18. Let S be the portion of the plane z = x contained in the half-cylinder of radius R depicted in Figure 17. Use Stokes' Theorem to calculate the circulation of F = (z, x, y + 2z) around the boundary of S (a half-ellipse) in the counterclockwise direction when viewed from above. Hint: Show that curl(F) is orthogonal to the normal vector to the plane. -R y R FIGURE 17

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Q: 2. Let C be the curve of intersection between the cylinder r2+y = 1 and r-2y+z = 0, oriented…

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Q: 18 Verify Stokes' theorem F.dr = (curl F)· dS C S where F = (x² + y – 4, 3xy, 2xz +z?) and S is the…

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Q: 3. Consider A(r) = (y, -x,0), and the line integral A• dr along the C boundary of the plane x + y +…

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Q: Consider the parametrization y: [0,47] → R, (t) = (t cos t, t sin t), of a counter-clockwise spiral…

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Q: Given the vector field F = (z, x, y), calculate the flux of the curl of F through the portion of the…

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Q: Let C be the boundary of the portion of the plane x+y+z=2 that lies in the first octant, oriented…

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Q: H=pz coup ap te'sing a4 tp'ar At point (1, ,0)

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Q: 2. Use Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

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Q: Let F = Use Stokes' Theorem to evaluate / F. dī, where Cis the curve of intersection of the…

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Q: Suppose that C is the curve of interesection between the plane z = x and the cylinder x² + y? be the…

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Q: Suppose F(x, y) : (e*, ev) and C is the portion of the ellipse centered at the origin from the point…

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Q: :) Use Stokes' theorem to evaluate RF dr where C F = z?i + yj + xk and C is the triangle with…

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Q: Given F(x, y, z) = xzi + 3yj + yzk be a vector function that passes through a surface S of z+x? + y?…

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Q: Use Stokes' Theorem to find the circulation of the vector fieldF = (a2 – y, 4z, r2) around the curve…

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Q: 1. Use Stokes' Theorem to evaluate curlF.dS where F =z'i+ (4y-y) j+2zk and S is the portion of z = r…

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