Use the Divergence Theorem to calculate the surface integral F. aS; that is, calculate the flux of F across S. F(x, y, 2) = 3xy²i + xe²j + z°k, S is the surface of the solid bounded by the cylinder y + z² = 1 and the planes x = -3 and x = 3.

Q: (iv) and (0, -1) to (1, 1). Find the circulation of F round the curve C, where F=e* siny i + e*…

A: Circulation of a vector field about a contour is the line integral of the vector field when it is…

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

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Q: b) Given F(.r, y. :) = (r + cosh 2)i+ (2y-3r*y)j-(r + 4y*:) k. Use Gauss's theorem to calculate F.n…

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Q: 1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the…

A: The vector field is,F¯=6x2+2xy, 2y+x2z, 4x2y3The region D is cut from the first octant by the…

Q: Use the Divergence Theorem to find the flux of F(x, y, z)=z³ i-x³j+y³ k across the sphere x² + y² +…

A: This is a problem of vector calculus.

Q: Evaluate | F dr where Cis the intersection of the surfaces y =x' and z =x' from the origin to…

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Q: tz = g(x, y) = f(3 cos(xy), y + e#") provided that f(3, 7) = 4, f1(3, 7) = 2, (3, 7) = 3. %3D Find…

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Q: Compute the surface integral of the function f(x, y, z) = 2xy over the portion of the plane 2x + 3y…

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Q: Let S be the surface given by x² + y²+2z² = 1, x > 0, y ≥ 0, z ≥ 0. Find the flux of (z, 0, 0)…

A: Step 1:The surface S given by x2+y2+2z2=1 is an ellipsoid, and we are considering the upper half…

Q: Use Stokes' Theorem to evaluate fF.dr if F(x,y,z) =( z-y)i +(x+z)]- (x +y)K and C is the curve of…

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Q: (x², y², z(x² + y²)) around any closed curve on the surface of Show that the circulation of F = the…

A: Circulation of vector field F around a close curve C = ∫CF.ds If parametric equation of curve C…

Q: (1) Let F(r, y, z) = (y², e², z). (a) Compute the divergence of F. (b) Use the divergence theorem to…

A: Answer

Q: Evaluate f (x-y) dV, where E is the solid E in the first octant bounded by the surface z=1-² and the…

A: In the given problem we have to evaluate the integral                                       where…

Q: Exercise 1. A student is asked to prove If m is even and n is odd, then m² + n² is odd. 2 The…

A: Let us answer above questions

Q: Integrate VX F, F = (3y, -xz, -yz) over the portion of the surface 2z = x² + y² below the plane z =…

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Q: Evaluate f. F dS where F(x,y, z) = (4x³z, 4y³z, 3z*) and S is the surface of the solid bounded by…

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Q: Evaluate ff F.n ds where F(x, y, z) = 12xi +24j+24x²k and is that portion of the paraboloid z = x² +…

A: We will find normal vector n and region of integral on xy plane to find value of integral .

Q: i) Find g1 (0, 4). ii) Find g2 (0, 4). iii) Find the equation of the tangent plane to the surface z…

A: Since g(x, y) = f(3cosxy,y+exy)

Q: Calculate F- dS where F = (3ry², re², z³) S is the surface of the solid bounded by the cylinder y? +…

A: The general form of a vector-valued function in three dimensions is given by F¯=F1,F2,F3, where F1,…

Q: 2) Find the area of the surface r(u,v)=u cos vi+u sin vj+2vk where 0su<3 and 0<vs 2n .

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