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: Compute Div(F) and Curl(F): F--(4y-1) i+yj+(-3y) k

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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: 3) Let S be the unit sphere x² + y² + z² = 1 and F(x, y, z) = (2x, y²,z²). Use the divergence…

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Q: 1. Get center point of homo geneu us plane limited by 9) y =4- x2 and x axis

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer one question at a…

Q: Q3// A surface S is defined by 2x2 + z = 1 and the bounded by the planes y = 0,y 2, x 0,z = 0 in the…

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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: f(x, y) = x - y“; R is the disk x +y < 1.

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Q: 3. Consider the surface S={(x, y, z)| = = e`(x²+x²) – e¯ªª, (x² + y²) ≤a²}. Use a simple way to…

A: Given surface S is S=x ,y,z  z=e-x2+y2 , x2+y2≤a2 To Find : Vector Area of S.

Q: Q17// A scalar filed V = x + y exists over a surface S defined by x2 + y2 + z² = 9 bounded by the…

A: Given scalar field is V=x+y The surface S is defined by x2+y2+z2=9 bounded by the plane x=0, y=0,…

Q: 7. Which surface is parameterized by ř(u, v) = (u, uv, v) for –1 < u <1 and –1< v< 1?

A: r→(u, v) = u, uv, v for -1≤u≤1 and -1≤v≤1The parametric curve is  Option C is correct.

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: If F(x, y, z) = xzi+3ryj-2zk, evaluate fF.dS using Gauss's theorem when S is the closed cylinder…

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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: Let M⊂R ^ 3 be a surface and f be a diff.bilir function on this surface. Show that if Q is this…

A: Given: Let M⊂R3be a surface and let f be a differentiable function on this surface. If Q is a 1-1…

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: Eind equatons of the following x²-2y²+2²+ yz =20, [4,6-3) W the tangent plane 6 the normal line to…

A: Solution: f(x,y,z)=x2-2y2+z2+yz-20

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: Consider the following subsets of real plane R² A = {(0,y): ≤ y ≤ 1} B = {(x, y): 0 ≤ x ≤ 1, y =…

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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¯

Q: Let S be the triangular prism with the triangular face in the plane y-3 removed. Let C be the…

A: Given curl F→=xi^-yj^ and the given plane is y=3 So, ϕx,y,z=3j^ Therefore, calculate curl F→·n^ curl…

Q: d(fv) = df ^ ¢v + fdý %3D

A: