65. Transform the surface integral curl(F dS into a line integral using Stokes' theorem, and evaluate the line integral: (a) F(x, y, z(y-, yz,-xz), S consists of the five faces of the cube 0 n is outward 2, unit normal x, y, z Answer: -4 (b) F(x, y, z) (xz,-y,x2y), S consists of three faces not in the xz-plane of the tetrahedron bounded by the three coordinate planes and the plane 3.x + y + 3z = 6. The unit normal n is outward of the tetrahedron. Answer: 4/3

Q: Let F(x, y, z) = (-y, xy, 0) and C be the curve of the intersection of z = x² + y² and z = 2x such…

A: Given clear answer below

Q: Let F(x, y, z)= (yz)i+(2x)j+(xy-x)k . (1) Calculate curl F. (ii) Evaluate the line integral F dr,…

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Q: 3. Use Stokes' theorem to evaluate the integral F. dr where C is the curve given by the intersection…

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Q: Use Stokes' Theorem to find the line integral of F = (xy, yz, xz) around the boundary of the surface…

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Q: 7. Evaluate the line integral f, F· dr where F(x,y, z) = x³i – y²j+ zk and C is the x2 + 2xy + y²…

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Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F = +z)i+ (x…

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Q: Use Stokes' Theorem to evaluate the integral: [cur S curl F. dS F(x,y,z)=ze' i+xcos (y) j+xzsin (y)…

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Q: Verify the formula fF-T ds = [ (curl F) -n ds in Stokes' Theorem by evaluating the line integral and…

A: According to the given information, it is required to verify the stokes theorem.

Q: Use Stokes' Theorem to find the line integral [((x² + 2y² ((x + 2y²)i + (y + 2²)j + (2 + 2x²)k) · dř…

A: Given that C is boundary of triangle with vertices 1,0,0,0,1,0,0,0,1 So, equation of the plane with…

Q: 3- Use Stokes theorem to compute the line integral SF · dr, where F = [e-3y, e², e-2*] and C is the…

A: Given that F=e-3y,ez,e-2x The objective is to find ∫CF.dr. Where C is z=2x2,0≤x≤2,0≤y≤1.

Q: 3- Use Stokes theorem to compute the line integral F . dr, where F = [e-3y, e, e201 and C is the…

A: By using the stokes theorem we need to find ∫CF·dr, where F=e-3y, ez, e-2x and C is the boundary of…

Q: Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented…

A: Please see the below picture for detailed solution.

Q: Evaluate the line integral ∮C F ⋅ dr using Stokes’ Theorem. Assume C has counterclockwise…

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Q: Use Stokes' Theorem to find the line integral [((x² + x + 2y²)i + (y + z³)j + (z + 2x²)k) · dr where…

A: To Evaluate: The integral ∫C(x+2y2)i+(y+z2)j+(z+2x2)k·dr. where C is the boundary of the triangle T…

Q: Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented…

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Q: Use Stokes’s Theorem to evaluate the circulation line integral ∫_c F·dr where         F(x, y, z) =…

A: Given vector field is Fx,y,z=z2-x+x2-yj+y2-zk Then,  ∇×F=ijk∂∂x∂∂y∂∂zz2-xx2-yy2-z=2y,2z,2x Given…

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

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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: 3. Consider A(r) = (y, -x,0), and the line integral A• dr along the C boundary of the plane x + y +…

A: The given vector function is A(r)=(y,-x,0)., the line integral ∮CA·dr is along the boundary of the…

Q: 387. Use the divergence theorem to calculate surface integral F. dS when F(x, y, z) = x²z³i+ 2xyz³j+…

A: Given that: F(x,y,z)=x2z3i+2xyz3j+xz4k and S is the surface of the box with vertices (±1,±2,±3).

Q: 14. Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field: (r…

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

A: Calculate the flux of the curl of the field F = z i + 5 x j + y k, by using the surface integral in…

Q: Use Stokes's Theorem to evaluate F(x, y, z) = yzi + (2 − 6y)j + (x² + y²)k, x² + y² ≤ 49 S: the…

A: Let's solve part 2.

Q: Evaluate the surface integal [[Curl F.ñ dS for the Sur face and F=xt-y²+ (x+4+z)Ñ and C is the bo…

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Q: 2. Evaluate the surface integral f, (curlF) · n dA directly for the given F and S, F =…

A: Here we have to surface integral ∬s(curlF).n dA

Q: 4. Use Stokes' Theorem to evaluate curl F · dS when F = ze i+ x cos yj+ xz sin y k and S is the…

A: Given : F = zeyi + x cos y j + xz sin y k and Sis the hemisphere x2 + y2 + z2 = 16, y ≥ 0.   To…

Q: 3) Use Stokes theorem to evaluate (| curl F-ndS. where F= (xyz, x, e" cos z), while S is a surface…

A: r(t) is the vector function for C, the curve around the base of the hemisphere (and therefore 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: Evaluate the flux integral ff. curl (z cos x i+yj+x sin y k) ds where S is the part of the outward…

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Q: 14. Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field: F =…

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Q: Evaluate the line integral F. dr where F(x, y, z) = x²y i+ x³ j+e² tan-¹ z k and C is the curve of…

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Q: Let F(x, y, z) = zi + xj – yk and the curve Cbe the boundary of that portion of the surface S above…

A: Given : vector field F→x, y, z=z i^+x j^-y k^ and the curve C be the boundary of that portion of the…

Q: Evaluate the line integral in Stokes' Theorem to find the surface integral [[ (V × F) · ndS, where…

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Q: 3- Use Stokes theorem to compute the line integral Sc F dr, where F = [e-3y, e², e2"] and C is the…

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Q: (3) Evaluate the flux integral where S is the surface given by 22 =r + y², 0< z< 4, oriented upwards…

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Q: Use Stokes theorem to evaluate ff.curl F.nds. where F = (xyz, x, e* cos z), while S is surface x² +…

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Q: please only answer part (b) thank you

A: For any vector field F bounded by a curve C with positive orientation and S is the smooth surface.…

Q: Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F= 3yi +…

A: Given, F→=3yi^+5-3xj^+z2-2k^r→=3sinφcosθi^+3sinφsinθj^+3cosφk^ So, the flux of curl F is, Flux of…

Q: Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented…

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Q: 3- Use Stokes theorem to compute the line integral ScF - dr, where F = [e-3,e²,e-2=] and C is the…

A: as per our company guidelines we are supposed to answer only first 3 sub-parts. Kindly repost other…

Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F = (y +z)i+…

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Q: 3- Use Stokes theorem to compute the line integral ScF · dr, where F = [e-3y, e², e-2"] and C is the…

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Q: 2). Use Stokes' theorem to evaluate the integral 6 F • dř for F(x, y, z) = (x – y)î + (y – z)ĵ + (z…

A: Formula: By Stoke's theorem we have ∮C F · dr¯  =∫t=t1t=t2 F¯(r(t))r'(t) dt, where r(t) is…

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

A: Given field : F=x2yi+2y3zj+3zk    surface S is r(r,θ)=(r cosθ)i+(r sinθ)j+rk    0≤r≤1,     0≤θ≤2π

Q: 2. Consider the triangle {(x, y)| x, y > 0, 2x + y < 1}. Let C be the boundary of this triangle…

A: Given triangle is x,y:x,y≥0,2x+y≤1. Let C be the boundary of this triangle oriented in the…

Q: 3- Use Stokes theorem to compute the line integral SF· dr, where F = [e-3y, e², e-2*] and C is the…

A: