F dS; that is, calculate the flux of F across S. Use the Divergence Theorem to calculate the surface integral (x3 y3)i (y3 z3)j + (z3 + x3)k, F(x, у, 2) = S is the sphere with center the origin and radius 2.

Q: Let S be the surface defined by the vector function R(u, v) = (2e" sin v, 2e" cos v, u²+u), where u…

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Q: Determine the flux of F(x, y, z) = across the surface with an upward orientation. Let the surface…

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Q: Verify Stokes' theorem where A = (x-z)i + (x³ +zy)j - 3x y²k and S is the surface of z=2-√√x² + y²…

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Q: Find the flow of F=xzi-yk through the upper part of plane z-1 in the x +y² +z = 4sphere. %3D

A: The general form of a vector-valued function in three dimensions is given by F=F1i+F2j+F3k, where…

Q: Your friends correctly calculate the gradient vector for f(x, y)=x²+y at the point (2,-3) as…

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Q: Sketch a surface representing a function f of two variables x and y. Use the sketch to give…

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Q: Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u ER…

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Q: Draw the parameterized surface

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Q: Let F(x, y, z) = xye" – 2° Ja and P(1,0, –1). Let S be the surface defined by the equation xye" –…

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Q: Verify Green's Theorem in the plane for F = xyî + x²ĵ and where C is the boundary of he region…

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Q: Verify Stoke's theorem for the vector point function yz2 (xz² .3 + ex over the surface S of the cube…

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Q: 3) Evaluate, to the nearest hundredth, 11. (V x F) dS where F = and S = {(x, y, z)| 2² − 1 = 3x² +…

A: Here we will use Stokes theorem  According to stokes theorem  ∫∫S∇×F.dS=∫CF.dr C is the  boundary…

Q: Let S be the surface defined by the vector function R(u, v) = (v + 4, 4u - v, uv²). Set up an…

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Q: Consider the space curve represented by the intersection of the surfaces. Represent the curve by a…

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Q: II. Let S be the portion of the cylinder x² - 6x + z = 4, oriented by upward-slanting normal…

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Q: Consider the function f(x, y,z)=x² - y² - z² + 4x+2z+4. a. (i) Sketch and describe the level…

A: given function fx,y,z=x2-y2-z2+4x+2z+4 claim- (i) sketch and describe the level surface for f=0 and…

Q: 4. Let F(r, y, z) =-ri+ yj + zk and o be the portion of the surface z = 8x-² - y? that lies above…

A: Given vector field is  F(x,y,z)=43xi+43yj+53zk and also the surface σ is portion of the sphere…

Q: Calculate the curl(F) and then apply Stokes' Theorem to compute the flux of curl(F) through the…

A: The given problem is to find the curl of the given  field and also to find the flux of curl of the…

Q: A normal vector to the tangent plane for the surface z = , at the point (1, 1, 1) is: y - Select…

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Q: Let F = 2x3i + 2y3j + 2z3k and let S be the surface consisting of the hemisphere x2 + y2 + z2 = 1, z…

A: Given, F=2x3i+2y3j+2z3k and  S be the surface consisting of the hemisphere x2+y2+z2=1, z≥0 and the…

Q: The gradient vector for the surface xy + xz + yz = 1 at the point (1, 1,0) is (1, 1,0). Select one:…

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Q: Let F(x, y, z) = xzi+yzj + xyk, Surface S is the part of the sphere x² + y² + z² = 4 that lies…

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Q: Suppose that the plane tangent to the surface z = f(x, y) at (2, 7, 8) has equation z + 8x – 2y =…

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Q: Show that the tangent plane to the quadric surface at the point (x0, y0, z0) can be written in the…

A: Given surface is x2a2+y2b2+z2c2=1 To find the equation of the plane, find the normal of the given…

Q: Let F = –9zi+ (xe*z – 2xe*² ) }+ 12 k. Find f, F •dÃ, and let S be the portion of the plane 2x + 3z…

A: Given, F→=-9zi^+xexz-2xex2j^+12k^ and S be the portion of the plane 2x+3z=6 0≤y≤4

Q: Find the flux of F= (y, z-y, z). across the tetrahedron with vertices (0,0,0), (1,0,0), (0, 1,0),…

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Q: JJs where S is the sphere x² + y2 + z2 = 1.

A: Given S is the sphere                         x2+y2+z2=1 To evaluate:             ∫∫S3x+3y+z2dS…

Q: Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u eR…

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