = 25 using t = y as the…

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