16.8withScannedExamples

# 16.8withScannedExamples - 16.8: STOKES’ THEOREM KIAM...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

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

Unformatted text preview: 16.8: STOKES’ THEOREM KIAM HEONG KWA 1. Stokes’ Theorem Let S be an oriented piecewise smooth surface whose boundary C is a simple, closed, piecewise smooth curve. Let ( x,y,z ) 7→ n ( x,y,z ) be the given orientation of S and let r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k , a ≤ t ≤ b, be a parametrization of C . Then C is said to have the induced positive orientation as the boundary curve of S if at each point (¯ x, ¯ y, ¯ z ) = ( x ( ¯ t ) ,y ( ¯ t ) ,z ( ¯ t )) of C at which r ( ¯ t ) exists and is nonzero, the cross product n (¯ x, ¯ y, ¯ z ) × r ( ¯ t ) always points inward in the direction of the surface S . It is customary to denote C by ∂S whenever it has the induced positive orientation. Theorem 1 (Stokes’ Theorem) . If F is a vector field whose compo- nents have continuous partial derivatives on an open region in R 3 that contains an oriented piecewise smooth surface S , then (1.1) Z ∂S F · d r = ZZ S ∇ × F · d S . There are a few interesting corollaries of Stokes’ theorem: (1) If S 1 and S 2 are two oriented piecewise smooth surfaces such that ∂S 1 = ∂S 2 , then ZZ S 1 ∇ × F · d S = ZZ S 2 ∇ × F · d S since the surface integrals are both equal to the line integral of F along ∂S 1 = ∂S 2 by Stokes’ theorem....
View Full Document

## This note was uploaded on 11/11/2011 for the course MATH 254.01 taught by Professor Kwa during the Fall '10 term at Ohio State.

### Page1 / 8

16.8withScannedExamples - 16.8: STOKES’ THEOREM KIAM...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online