(5) Let C be the closed curve given by the intersection of the hyperbolic paraboloid z...

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(5) Let C be the closed curve given by the intersection of the hyperbolic paraboloid z = yr and the cylinder x + y = 1 oriented anticlockwise as viewed from above. Let H be the part of the same hyperbolic paraboloid that lies inside the cylinder, with orientation determined by a unit normal vector with positive k component. Define the vector field. F(x, y, z) = xyi + xj + xyk. (a) Evaluate the line integral F.T ds without using Stokes' Theorem, where T is a unit tangent vector to the curve C. (b) Evaluate the surface integral (VxF) n dS without using Stokes' Theorem. Note: You should obtain the same answer in (a) and (b). In this question you are confirming the result of Stokes' Theorem for this particular example. (5) Let C be the closed curve given by the intersection of the hyperbolic paraboloid z = yr and the cylinder x + y = 1 oriented anticlockwise as viewed from above. Let H be the part of the same hyperbolic paraboloid that lies inside the cylinder, with orientation determined by a unit normal vector with positive k component. Define the vector field. F(x, y, z) = xyi + xj + xyk. (a) Evaluate the line integral F.T ds without using Stokes' Theorem, where T is a unit tangent vector to the curve C. (b) Evaluate the surface integral (VxF) n dS without using Stokes' Theorem. Note: You should obtain the same answer in (a) and (b). In this question you are confirming the result of Stokes' Theorem for this particular example.

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a Line Integral The line integral can be evaluated using the formula C F T ds ab Frt rt dt where C is the curve parameterized by rt F is the vector fi... View the full answer