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Using Stokes’s Theorem In Exercises 7-16, use Stokes’s Theorem to evaluate
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Multivariable Calculus
- Use Stokes’ Theorem to evaluate ∫ F*dr where C is oriented counter-clockwise as viewed from above. F(x,y,z) = yi-zj+x2k C is the triangle with vertices (1,0,0), (0,1,0), and (0,0,1) Note: The triangle is a portion of the plane x+y+z=1arrow_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_forwardcheck 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_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_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_forwardFlux across curves in a vector field Consider the vector fieldF = ⟨y, x⟩ shown in the figure.a. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ π/2.b. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for π/2 ≤ t ≤ π.c. Explain why the flux across the quarter-circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter-circle in the fourth quadrant equals the flux computed in part (b).e. What is the outward flux across the full circle?arrow_forward
- Using 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 the curl integral in Stokes’ Theorem to find the circulation of the field F around the curve C in the indicated direction. F = (x2 + y)i + (x + y)j + (4y2 - z)k C: The circle in which the plane z =-y meets the sphere x2 + y2 + z2 = 4, counterclockwise as viewed from above.arrow_forwardLine integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨x, y⟩ on the parabola r(t) = ⟨4t, t2⟩ , for 0 ≤ t ≤ 1arrow_forward
- Work integrals Given the force field F, find the work required to move an object on the given oriented curve. F = ⟨x, y, z⟩ on the tilted ellipse r(t) = ⟨4 cos t, 4 sin t, 4 cos t⟩, for 0 ≤ t ≤ 2πarrow_forwardConsider 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_forwardOutward flux of a radial field Use Green’s Theorem to compute the outward flux of the radial field F = ⟨x, y⟩ across the unit circle C = {(x, y2: x2 + y2 = 1} (see figure). Interpret the result.arrow_forward
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