stokes_theorem

# stokes_theorem - MIT OpenCourseWare http/ocw.mit.edu 18.02...

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document
V13. Stokes' Theorem 1. Introduction; statement of the theorem. The normal form of Green's theorem generalizes in 3-space to the divergence theorem. What is the generalization to space of the tangential form of Green's theorem? It says where C is a simple closed curve enclosing the plane region R. Since the left side represents work done going around a closed curve in the plane, its natural generalization to space would be the integral \$ F dr representing work done going around a closed curve in 3-space. In trying to generalize the right-hand side of (I), the space curve C can only be the boundary of some piece of surface S - which of course will no longer be a piece of a plane. So it is natural to look for a generalization of the form gjc F . dr = /L(something derived from F)dS The surface integral on the right should have these properties: a) If curl F = 0 in Bspace, then the surface integral should be 0; (for F is then a gradient field, by V12, (4), so the line integral is 0, by Vll, (12)). b) If C is in the xy-plane with S as its interior, and the field F does not depend on z and has only a k-component, the right-hand side should be These JL curl F dS . things suggest that the theorem we are looking for in space is Stokes' theorem For the hypotheses, first of all C should be a closed curve, since it is the boundary of S, and it should be oriented, since we have to calculate a line integral over it. S is an oriented surface, since we have to calculate the flux of curl F through it. This means that S is two-sided, and one of the sides designated as positive; then the unit normal n is the one whose base is on the positive side. (There is no "standard" choice for positive side, since the surface S is not closed.) cubical surface: no boundary It is important that C and Sbe compatibly oriented. By this we mean that the right-hand rule applies: when you walk in the positive direction on C, keeping S to your left, then your head should point in the direction of n. The pictures give some examples.
2 V. VECTOR INTEGRAL CALCLUS The field F = Mi + N j + Pk should have continuous first partial derivatives, so that we will be able to integrate curl F. For the same reason, the piece of surface S should be piecewise smooth and should be finite- i.e., not go off to infinity in any direction, and have

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.

{[ snackBarMessage ]}

### Page1 / 7

stokes_theorem - MIT OpenCourseWare http/ocw.mit.edu 18.02...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online