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: 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: C:1-1,4), (1,2) (-レ-1),(ル-) the boundary of the rectonale that accepts points as a corner…

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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: at a A- Find the point on the curve r(t) = (5 sin t)i + (5 cos t)j + 12tk distance 26 units along…

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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: 28. Show that the circulation of F(x, y, z) = (x², y², z(x² + y²)) around any curve C on the surface…

A: Given figure Stoke's Theorem Here take single integral over closed path C and double integral over…

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: 3-) Find the pointls) to point Az(d,0) on x-3~y2 that are closest メ=2 25 +1,5 0.5 + - -ロ5 + 8.5 to…

A: here we have to find the point on the curve x=3-y2 that are closest to point A(1,0) let point is…

Q: The tangent plane and the normal line of the surface y = x² + z – 6 at the point (2, -1, 1) have the…

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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: 1) Prove Green's Theorem for F = 3xy'i - 4xj around the curve C bounded %3D 160 by y = 2x and y =…

A: These questions are about green's theorem around any given curve c.

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: det R have the inner Euclidian fipace product 4=(5,-1) and let X= L1,1) and Find l lyll and Kxy>…

A: Given that x=(1,1) and y=(5,-1)

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: Let C be the simple closed curve which consists of the line segment from (1,1) to (-2,4) followed by…

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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: 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: .72+40

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Q: d(fv) = df ^ ¢v + fdý %3D

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