In Exercises use the surface integral in Stokes' Theorem to calculate the circulation of the field F around the curve C in the indi- cated direction. 7. F = x²i + 2xj + z²k C: The ellipse 4x² + y² = 4 in the xy-plane, counterclockwise when viewed from above 8. F = 2yi + 3xj – z²k C: The circle x² + y? = 9 in the xy-plane, counterclockwise when viewed from above 9. F = yi + xzj + x²k C: The boundary of the triangle cut from the planex + y + z = 1 by the first octant, counterclockwise when viewed from above 10. F = (y² + z?)i + (x² + z²?)j + (x² + y²)k C: The boundary of the triangle cut from the plane x + y + z = 1 by the first octant, counterclockwise when viewed from above

Q: use the surface integral in Stokes’ Theorem tocalculate the circulation of the field F around the…

A: Use the surface integral in Stokes’ Theorem to calculate the circulation of the field F around the…

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

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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 outflow of the field F(x,y,z)=(e^y−xz^2,x−yx^2,y−zy^2), across the edge of the solid contained…

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Q: Compute the line integral of the vector field F(r, y, z) = (x + 2y + 3z) i+ (2x + 5y)j+(z – 3y)k…

A: As per policy, we will solve the Question 4 only please repost it to get solution of Question 5. In…

Q: (a) Given the vector field F = xy + 2yzý + 3zx 2, calculate the line integral around the closed path…

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Q: Let C be the ellipse in which the plane 2x + 3y - z = 0 meets the cylinder x2 + y2 = 12. Show,…

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Q: find the circulation and flux of the field F aroundand across the closed semicircular path that…

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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: Given that F(x, y, z) = (ey, xey + ez, ye²) is a conservative vector field. a. Find a potential…

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Q: 3) Given the vector field F = circulation around the triangle formed by the intersection of 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: Find the circulation and flux of the field F = - xi – yj around and across the closed semicircular…

A: The circulation of a vector F=M i+ N j along the curve C is defined by ∮CF·T ds=∮CMdx+Ndy. Here…

Q: Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F = 4zi +…

A: The given question can be solved as shown in step2

Q: Let f(x, y, z) = (x² + y² + z²)¬/2. Show that the clockwise circulation of the field F = Vf around…

A: Calculate: F=∇f   =∂∂xx2+y2+z2-1/2,∂∂yx2+y2+z2-1/2,∂∂zx2+y2+z2-1/2   =-x2+y2+z2-3/2x,y,z

Q: Let a, b > 0. Verify Stokes' Theorem f F - dĩ = SS, curl(F) - ndS for 2xz the vector field F(x, Y,…

A: Let a, b>0. Verify Stokes' Theorem ∮F→· dr→ =∫∫σ curl F→. n dS for the vector field F→x, y,…

Q: find the circulation and flux of the field F aroundand across the closed semicircular path that…

A: The given vector field is Fx,y=-yi+xj.

Q: Use the Stokes' Theorem to caleulate the circulation of the field F = -6y*7 + 6x*J° + 7z*R° around…

A: Given field, F→=-6y3i^+6x3j^+7z3k^ where C is the curve of the paraboloid x2+y2=z cut by the…

Q: Evaluate the surface integral of D enclosed by the rectangular parallelepiped formed by the planes x…

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Q: Calculate the flux of the field F across the closed plane curve C where F=xi+ vj and the curve is…

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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: Find the counterclockwise circulation and outward flux of the field F = xyi + y2j around and over…

A: Theorem used:

Q: Consider the vector field F = a?i+ ?j+ y²k Calculate the divergence: Calculate the curl, Suppose S…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, y, - 2z) across the…

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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: Find the flux of the field F up through the part of the 1+ xy paraboloid z = 1 – x² – y? that…

A: FORMULA: Flux=∬F.n^ dS According to Stokes theorem, ∬F.n^ dS =∬∇×F . n^ dS According to the…

Q: 3. Use the surface integral is Stokes' Theorem to calculate the flux of the curl of the field F…

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Q: Let ƒ(x, y, z) = (x2 + y2 + z2)^(-1/2). Show that the clockwise circulation of the field F = ∇ƒ…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (x,yz) across the…

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Q: Given the following vector field and oriented curve C, evaluate F•Tds. F= (x,y) on the parabola r(t)…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (-3x, y, 8z) across the…

A: Given:F=-3x,y,8zS=x,y,z:x2+y2+z2=6

Q: Given the radial field (x, y, z) (x² + y² + z²)³/2 |r|3/2 Find the outward flux across the sphere of…

A: Image is attached with detailed solution.

Q: - For the vector field in the xy plane: F= (zi - v)), use Stokes theorem to compute the line…

A: We will use stokes theorem for 2d

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: 2. Find the surface integral SS. ř · dS for the vector field F = xyî + yzĵ + zxk where S is part of…

A: To find Surface integral ∬SF→·dS→: The Surface S is of the form z=g(x,y) and is oriented upwards.…

Q: The field F = -y³i + (x³ + e=y)j + 0k is the velocity field of a flow in space. Determine the…

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Q: Find the outward flux of the field F = xz i + yz j + k across the surface of the upper cap cut from…

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Q: 18. F = 1 + i + (tan' y)j x² + y? = 1

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Q: Evaluate the flux integral , curl (z cos xi + yj+x sin y k) dS where S is the part of the outward…

A: By  stoke's theorem ∫∫sCurl F.dS  =∫∂sF.ds

Q: 1. Use Green's Theorem to find the counterclockwise circulation of the field F around the curve C as…

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Q: Consider the closed, solid region E between the paraboloid z = 4 – x? – y? and the xy-plane, and…

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Q: Find the flux of the field F = z k across the portion of the sphere x2 + y2 + z2 = a2 in the first…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, 3y, - 3z) across…

A: Let's find.

Q: (4) Use Stokes' Theorem to calculate the work-done by the force field F(r, y, z) = (x² + z²)î + (y²…

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

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Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F = by the…

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Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F =xi+ 5xj +zk…

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Q: 03 Verify Stokes theorem for the vector field F= 2(-+x)i-zyi-zy'k over the upper hlf surface of…

A: To find- Verify Stokes theorem for the vector field F→ = 2-y2 + xi→ + z2yj→ - zy2k→ over the upper…