1: Let F-(3x, y, z). Evaluate c. [F. 2 F-ds where S is the closed surface given by the concentric cylinders 2² + y²-1,2² + y²-9 with 0 ≤z≤ 2, top annulus 1≤ x² + y² ≤9, z-2 and bottom annulus 1 ≤ 2²2 + y² ≤9, z-0 a) Without using Divergence (Gauss) theorem, b) Using Gauss theorem. Y

Q: Estimate the area under the line y= 2x+1 using six rectangles and right endponits over the interval…

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Q: 6. Evaluate (7xi - zk) · ndA over the sphere S : x² + y² + z² = 4 (u) by the Divergence Theorem, (b)…

A: The divergence theorem states that  ∬SF·ndA=∭V∇·FdV Where, F is the vector field, V is the given…

Q: 4) Show that the surfaces 3X² + 2Y²+Z² = 9 and x² + y² + Z²-8X-6Y-8Z + 24 = 0 have a common tangent…

A: We have to prove both surfaces have common tangent at P(1,1,2).

Q: 2) Evaluate the vategral if cis any Seetion al Smooth Curve from 10,0,0) to (lilil). .(-3p²+2y-Z) da…

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Q: f(x, y) = 2x2 +9y2 on the disk bounded by the circle x² + y2 = 9 A) Absolute maximum: 18 at (3, 0)…

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Q: Define p: R^2 x R^2 to [0, infinity) by the rule p(x1,y1),(x2,y2)=squareroot((x1-x2)^2+(y1-y2)^2)…

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Q: (a) Compute directly I = S Ss F-dS, where S is the surface of the sphere x? +y? +z? = 4 and F =…

A: X²+y²+z²=4=2² Therefore radius of sphere is 2.

Q: Please sketch the following (do it by hand and not software): a) The space curve defined by r(t) =…

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Q: 1) ZX + 9 At what points on the surface XY + YZ + is the tangent plane parallel to the YZ plane? 0

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Q: 2x2 If we approximate f(-/4.1,/49.1)' Suppose we have a surface f (x,y) ,which of the following is…

A: I have appended the formula used in the solution part.

Q: A) Show that the function u(x; y) = 5x4y - 10 x²y³ + y5 is harmonic and determine the harmonic…

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Q: 9) of which of the following is the groph X² y = = A) 24 B) c) D) 24 ²4

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Q: JJs where S is the sphere x² + y2 + z2 = 1.

A: Given S is the sphere                         x2+y2+z2=1 To evaluate:             ∫∫S3x+3y+z2dS…

Q: Use Divergence Theorem to evaluate .v²z²i +z²x²j+x²y²k).ds, where S is the upper part of the sphere…

A: To evaluate the below integral. ∫∫sy2z2i+z2x2j+x2y2kds¯