Verify the conclusion of Green's Theorem by evaluating both sides of each of the two forms of Green's Theorem for the field F = 5xi-7yj. Take the domains of integration in each case to be the disk R: x² + y²sa² and its bounding circle C: r= (a cost)i + (a sin t)j, Ost ≤2x. Click here for the two forms of Green's Theorem. The flux is. (Type an exact answer, using as needed.)

Q: Find the circulation and flux of the field F = -5x²i-5y²j around and across the closed semicircular…

A: A vector field F=-5x2i^-5y2j^ around the semicircular path consists of a semicircular arch…

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: 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: Compute the curl of the following vector field. F= (3z sin y, 3xz cos y, 6xz sin y) The curl of F is…

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Q: Find the outward (normal away from the z-axis) flux of the field F = y2î + xzî – k across the cone z…

A: In this, we find the flux of the given function across the given cone z=3√(x^2+y^2).

Q: Find the circulation and flux of the field F = -x²i-y²j around and across the closed semicircular…

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Q: Evaluate , (V. A) dV for the vector field A = x²yi +xz³j- y²z²kwithin the volume of a cube defined…

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Q: Let r = (x2 + y²)/2 and consider the vector field F = r4(-yi + j), where r + 0 and A is a constant.…

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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: Determine which theorem applies to simplify the following. It is possible that more than one will…

A: Given  Find the work done by the vector field Fx,y,z=yzcosxyi^+xzcosxyj^+sinxyk^ along the…

Q: ) Verify Stokes' Theorem for the hemisphere S : x²+y²+z² = 9, z > 0, %3D its bounding circle C: 1?…

A: Stokes theorem  If a smooth surface S bounded by closed ,smooth boundary curve C oriented in…

Q: Determine whether the line integral of each vector field (in blue) along the oriented path (in red)…

A: it is known that  line integral F·dr will be positive if both F and dr have same direction. and if…

Q: If F = Pi+Qj+ Rk, prove that div (curl F) = 0

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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: Use the surface integral in Stokes Theorem to calculate the flux of the curl of the field F across…

A: Given the field is                and

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

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Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

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Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F = (y²…

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Q: For what values of a and d does the vector field F = ⟨ax, dy⟩ have zero flux across the unit circle…

A: A vector field, have zero flux. Across the unit-circle centered at the origin and oriented…

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: Find the outward flux of the field F = 8xyi + 4yzj + 2xzk across the surface of the cube cut from…

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Q: Prove that o curl f· dS = f pf • dr - f v o xf - d$ %3D

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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…

Q: Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F across…

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