Q.26 Let S, be the hemisphere x? + y? + z? = 1, z > 0 and S2 be the closed disc x? + y? <1 in the xy-plane. Using Gauss' divergence theorem, evaluate || F.d5 where F = z?xỉ +( + tanz)j + (x²z +y²) k and S = S,US2. Also evaluate the 3 integral F.dS.

Q: 20. Let S be an oriented surface parametrized by r(u,v). Define the notation do = r, du × r, dv so…

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Q: 6. For any closed surface S, prove that || CurlF.N dS=0.

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Q: Let L be the line segment with endpoints at (1,2,3) and (-1,1,4). A light source at is at the point…

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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: B- Consider the following subsets of real plane R² Determine with graph: 1- Arcwise connected A =…

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Q: ,A, B,C be 4 points in R², no three of which colinear. Fix A' e VA, B' e VB and C' e VC h that AB ||…

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Q: Q2/ Find T, N, and k for the space curve r(t) = (3 sin t)i +(3cos t)j + 4tk

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Q: Let R be the annulus defined by {1 ≤ x² + y² ≤ 4} and let C₁ and C₂ be the circles of ratius 2 and 1…

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Q: The distance of a point P(x,y,z) from the xy plane is d = |x| how do we show the work to prove this?

A: To show the distance of a point P(x, y, z) from the xy-plane is d = |x|, it can use the distance…

Q: 5. Among all the points on the graph of z = 10 – x² – y² that lie above the plane r+2y +3z = 0, find…

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Q: aces x2 + y2 pint (2, -1, 2)

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Q: ₂2 Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

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Q: 12. Let C represent the curve of intersection between the cylinder x² + y² = 4 and the plane z = 10…

A: The plane z=10-x will cut the cylinder x2+y2=4 to give an ellipse.

Q: F. dS, where F(x, y, z) = z²xi + 2) + (x2z + y2)k and S is the top half of the sphere x² + y² + z? =…

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Q: D) Evaluate f (r² + y²) dady where D is the disk z² + y² ≤ 4. Usc polar coordinates. (Ans: 647/5) E)…

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Q: II1. Recall that using vector projections, we showed that the dis- tance from a point (x1, Y1, 21)…

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Q: 2. Consider the line r(t) = (4t+1,3, 2 - t) and the ellipsoid + + z? = 1. What is the distance from…

A: Consider the line  and the ellipsoid To find  distance from the point (-3,3,3) on the line to the…

Q: (y - eCos(e25! dx + (x2 + y+1) dy, where C is the circle r = 2.

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Q: One model for expenditures at high school gives the cosr C, in dollars per pupil, as a function of…

A: The quadratic equation, y=ax2+bx+c can be expressed in the vertex form as y=a(x-h)2+k, where, the…

Q: A rental truck costs $45.45 for a day plus 94¢ per mile. (a) If x is the number of miles driven,…

A: Given that, a rental truck costs $45.45 for a day plus 94c per mile.

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: a Find the distance from the point pC,,1) to the plane 2K -3y+77=1

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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: 6. Let S denote the disc of radius 2 that is centered at the point (0, 0,3) and parallel to the X-Y…

A: Given that S denote the disc of radius 2 that is centered at the point  and parallelto the XY-plane,…