1. Verify Stokes' Theorem for F = (2x - y)i - yz²j - y² zk, where S is the upper half surface of the sphere x2 + y2 + z2 = 1 and C is its boundary. Let D be the projection of S on the xy-plane.

Q: 1. Let F(x, y, z) = yi+ xj+ yzk and the surface S be the part of the paraboloid z = 4 – x2 – y? that…

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Q: Verify Stokes' Theorem if o is the portion of the sphere x + y +z² =1 for which z20 and F(x,…

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Q: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

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Q: 4. Use Stokes’ theorem to evaluate fa F. dr, where F(x,y, z) = (xy,yz, zx), and C is the triangle…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: parametrizes the portion of the sphere x² + y² + z² = 9 that is above the xy-plane and below the…

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Q: 16. Let F(x. y, z) = xyi+ xzj+yzk and C be the curve of intersection between the sphere z² +7+z² =…

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Q: Evaluate the surface integral fs F. dS where F = (4x, -2z, 2y) and S is the part of the sphere x² +…

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Q: Calculate F dS if F(x, y, z) = x²yei+ xyj – 2xye k, where S is the surface of the pox bounded by the…

A: Given-F(x,y,z)=x2yezi+xy2j-2xyezk To find- ∬SF.dS ,where S is the surface of the box bounded by the…

Q: Evaluate || f (x,y,z) dS where S is the portion of the sphere centered at the origin of radius two…

A: we have the equation of sphere x2+y2+z2=22s is the portion bounded between xy plane and z=1 in the…

Q: a) Evaluate x²y? ds, where the curve C is the intersection between the surface z = := V1– y? and the…

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Q: Let H be the segment of the hyperbola x-y°= | Connecting (1,0) to (2, J3). Evaluate the line,…

A: Given  H be the segment of hyperbola x2-y2=1 connecting the points (1,0) to (2,3)  we have to…

Q: Let (S) be the surface given by the following equation. x²+y² = 1+ z?. An equation of the tangent…

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Q: Find a parametric representation for the part of the cylinder y2 + z² = 36 that lies between the…

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Q: Let (S) be the surface given by the following equation. x²+y? = 1+ z?. An equation of the tangent…

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Q: Determine the projection of the following surface on the xz plane. (xz trace) 4x – + = 0

A: Given equation of surface is 4x-y24+z28=0

Q: (4) Use Stokes' Theorem to evaluate 11. Vx (yz, xz, xy) dS where S is the surface S that is the part…

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Q: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder x² + (y − 1)² = 1.…

A: For the given curve we have the following vector valued function: R⇀=2cost. 1+sint,…

Q: Use Stokes's theorem to evaluate o F· dr, where F(x, y, 2) = xy² i+x²yj+yz k, where C is a…

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Q: EXERCISES 2.10 1. Use Stokes' Theorem to evaluate F.dr for E = (xz,-y,x²y), where C is the boundary…

A: Applying stokes theorem we solve it

Q: Attached picture is the question

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Q: 9. If F = (x – z) î + (x³ + yz)ĵ + 3xy² k and S is the surface of the cone z = a – Va? +y?) above…

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Q: 2. Use Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

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Q: 1. The part of the sphere x? + y² + z² = a² that lies within the cylinder x² + y² : = ax and above…

A: To Determine :-  The part of the sphere x2 + y2 + z2 = a2 that lies within the cylinder x2 + y2 = ax…

Q: 11. Let F = (-xy + xz²)i + (4xz + y²)j + e² k. Use Stokes' Theorem to calculate fcF. dr, where C is…

A: We have to find∫cF→.dr→.

Q: Let F = < 4x + y², 5y + z, 9z + x So F. dī, Use Stokes' Theorem to evaluate C where C is the…

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Q: Let (S) be the surface given by the following equation. x²+y? = 1+ z?. An equation of the tangent…

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Q: The tangent plane to the graph of the surface z=In(x2+y2) at the point (1,0,0) contains point…

A: Given function is, z=lnx2+y2

Q: Evaluate G if G is a solid in the first octant bounded by the plane y +z = 2 and the surface y = 1–…

A: Use the following formulae, to obtain the solution. a-b2=a2-2ab+b2∫xndx=xn+1n+1+C

Q: Calculate F- dS if F(x, y, z) = x²ye²i+ æy?j – 2xye*k, where S is the surface of the box bounded by…

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Q: b) Use Gauss' theorem to evaluate RR F. ndS where F = (x² - y)i + y³z²j + (xy – y²z³)k UTM UTM and o…

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Q: 1. Find a parametric representation for the surface: the part of the sphere x2 + y? + z2 = 4 that…

A: According to the given information, It is required to calculate the parametric representation for…

Q: Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface…

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Q: 5. Use Stokes theorem to evaluate J[V x F-ds over the surface of the paraboloid z = 5-x² - y² above…

A: If S  be   smooth surface bounded by a  closed , positively  oriented  smooth boundary curve C with…

Q: If the tangent plane to the surface z = +y is parallel to the plane r+ 2y-z 0 at the point (1, a,…

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Q: where F(x, y, z) = (ry, x²y, y) Use Gauss's theorem to calculate where is the paraboloid z = 4 – x²…

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Q: Use Green's theorem to calculate the work done by force F on a particle that is moving…

A: GivenF(x,y)=x32-8yi+5x+8yj,C: Boundary of a triangle with vertices (0,0), (5,0) and (0,5)To…

Q: 7. Consider the Surface S:z = 52² + 5y?. An equation of the plane tangent to Sat the point P (3, 3,…

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Q: #6. Use Stokes' Theorem to evaluate S. F dr, where F(x, y,2) = i+(x + yz)j+(xy-V) k and C is the…

A: Given that, Fx,y,z=i+x+yzj+xy-zk and C is the boundary of the part of the plane 3x+2y+z=1 in the…

Q: 6. If S is the surface of the sphere x + y² + z? = 9. Prove that curl F- ds = 0. %3D

A: Solution:-

Q: Which of the following is a defining equation for the tangential plane to the surface defined by…

A: To find  An equation for the tangential plane to the surface defined at the point (1, 1,1)

Q: 1. Evaluate the surface integral [[' xz² ds ; o is the part of the cone z = x² + y² that lies…

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Q: (a) Given that F(x,y,z) = (3x² + 3y²)i – 2xyj + yzk, evaluate S, F·dr for which C is the boundary of…

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Q: The parametric representation of the surface y2 - 4y + z2 = 0,0 <x< 3 %3D is a) F(u, v) = (v, 2 + 2…

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Q: 5. Prove that the surface E, = {(x, y, z)|a² + y² = 4} is locally isometric to the surface E2 = {(x,…

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Q: Consider the closed surface S consisting of the graph z = 1 - x² – y? with z 2 0, and also the unit…

A: Surface integration

Q: Let S be the upper surface of the sphere x- 2x+y+z = 0 and let r be the boundary. Find [.curi(F) -…

A: S is upper surface, the equation  x2-2x+y2+z2=0  x-12+y2+z2=1  Hence, is a sphere centered at (1, 0,…

Q: 1.Find the area of the surface. a. The part of the plane 2x + 5y + z = 10 that lies inside the…

A: Note : As per our guidelines we can supported to answer only first three subparts

Q: Evaluate F. dS, where F = 6xy²i + 6x²yj + 2z°k and S is the surface of the unit sphere.

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Q: Use Stoke’s Theorem to evaluate the line integral ∮C(z2−y2)dx +(x2−z2)dy +(y2−x2)dz, where the curve…

A: Given equation of the curve is ∮C(z2-y2)dx+x2-z2dy+(y2-x2)dz ,the equation of the paraboloid…