Let (x, y, z) = (x 2 + y 2 + z 2 ) -1/2 . Show

Let ƒ(x, y, z) = (x² + y² + z²)^-1/2. Show that the clockwise circulation of the field F = ∇ƒ around the circle x² + y² = a² in the xy-plane is zero

a. By taking r = (a cos t)i + (a sin t)j, 0 ≤ t ≤ 2π, and integrating F · dr over the circle.

b. By applying Stokes’ Theorem.

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THEOREM 6-Stokes' Theorem Let S be a piecewise smooth oriented surface having a piecewise smooth boundary curve C. Let F = Mi + Nj + Pk be a vector field whose components have continuous first partial derivatives on an open region containing S. Then the circulation of F around C in the direction counterclockwise with respect to the surface's unit normal vector n equals the integral of the curl vector field VX F over S: fF.dr = [[ S Counterclockwise circulation VXF ndo Curl integral (4)