(b) Verify Stokes' theorem on a region which is a portion of the sphere x? +y? +z² = 16 which lies above the plane z 2 and with F(x, Y, z) = x²i – z²j + yzk.

Q: Use Stokes’ Theorem to evaluate the flux integral ∫∫(∇×F)·ndS, where F =〈2y, −z, x+y −z〉, S is the…

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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: How do you calculate the area of a parametrized surface in space? Of an implicitly defined surface…

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Q: 5. Using Stokes' theorem, find I (VxF) ndo, where F yi+xj+(y+z) k and S is the portion of the…

A: We have to find I=∫∫S∇×F·ndσ using Stokes' theorem.

Q: 1: Describe the solid bounded by the sphere ² +y² + z² = 9 and the pla 2 over the xy-plane using %3D…

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Q: 5. Use polar coordinates to find the volume of the solid enclosed by the hyper- boloid –r² – y² + z²…

A: To obtain the limits of integration for z, rewrite the equation -x2-y2+z2=1, and find z.…

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: The solid region bounded on the sides by the paraboloid z = 4-x-y, above by the plane z= 3, and…

A: The Equation of paraboloid: z = 4-x2-y2The equation of upper plane: z = 3The equation of lower…

Q: 1. Verify Stokes' Theorem for F = (2x - y)i - yz²j - y² zk, where S is the upper half surface of the…

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Q: AREA UNDER THE PLANE 6x +4y +z = 12 above the disk W/ BOUNDARY CIRCLE

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Q: 3. Use Stokes' theorem to evaluate the line integral F· dr where A = -yi + xj and C is the boundary…

A: Stoke's theorem given by ∫CF.dr=∫∫Scurl A×n^ds According to the question, A=-y i+x j…

Q: Completely set up (but do not solve!) the surface integral over the given oriented surface: F = (y°,…

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Q: Evaluate the line integral C F· dR, where F (x, y, z) =  and C  is the positively oriented curve…

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Q: 3.) Verify the conclusion of Stokes' Theorem given F(x,y,z)=(x-y,y-z,z–x) where S is the surface cut…

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Q: 2. Use a parametrization to express the area of the given surface as a double integral and then…

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Q: #2. Use polar coordinates to find the volume of the solid bounded by the plane z = 0 and the…

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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: 1. Find the area of the surface the part of the plane 3x + 2y + z = 6 that lies on the first octant

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Q: Evaluate the surface integral. x² + y2 + z• ds S is the part of the cylinder x2 + y2 = 25 that lies…

A: Given, Where, S is the part of the cylinder x2 + y2 = 25 which lies between the planes z = 0 and z =…

Q: 8. Using spherical coordinates, evaluate the triple integral x² + y² + z² dV D where D is the…

A: A scalar function f(x,y,z) is a rule that is used to associate a number or quantity with a…

Q: 63. Verify Stokes' theorem for A = (y - z +2)i + (yz +4)j – xz k, where S is the surface of the cube…

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Q: (b) Verify Stokes' theorem on a region which is a portion of the sphere r2 +y² +2 16 which lies…

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Q: 4. Find the volume of the solid bounded by the plane z = 0 and the paraboloid z=1-x² - y². (if using…

A: Solution :-

Q: Verify Stokes' theorem for the vector ficld v=x'i+x' yj. C is the boundary of the rectangle whose…

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Q: Ise Stokes' Theorem to evaluate F. dr where C is oriented counterclockwise as viewed from above.…

A: Using stokes' Theorem we have to evaluate given integral

Q: 4) Evaluate the surface integral [[F•dS where F(x,y,z)= (yz,-xz,z)and S is the piece of the…

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Q: (ii) Use Stokes' Theorem to evaluate SF · dr, where F(x, y, z) = x²zi+xy²j+z²k and C is the curve of…

A: Given function is F(x,y,z)=x2z i^+xy2 j^ +z2k^ . Plane: x+y+z=1 Cylinder: x2+y2=9 The objective is…

Q: Integrate F =-(y sin z)i + (x sin z)j + (xy cos z)k around the circle cut from the sphere x2 + y2 +…

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Q: Use Stokes' Theorem to evaluate the line integral (y + 2z)dr + (x + 2z)dy + (x+ 2y)dz where C is the…

A: Given intgeral. ∫Cy+2zdx+x+2zdy+x+2ydz The Curve is formed by the intersection of the sphere…

Q: Verify Stokes' Theorem if F(x, y, z) = (x – y)i + (y – z)j+ (z – x)k and S be the portion of the…

A: Solution: The objective is to verify the Stokes theorem for the given problem

Q: Let F (x, y, z): = (x + 2y2, y + z2, z + 2x2), and let C be the curve that results from the…

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Q: Verify Stokes' theorem on a region which is a portion of the sphere r+y+2= 16 which lies above the…

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Q: Use spherical coordinates to evaluate the triple integral ∫∫∫ √(x^2 + y^2 + z^2) dv where G is the…

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Q: Verify Stokes' theorem for A = (p a, + cos o a, + a:) on the upper half-surface of the sphere x + y?…

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Q: Use Stokes' Theorem to evaluate F. dr where C is oriented counterclockwise as viewed from above.…

A: Given that  Fx,y,z=i+x+yzj+xy-zk and C is the boundary of the part of the plane 5x+4y+z=1 in the…

Q: Verify stokes theorem for F = (2x- y)i – yz'j - y zk over %3D the upper half surface of the sphere…

A: Given the field, F¯=2x-yi¯-yz2j¯-y2zk¯ S is the upper half surface of the sphere x2+y2+z2=1 bounded…

Q: The region cut from the solid elliptical cylinder x2 + 4y2≤ 4 by the xy-plane and the plane z = x +…

A: Given that, z=x+2 , x2+y2≤4

Q: the tangent plane of the surface [ z=x^5+e^(xy)cosyx] at the point (1,0) O z=5x-y-3 O z=5x+y-3 O…

A: Above question is solved.

Q: 5. (a) Evaluate the surface integral / :² dS where o is part of the cone z = V + y that lies between…

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Q: - S is the surface with parametric equations x = 3u+v, y = 3u-v, z = uv for (u, v) in the region D…

A: We will find out the required value.

Q: verify Stokes § = (a%-y)i - yz'i over the uppea hazf SOD Face Of the sphere =1 bounded by the

A: Given that,  F→=(2x-y)i→-yz2j→-y2zk→ To verify Stoke's theorem: ∫CF→·dr→=∬Scurl F→·n^dS Here C is…

Q: Use Stokes' Theorem to evaluate F. dr where C is oriented counterclockwise as viewed from above.…

A: Stokes's theorem ∬ScurlF.dS=∫CF.dr where, C is the boundary of the surface oriented…

Q: (a) The part of the surface z = 1+ 3x + 2y² that lies above the triangle with vertices (0,0), (0,…

A: a) Given part of surface is  z=1+3x+2y2, which lies above the triangle with vertices (0, 0), (0, 1),…

Q: 2)Use Stokes' Theorem to evaluate | F.dr where F =2xyi- yj+2xzk and S is the first octant surface of…

A: The given problem is to evaluate the line integral in the surface of first octant with upward…

Q: Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points…

A: Consider the provided question,

Q: What is the center of gravity of the solid formed when the plane region bounded by 2x+y-2= 0, y = 0…

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Q: Use Stokes' Theorem to Calculate . curl F.ds F (x, y,z) = x*y³ zî+ Sin (xyz)j + xyz k where o is the…

A: Consider the provided question, F→x,y,z=x2y3zi^+sinxyzj^+xyzk^ that lies between the planes y = 0…

Q: (a) Find the surface area of the plane 3x +2y + z = 6 that lies in the first octant.

A: If z=fx, y then surface area is, AS=∫∫D1+zx2+zy2dA

Q: Evaluate the surface integral. || x2yz ds, s is the part of the plane z = 1+ 2x + 3y that lies above…

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