Suppose that surface o is parameterized by r(u, v) = (u cos (2v), u sin(2v), v), 0≤u≤ 5 and 0 ≤u< ¹T and f(x, y, z) = x² + y² + z². Set up the surface integral (you don't need to evaluate it). [[ f(x, y, z) ds LI [ƒ ƒ (x(u, v), y(u, v), z(u, v)) R Ər du Ər მა dA du dv

Q: Evaluate the integral |(2e' – 2x) dy +(2xy +1) dx using Green's theorem where C is C 64 ans:-- 3…

A: Using Greens theorem the integral over a closed path can be converted into a corresponding surface…

Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

Q: 4. Let S be the surface in R³ defined by 2² - y² +22² = 9. (b) Does the point (2, 1, 3) lie in the…

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Q: 6. Evaluate the line integral F · dr where F = (x³ —y¹, ex²+z², x² – 16y² z²) and C is the curve x²…

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Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

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Q: 2. Consider the line integral y² dx + x² dy where C is the boundary of the half disk x² + y² ≤ 1, y…

A: The given problem is to evaluate the given line integral of vector field over the given path of…

Q: The value of the line integral S(2xy dx + 2x?ydy + dz) C along a path joining the origin and the…

A: We have to find the value of given line integral.

Q: Which of F(x, y) = (-y, x) and G(x, y) = (y,x) is conservative in R2 and why? If again F(x, y) =…

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Q: Let P be the curve that follows the helix r = from (1,0,0) to ( 1/2,√3/2 , π/3 ), and then returns…

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Q: F(x, y) = (4x³,3y²) along a path from (0, 0) to (2, 6). Find S F · dř. a) using a parabolic path. b)…

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Q: 1. Consider the surface defined by z = x2 + e9y ln(x-y). Let f(x,y) =  x2 + e9y ln(x-y). ○ Compute…

A: The given problem is to find the gradient vector of the given surface at point and to find the…

Q: 5. Let F = (-y, x), and let C be the line segment from (0, 2) to (1,5). (a) Parametrize the line…

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Q: 2, z) = 5 and the surfac e at (1,2,3). Evaluate = reasons).

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Q: Use Green's theorem to evaluate the line integral r ry dr+r² where C is positively oriented curve…

A: Introduction: The process of determining the area of the region under the curve is known as…

Q: 5. Consider the saddle surface r = y- 2, find the point where the tangent plane is parallel to x+y+z…

A: If two vectors a1, b1, c1 and a2, b2, c2 are parallel, then it can be written that a1, b1, c1=ka2,…

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: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду

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Q: 4. Let f(x, y) = sin(x²y) + vx + e+y. Find the equation of the tangent plane to the surface z f(x,…

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

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Q: e 1. Consider the surface defined by z = x² + eln(x - y). Let f(x,y) = x² + ³y ln(x−y). (a) Compute…

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Q: Given the line integral | (2x2° + 6y) da + (6x – 2yz) dy + (3a² z² – y²) dz. Let C be the path that…

A: Note that the following integral reduces like this :

Q: Let C be the positively oriented square with vertices (0, 0), (2,0). (2, 2). (0, 2). Use Green's…

A: Green's Theorem:  For a positively oriented curve C∫CP dx+Q dy=∫∫D∂Q∂x-∂P∂y dAfor the region D…

Q: Use Gauss' trick to determine the sum 1+3 + 5 +... + (2n – 3) + (2n – 1) + (2n + 1

A: Gauss's trick is very significant in determining the sum of a series. This method includes the…

Q: Evaluate the line integral |(3="y + (1- y):) dz + (z² – z2) dy + z(1 – y) dz where C is any curve…

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Q: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y

A: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y