7. Use Stokes's Theorem to write S.(2y² + 6x*)dx + (2x² + 7y³)dy+ (2z° – yz)dz as a surface integral where C is the boundary of the portion of the paraboloid z = 72 – 6x2 – 6y² that is above the triangle with vertices at (0, 0), (3, 0) and (0, 3) oriented counterclockwise as viewed from above. Then set up a double integral in tems of x and y that could be used to evaluate the surface integral. (Do not integrate.). Stokes's Theorem: Let S be an oriented surface with unit normal vector Ñ, bounded by a piecewise smooth simple closed curve C with a positive orientation. If F is a vector field whose component functions have continuous partial derivatives on an open region containing S and C, then [F dĩ = (ExF)-Ñ ds .

Q: 5. Find a parametric representation for the part of the sphere x + y + z2 = 4 that lies between the…

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Q: 2. Find the surface area of the part of the paraboloid z = x² + y² lying inside the cylinder x² +…

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Q: Find the equation of tangent plane to this surface at the point where x :1 and y = -1.

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Q: Verify Stokes' theorem where A = (x-z)i + (x³ +zy)j - 3x y²k and S is the surface of z=2-√√x² + y²…

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Q: c) Use Stokes' theorem to evaluate F. dr where F = z' i+yJ + xk and C is the triangle with vertices…

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Q: Let S be the portion of the plane 2x + 3y + z = 2 lying between the points (-1, 1, 1), (2, 1, −5),…

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Q: 2)Use Stokes' Theorem to evaluate [F.dr where E2xyi, y² j‡ 2xzk and Sis the first octant с surface…

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Q: 4. Consider the surface defined by the portion of the cone z = -Vx2 + y? that lies below the…

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Q: c) Use Stokes' theorem to evaluate where F(x, y, z) = (x²+y²z) i+(x²+xy²) j+(x²+z²) k, and C is…

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Q: 1. Find the smallest distance from the point (0, c, 0) to the surface y = x² + z².

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Q: 2. a) Give a representation of the cylinder y2 + z² = 16,-2 ≤ x ≤ 3 as a parametric surface. b) Find…

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Q: Find a parametric representation for the surface S, where S is the part of the cylinder x2+y2 = 4…

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Q: Locate and classify the stationary points on the surface z=xy2-2xy-2x2-3x, remembering that  x, y ,…

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Q: prove hat for 4 x² + y² + z² + 6xyz + 2x² - y² +z²+ 4yz + 3zx = 0, rigin is a conic node. the locus…

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Q: Find the tangent plane to the elliptic paraboloid z = x² – 2y at the point (1,1,1). - O A. z = 2x –…

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Q: Let the surface xz – yz³ + yz? = 2, then - the equation of the tangent plane to the surface at the…

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