15.7 3 Use the surface integral in Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. F = yi+xzj+x²k C: The boundary of the triangle cut from the plane 10x + y + z = 10 by the first octant, counterclockwise when viewed from above. The circulation is (Type an integer or a fraction.) (...)

2. Find the circulation and flux of the field F(r.y) = (-y,r) along and across the curve (cost, 4 sin t), tE [0, 2]

Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F across the surface S in the direction away from the origin. F = 4yi + (5-2x)j + (z² - 2)k S: r(0, 0) =(√6 sin cos 0) i + ( √5 sin ô sin 0)j + (√√6 cos ¢) k, 0≤ ≤r/2, 0≤0≤2r The flux of the curl of the field F across the surface S in the direction of the outward unit normal n is (Type an exact answer, using as needed.)

5. Discuss real world quantities that the circulation of a vector field around a closed loop could measure? Now let F (1, 1). Sketch the vector field F at various points on the edges of the square. Based on your sketch, what value would you expect for the circulation of F around the perimeter of the square. Use Stokes' theorem to find the circulation of F for any closed loop C in space. yzi + zxj + yk. Draw a unit square in the xy plane with one corner at the origin and another at

Use the surface integral in Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction.

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