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Using Stokes’s TheoremIn Exercises 7–16, use Stokes’s Theorem to evaluate
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Chapter 15 Solutions
Calculus
- 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_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_forwardUse 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_forward
- Use 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_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_forwardFlux 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_forward
- Consider the vector field F and the curve C: F(x, y, z) = (x + y)i + (y + z)j + (z + x)k, C is the triangle with vertices (9, 0, 0), (0, 9, 0), and (0, 0, 9), oriented counterclockwise as viewed from above. Use Stokes' Theorem to evaluate intergal sign(c)F · dr .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_forwardLet F(x,y,z)=(4x-5yz, xy, z^2). Calculate curl F. Let H(x,y,z)=(0, -5y, y+5z). Use Stokes' Theorem to calculate: ∬_S(H*ndσ) where S is the rectangle {(x,y,z): 3≤5≤, 1≤y≤2, z=4}arrow_forward
- Find the flux of the vector field F across the surface S in the indicated direction.F = x 2y i - z k; S is portion of the cone z = 4 square root of x^2+y^2 between z = 0 and z = 1; direction is outward a)-1/24 pi b)-1/8 pi c)1/24 pi d)-1/48 piarrow_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_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_forward
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