Use Stoke's Theorem to determine the circulation of the vector field F (x, y, z) =(3y²) î – (cos z)j + (y – 2e*)k around the square on the x-y plane defined by R: – 1< x <1 ; -1

Q: What is the (numerical) value of the 3-form field (x² + y + z?) dx A dy A dz on the ordered triple…

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Q: Use the divergence theorem to find the outward flux of the vector field F(x, y, z) = 4x²i + 5y²j +…

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Q: Evaluate the surface integral ∬SF⋅dS for the given vector field F and the oriented surface S. In…

A: Compare the given equation with general equation and find the value of P, Q and R.

Q: Use the Divergence Theorem to compute the net outward flux of the field F=(2x, 3y, -z)  across the…

A: To calculate the net outward flux of the given vector field across the given surface

Q: The vector field F = z tan-(y²) i + z³ In(x² + 1) j + (cos z + ay) k has a vector potential with…

A: The solution are next step is

Q: The following figure shows a half-cylindrical closed surface, defined by the cylinder x² + y² = 4…

A: As per the question we have to find the total flux of the vector field V through the flat top…

Q: Compute the flux of the vector field F(x, y, z) = (2y, 3, –2xz) through the surface S, which is the…

A: Given vector field is, Fx,y,z=2y,3,-2xz. The given surface is, y=x2+z2  and  x2+z2≤1. Now for the…

Q: The vector field F = ( , ,2²) has a scalar potential in the region R= {(r, y, z) E R³ : x> 0}

A: ANSWER: TRUE

Q: Find the curl of the vector field F(x, y, z) = e" sin yi – e" cos yj O -2e" cos yk O -2e" cos y O…

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Q: Find the divergence of the vector field F. F(x, y, z) = In(9x2+ 6y² )i + 54xyj + In (6y? + 5z?)«…

A: To find the divergence of given vector field

Q: Consider the vector field F(x, y, z) = (2ry, x², 1). (c) Calculate the line integral / F - dĩ where…

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Q: I. Use the Divergence Theorem: (² F. ds = Ifv. Fav SSS E = to calculate the flux of the field F…

A: Divergence theorem uses  Double integration converts into triple integration.

Q: Evaluate the line integral of the vector field F(x, y, z) = (x²y³, e™y+z, x+ z²) around the circle…

A: Solution is as below

Q: 3. Consider the vector field F = (y, -x, x² + y?). Find the flux of F through S if: 9– x2 – y? above…

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Q: Use the divergence theorem to calculate the flux of the vector field F(x, y, z)=-3ry-ly:j-Szak…

A: To find- Use the divergence theorem to calculate the flux of the vector field F→x, y, z = -3xyi^ -…

Q: Compute the flux of the vector field F(x, y, z) = 2xj+ 4yk through the surface S, where S is the…

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Q: Consider the surface integral of the vector field F = [xy?, 3x², x²z] over a surface S, where S…

A: We will use of gauss divergence theorem

Q: Evaluate the line integral F. dr, with vector field F(x, y, z) = sin xi + cos yj + xz k, and path C…

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Q: Use the divergence theorem to find the outward flux F ndS of the vector field F = cos(8y + 8z) i + 4…

A: …

Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F = (4x…

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Q: Find the curl and divergence of the vector field. F(x,y,z)=x^2yz i -x x^3y k

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Q: Q2) Find the flux of the vector field F(x,y,z) = (x,y, z) , where the surface S is the portion of…

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Q: Find the divergence of the vector field F→(x,y,z)=7sinxi→−cos(3y)j→+z^9k→.   Enter the exact…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F=&lt;x,2y,-z&gt; across the…

A: From the Divergence theorem, for a positively oriented boundary surface (outward Normal) of region E…

Q: Find the flux of the field F =-x i - y j + z2 k outward (normal away from the z-axis) through the…

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Q: Compute the flux ∫F · n ds of F across the curve C for the given vector field and curve using the…

A: Solution; Consider the given function: F(x, y) = xy, x-y = P, Q P = x y Q = x -y and the boundaries…

Q: Compute the flux of the vector field F 3yi + 4j – 3xzk through the surface S, which is the surface y…

A: We apply formula for finding this

Q: The vector field F=(x+2y)i+(2x+y)j is conservative. Find a scalar potential f and evaluate the line…

A: Given function is, F=x+2yi+2x+yj

Q: Let ryz , (교2 + y"), 17 F = ( x2 +y? be a vector field on U = {(x, y, z) E R³ | (x, y) # (0, 0)}.…

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Q: Consider the vector field F = 2xy i + (x2 + 2y) j. Verify the Green's Theorem in the closed region…

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Q: Let E be the solid region bounded by the upper half-sphere x2 + y² + z² = 4 and the plane z = 0. Use…

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Q: 4. Use the Divergence Theorem to find the outward flux of the vector field F(x,y, z) = v3;1 v3;173…

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Q: Consider the vector field f (x, y, z) = xz i + yz j+ x²y² k . Let S be the surface of the sphere of…

A: Using parametrization of the surface and curve

Q: Given the vector field F (x, y, z) = 2xyi + (x² + z?) j +2yzk a. Verify that the vector field…

A: Ans :A)

Q: Find the divergence of the vector field F. F(x, Y, z) = In( 9x²+ 8y² )i + 72xyj + In(8y² + 7z? )«…

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Q: 2.4) Use the divergence theorem to find the outward flux of the vector field F(x, y, x) = x³i+ y³j+…

A: The divergence of the given vector field is  ∇·F=∂∂xx3+∂∂yy3+∂∂zz3=3x2+3y2+3z2 Let, S be the surface…

Q: Compute the flux of the vector field F = zi + 8xk through the parameterized surface S oriented…

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Q: The vector field F = ( z, z²) has a scalar potential in the region R = {(x, y, z) E R° : x² + y? >…

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Q: Consider the surface integral of the vector field F= [xy², 3x², x²z]over a surface S, where S…

A: From Gauss divergence theorem, we have ∬SF→⋅dS=∭VdivF→dV   The vector field is F→=xy2,3x2,x2z.…

Q: Show that the vector field F = 2.xyi +(x² – 2 y)j is conservative and evaluate the line integral…

A: Given the vector field

Q: Verify Stoke's theorem for the vector field F = (2x - y) I– yz"J - y z K over the upper half surface…

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Q: Find the flux of the curl of the vector field F(x, y, z) = [(z2 −4) cos(xyz), 3x+y,(z2 −4) sin(ex…

A: Surface is oriented outwards. And -2 &lt;= z &lt;= 2

Q: Verify Stokes' Theorem holds for the vector field F = yz i + x k where S is the portion of the plane…

A: The given vector field is F=yzi^+xk^ where S is the portion of the plane 3x+2y+6z=6 in the first…

Q: 2. Determine the outward flux of the vector field F = (xy)î + (y²)ĵ over the region R depicted %3D…

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Q: Find the divergence of the vector field V(x, y, z) = -2xi + (y + 5 cos(x))j+ (7z + e8xy) k.

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Q: Consider the one-form w = x' cos(y) dx + xe13% dy+4z² dz on R3. The associated vector field is: F(x,…

A: For the solution follow the next step.

Q: Determine whether the vector field…

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Q: Let C be the helix that winds around the cylinder x2+y? = 1 (counterclockwise viewed from the…

A: here i used parametric equation of helix and done simple calculation like dot product and after that…

Q: Find the divergence of the vector field F. F(x, y, 2) = In(7x2+ 6y? )i + 42xyj + In(6y2. + 72² )«…

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Q: Evaluate F Î ds for the vector field F(x, y) = yì – xj counterclockwise along the circle x² + y² = 1…

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