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Using Stokes's Theorem In Exercises 7-16, use Stokes’s Theorem to evaluate
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Calculus: Early Transcendental Functions (MindTap Course List)
- Use Stokes's Theorem to evaluate C F · dr. C is oriented counterclockwise as viewed from above. F(x, y, z) = (cos(y) + y cos(x))i + (sin(x) − x sin(y))j + xyzk S: portion of z = y2 over the square in the xy-plane with vertices (0, 0), (a, 0), (a, a), and (0, a)arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forwardind the flux of the following vector fields across the given surface. Assume the vectors normal to the surface point outward. F = r/ | r | across the sphere of radius a centered at the origin,where r = ⟨x, y, z⟩arrow_forward
- Use Stokes' Theorem to evaluate Use Stokes' Theorem to evaluate ∫C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = yzi + 3xzj + exyk, C is the circle x2 + y2 = 4, z = 6.arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (7, 0), and (0, 4)arrow_forwardUse Stokes' Theorem to evaluate Use Stokes' Theorem to evaluate S curl F · dS. F(x, y, z) = zeyi + x cos(y)j + xz sin(y)k, S is the hemisphere x2 + y2 + z2 = 9, y ≥ 0, oriented in the direction of the positive y-axis. F(x, y, z) = zeyi + x cos(y)j + xz sin(y)k, S is the hemisphere x2 + y2 + z2 = 9, y ≥ 0, oriented in the direction of the positive y-axis.arrow_forward
- check the stokes theorem for vactor field A=(x)i+(y)j+(2xy)k where S is the lower hemisphere x2+y2+z2=4 and z<=0arrow_forwardRain on a roof Consider the vertical vector field F = ⟨0, 0, -1⟩, correspondingto a constant downward flow. Find the flux in the downward direction acrossthe surface S, which is the plane z = 4 - 2x - y in the first octant.arrow_forwarda) Find the work done by the force field F on a particle that moves along the curve C. F(x, y) = (x2 + xy)i + (y – x2 y)j C : x = t, y = 1/t (1 ≤ t ≤ 3)arrow_forward
- Flux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.arrow_forwardLet F = curl(A). Which of the following are related by Stokes' Theorem? (a) The circulation of A and flux of F (b) The circulation of F and flux of Aarrow_forwardUsing Green's Theorem on this vector field problem, compute a) the circulation on the boundary of R in terms of a and b, and b) the outward flux across the boundary of R in terms of a and b.arrow_forward
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