Solution the curve c is as shown c s n the

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Unformatted text preview: evaluate the line integral y dx + z dy + x dz C where C is the intersection of the sphere x2 + y 2 + z 2 = a2 and the plane x + y + z = 0. Assume that C is oriented anticlockwise, when viewed from a point on the z -axis with z > 0. Solution: The curve C is as shown: C S n The intersection C is a circle of radius a lying in the plane x + y + z = 0. Let F = y i + z j + x k and let S be the region bounded by C in the plane x + y + z = 0. Using Stoke’s theorem, y dx + z dy + x dz = C C F · dr = ( S 4 × F) · n dS where n is the unit normal to S with positive k-component. Now i ∂ ∂x ∂ ∂y ∂ = − i − j − k. ∂z y × F= j k z x On S1 , the unit normal is the normal to the plane x + y + z = 0: n= so that ( i+ j+ k √ 3 3 × F) · n = − √ . 3 Hence y dx + z dy + x dz = C 5 3πa2 3 − √ dS = − √ . 3 3 S...
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This document was uploaded on 04/08/2014.

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