4. Suppose that S is a closed surface and F is a vector field defined on all of R. It is a fact that curl F dS = 0. Give explanations of this fact using... (a) ... Stokes' Theorem.

Q: 2. Let vector field F=(xz,-y,z) and surface, S is a cylinder x+y = 4 for 3<z<0 with %3D outward…

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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: 5) By using Stokes' Theorem, find the work performed by the vector field Ex² + 4xy³i+ y²xk,on a…

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Q: Use the Stoke’s theorem to show that: for any closed curve" C "and differentiable scalar fields "f"…

A: Given  that   be a closed  curve  and  be  differtentiable functions.Now,recall  Stoke's  theorem :…

Q: Let S be the part of paraboloid z = 9 - x² - y² with z > 0. Verify Stokes' theorem for vector field…

A: Let S be the part of paraboloid z=9-x2-y2 and vector field Fx , y , z=4zi^+5xj^+2yk^. To Verify the…

Q: 3. Consider the vector field shown below. (a) Is ſe F · di positive, negative, or zero? (b) From…

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Q: Use (a) parametrization; field (b) Stokes' Theorem to compute fF. dr for the vector F = (x²+z)i +…

A: Given:F=(x2+z)i+(y2+2x)j+(z2-y)kC: intersection of x2+y2+z2=1 and z= x2+y2

Q: The surface H is defined by (x² + y²)² + z = 1; z ≥ 0. The vector field A(r) is given by A = (x, y,…

A: The surface H  is defined by    .The vector field  is given by   .We have to sketch the surface H.

Q: 3. The vector field F below is in the xy-plane and looks the same in all other horizontal planes. In…

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Q: Let F = (0, x), a 2D vector field. Compute fF . T ds, where C is the top half of the circle of…

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Q: 00 Consider the vector field F and the curves pictured below. Then Jc F Se, F- dr > 0, and f F dr +…

A: Given the vector field F→ and the curves,

Q: Consider F = ⟨x, xy, xyz⟩. Does there exist a vector field G such that curl G = F? Why or why not?

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Q: Check whether Green's theorem can be applied to evaluate work done by the force field F(x, y) =…

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Q: Problem 9. Let F(r, y) = (ry,x²) be a vector field over ry-plane. Use Green's Theorem to evaluate…

A: We need to use the Green's theorem to evaluate the integral. Use the concept of Green's Theorem…

Q: 4. True/False: curl(divF)) is not a meaningful expression. (Justify your answer.)

A: Relation between Crul and div of vector quantities

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: Suppose C is the closed, oriented curve in the figure. Sketch each vector field F given below and…

A: The vector fields are as follows:To Find:Determine whether the above vector field is positive,…

Q: 6. a. Sketch a vector field F and a closed curve c with (make it obvious): SF dF >0. b. What can you…

A: Let us consider a vector field F and C be the closed curve shown below. Since all arrows are in the…

Q: Verify that Stokes' Theorem is true for the vector field F = (-y³, x³) along the circle of radius R…

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Q: nce we do not have our path as a vector, we can use Stoke's Theorem to find th nd the "curl" of the…

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Q: 2) Check Stoke's theorem using the vector field A = origin in the xy plane. ayx + bxy, where a and b…

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Q: A vector field is given as F = âr2r“ + a¿r²CoSo + âz5z. Verify Stokes theorem for this vector field…

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Q: (a) For all vectors ū, v, w e R³, w x (i x ũ) = (w x i) ×ũ. True False (b) Let F : R → R³ be a…

A: We know that  w × (v × u) =(w.u)v - (w.v)u div(curl F) = 0  And curl(gard f) = 0

Q: Determine the vector field of F. F(x, y) = yi – xj 1 1

A: We need to determine the vector field of Fx,y=y i^-xj^    x,y Fx,y=y i^-xj^ =y,-x 1,0 0,-1…

Q: 8. Use (a) parametrization; (b) Stokes' Theorem to compute fF.dr for the vector field (x² + z)i +…

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Q: Find the level surfaces of the scalar field U = ear, where the vectors are: a = (2, 2, 1) and r = =…

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Q: 1. Consider the function f (x, y, z) = y- + y² +z? Determine the gradient vector field of f, F =Vf…

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Q: to be careful to choose a parametrization r(u, v) that determines the correct counterclock- wise…

A: As per the question we are given a statement regarding the Stoke's Theorem and we have to say if it…

Q: A4. Which of the following vector fields is not the curl of another vector field? A. F=zi+=j+yk B.…

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Q: b) Evaluate both sides of Stokes's theorem for the vector field B = r cos o î + sin o by evaluating…

A: Given the vector field is B=rcosϕr^+sinϕϕ^ and the given figure is

Q: A vector field in polar coordinates has the form F(r.0) = f(r.0)u, +g(r,0)ug, where the unit vectors…

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Q: d) Verify the Stoke's theorem for vector field E = xyx – (x² + 2y²)ỹ by evaluating the following: i.…

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Q: Let the following vectors of F and Ğ be a vector field, that is from some given domain of pC R' to…

A: See the attachment

Q: H.W: Select the divergence of F(x, y) i+ (2x – 3y)j. %3D 1 (b) -* +2, y2 1 (c) (a) 3, (d) -2. - - -

A: Multiple:   Divergence of a vector field find by taking the scalar product of the vector operator  ∇…

Q: Verify Stokes's Theorem for the following vector field E = xyâg – (x² + 2y²)ây for the contour shown…

A: If   S is  a smooth surface that is bounded by a simple, closed, smooth boundary curve C which…