Use Stokes' Theorem to evaluate ∮ C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = xyi + yzj + zxk, C is the boundary of the part of the paraboloid z = 1 − x^2 − y^2 in the first octant.
Added by Kevin J.
Step 1
Step 1: Parametrize the curve C Since C is the boundary of the part of the paraboloid z = 1 - x^2 - y^2 in the first octant, we can parametrize it as follows: x = r*cos(theta) y = r*sin(theta) z = 1 - r^2 where 0 <= r <= 1 and 0 <= theta <= pi/2. Show more…
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Use Stokes' Theorem to evaluate ∫_C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = xyi + yzj + zxk, C is the boundary of the part of the paraboloid z = 1 - x^2 - y^2 in the first octant.
Use Stokes' Theorem to evaluate $ \int_C \textbf{F} \cdot d\textbf{r} $. In each case $ C $ is oriented counterclockwise as viewed from above. $ \textbf{F}(x, y, z) = xy \, \textbf{i} + yz \, \textbf{j} + zx \, \textbf{k} $, $ C $ is the boundary of the part of the paraboloid $ z = 1 - x^2 - y^2 $ in the first octant
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