Let F(x, y, z) = yi - zj + xk be a vector field. Set up, but do not evaluate, the double integral that is equivalent to the surface integral ff, F. dS, where S is the plane, oriented towards the positive z-axis, with the following over the planar domain D: parametrization x=-U, z = 4u - 3v. y = u + v, Express the integrand of the double integral as a linear expression involving u and v.

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Q: Evaluate F. dS , where F = and S is the helicoid with vector equation r(u, v) = 0 < u < 1, 0 < v<T…

A: since, y,-x,z3 and, ru,v=u cosv, u sinv,v So, F⇀ru,v=u cosv,-u sinv, v3 Now the normal upward vector…

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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…

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A: Concept Used:- Divergence Theorem Flux=∫∫∫∇.F dVwhere ∇.F is the divergence of FVolume of the…

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Q: Which graph represents the vector field F = 7i+ yj?

A: Given

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A: We have to find ∫∫F dS.

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