Assume that u and v are continuously differentiable functions. Using Green's theorem, prove that SS' S D Ux Vx |uv|dA= udv, Wy Vy C where D is some domain enclosed by a simple closed curve C with positive orientation.
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- Use Green's Theorem to evaluate ∫C F·dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = ‹y2cos(x), x2 + 2ysin(x)›C is the triangle from (0, 0) to (2, 6) to (2, 0) to (0, 0).Dierentiable curves with zero torsion lie in planes That a sufficiently di¡erentiable curve with zero torsion lies in a plane is a special case of the fact that a particle whose velocity remains perpendicular to a fixed vector C moves in a plane perpendicular to C. This, in turn, can be viewed as the following result. Suppose r(t) = ƒ(t)i + g(t)j + h(t)k is twice di¡erentiable for all t in an interval 3a, b4 , that r = 0 when t = a, and that v # k = 0 for all t in 3a, b4 . Show that h(t) = 0 for all t in 3a, b4 . (Hint: Start with a = d2r/dt2 and apply the initial conditions in reverse order.)Use Stokes's Theorem to evaluate F · dr C . In this case, C is oriented counterclockwise as viewed from above. F(x, y, z) = 2yi + 3zj + xk C: triangle with vertices (5, 0, 0), (0, 5, 0), (0, 0, 5)
- Stokes’ Theorem on closed surfaces Prove that if F satisfies theconditions of Stokes’ Theorem, then ∫∫S (∇ x F) ⋅ n dS = 0,where S is a smooth surface that encloses a region.Line integrals of vector fields on closed curves Evaluate ∮C F ⋅ dr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. F = ⟨y, -x⟩; C is the circle of radius 3 centered at the origin orientedcounterclockwise.Let f(z)=|z+7|^2 Part (a) Using Cauchy-Riemann equations, show that f(z) is not differentiable everywhere. Part (b) Find a point where f(z) is differentiable, if there is any. Part (c) Find a point where f(z) is analytic, if there is any.
- Pseudometric spaces(a) Find the minimum and maximum xcoordinates of points on the cardioid r =1−cosθ.(b) Find the minimum and maximum ycoordinates of points on the cardioid in part (a).A smooth curve is normal to a surface ƒ(x, y, z) = c at a point of intersection if the curve’s velocity vector is a nonzero scalar multiple of ∇ƒ at the point. Show that the curve r(t) = sqrt(t) i + sqrt(t) j-1/4(t+3)k is normal to the surface x2 + y2 - z = 3 when t = 1.
- Conservative fields Use Stokes’ Theorem to find the circulationof the vector field F = ∇(10 - x2 + y2 + z2) around anysmooth closed curve C with counterclockwise orientation.Work integrals Given the force field F, find the work required to move an object on the given oriented curve. F = ⟨y, -x⟩ on the line segment y = 10 - 2x from (1, 8) to (3, 4)