4. Show that the line integral of F = [0, z] around a closed curve in the r- y plane, oriented as in Green's Theorem, measures the area of the region enclosed by the curve. Additionally, find two more vector fields G, H that satisfy the same property as F: the vector circulation around a simple closed curve C equals the area of the region enclosed by C. Explain.

Q: Let F = 4ai + 1yj and let C be the closed path around the circle x² + y² = 9, traversed once…

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Q: Let S be the part of paraboloid z = 9 – x2 - y? with z 2 0. Verify Stokes' theorem for vector field…

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Q: State the theorem for Vector Surface Integral.

A: More precisely, the divergence theorem states that the surface integral of a vector field over a…

Q: The vector surface integral j L F • dS is equal to the scalar surface integral of the function…

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Q: For vector A = , calculate the surface integral (f, A · ds)

A: In this question, we find the surface integral for the given vector function A=<x2yz, xy2z,…

Q: Your friends correctly calculate the gradient vector for f(x, y)=x²+y at the point (2,-3) as…

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Q: suppose gradient f(a,b) =4i + 3j. find a unit vector u such that a) Du f(a,b)=0 b) Du f(a,b) is a…

A: Please see the below picture for detailed solution.

Q: Answer the following True or False: Let f(x, y) be differentiable at (zo, 30). Then the vector is a…

A: The equation of the tangent plane to the surface f(x, y, z)=0 at x0, y0, z0 is defined as: fxx0, y0,…

Q: II. Let S be the portion of the cylinder x² - 6x + z = 4, oriented by upward-slanting normal…

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Q: Use Stokes' Theorem to find the circulation of F = 4yi +7zj +3ck around the triangle C' obtained by…

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Q: A vector is said to be conservative when it's: curl equal to zero Laplacian equal to zero divergence…

A: We know that ,  A conservative vector field , is a vector field  F  whose line integral  ∫C F·ds…

Q: Let S be the surface defined by the vector function R(u, v) = (v + 4, 4u - v, uv²). Set up an…

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Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

A: The surface area of the helicoid can be computed using the surface integral formula:   ∫∫Rru×rvdA…

Q: The tangent plane at a point Po(f (uovo)g (uovo),h (uo.vo)) on a parametrized surface r(u,v) =…

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Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: Let be the curve given by the intersection of the plane z = 2 and the cone z = √² + y² counter-…

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Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) (u cos v, u sin v,…

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Q: Let F = 3 x R, where is a constant vector. Calculate F. dR C on the line segment C connecting (1, 3,…

A: According to the given information, it is required to calculate the integral.

Q: Calculate the curl(F) and then apply Stokes' Theorem to compute the flux of curl(F) through the…

A: The given problem is to find the curl of the given  field and also to find the flux of curl of the…

Q: Let vector field F =(x,y,z²) across the surface S which is an upper hemisphere x²+ y² + z² = 4 . 1)…

A: Let vector field F =(x,y,z²) across the surface S which is an upper hemisphere x²+ y² + z² = 4 . 1)…

Q: (b) Given that a is the surface of the solid G enclosed by cone 2-4-√√². and plane z -0. (1) Compute…

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Q: F. Sketch the region onto which the sector r ≤ 1,0 ≤ 0 ≤/4 is mapped by the transfor- mation (a) w =…

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Q: 7 Use mathematical induction fo derive the following formeld for gll nzi!

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Q: Let vector field F=(xz,-y,z) and surface, S is a cylinder x² + y² = 4 for 3≤z≤0 with the outward…

A: According to Stoke's theorem If S is a smooth surface  bounded by a closed, smooth boundary curve C…

Q: Evaluate the line integral by stokes theorem where F(vector)=[ 0, xyz ,0] , C is the boundary of a…

A: Calculate curl:  curlF=ijk∂∂x∂∂y∂∂z0xyz0         =i-xy+kyz

Q: Find the flux of F across S, 1.1. S F. N dS where N is the upward unit normal vector to S. F(x, y,…

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Q: b) F(x,y) represents a velocity field of a fluid over a surface S defined by z= 6-3x-2y. If the…

A: The flux over the surface S in the first octant is given by the following integral: Flux=∫SF⋅ndS…

Q: Use Green's theorem for the work done by the force F= (x - y)i + (x + y)j, for the shaded region R…

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