Use the Divergence Theorem to compute the surface integral F-nds, where F(x, y, z) = (x³ - y³, y³ — 2³, 2³ - 2³) and S is the unit sphere x² + y² + z² = 1 S (the surface integral is thus the net outward flux of the vector field F across S). (Hint: use spherical coordinates to evaluate the triple integral).

Q: The vector equation r (u, v) = = u cos vi+usin vj + vk, 0 ≤ v ≤ 10m, 0 ≤ u ≤ 1, describes a helicoid…

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Q: 2. Verify Stokes's Theorem for F(x, y, z) = y i+ z j +x k and the plane x + y + z = 2 in the first…

A: Given, Fx,y,z=yi^+zj^+xk^x+y+z=2 So, curl F is calculating by, Curl F→=∇→×F→Curl…

Q: 2. Verify the Stokes's Theorem for the calculation of the work done by the vector field: F = (r,y²,…

A: Stokes theorem is verified

Q: H.W: Let V = {(x, y): x 20} with standard operation on R2. Is Va vector space?

A: A vector space is a set V that satisfies the following 8 conditions. Here two operations "+" and "."…

Q: Calculate the surface integral ( i.e. f(y x F) · dS )of the curl of the vector field: F(r, 6, 2) =…

A: The given vector field is  F→(r, ϕ, z)=1π8ϕ^+2z^

Q: Evaluate /| cos(x? + y²) dA, where D is the D disk x? + y <I Select one: a. -n O b. A O c. O O d. 2…

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Q: Vector Field: F(x,y) = (xv, 3x, – 2y) and Surface: z = 4 - x - 3 y? Use the normal vector formula to…

A: Given surface z=4-x-3y2

Q: Find a vector parametric equation for the ellipse that lies on the plane 4x - 4y + z = -7 and is…

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Q: Compute the flux of the vector field F = : 4xi + 4yj through the surface S, which is the part of the…

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Q: -y Consider the vector field F = (y²¹, ²²), and the curve defined by 1% + 2 + 2 = 2 and z = c.…

A: Consider the vector field .The curve is given by  and z=c.Substituting, we getThe curve is an…

Q: Compute the unit normal vector field that will give the surface parametrized by r(u, v) = (e"…

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Q: The upper half of the unit sphere x2+y2+z2=1 is z=Square root of 1-x2-y2. Find its centroid

A: Solution Given The intersection gives z2=1-x2+y2we need to find the centroid

Q: (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of…

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Q: Evaluate the surface integral [[ (a² + y²) dS where S is the surface with vector function r(u, v) =…

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Q: a) Find the curl V×Fif F = (y',2xy,yz)

A: Since you have posted more than one question, we solve the first one for you. For the remaining…

Q: The part of the plane with vector equation r(u, v) = (u + v, 2 – 3u, 1 + u – v) that is given by 0 <…

A: This is a problem of surface integral.