# Solution the curve c is as shown c s n the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 04/08/2014.

Ask a homework question - tutors are online