differential geometry w notes from teacher_Part_8

differential geometry w notes from teacher_Part_8 - 1.3....

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1.3. TANGENT VECTORS AND MAPPINGS 15 Definition 1.3.1 A tangent vector at a point p M of a manifold M is a map that assigns to each coordinate chart ( U , x ) about p an ordered n-tuple ( X 1 , . . . , X n ) such that X i = n X j = 1 x i x j ( p ) X j . 1.3.2 Vectors as Di ff erential Operators Let f : M R be a real-valued function on M . Let p M be a point on M and X be a tangent vector at p . Let ( U , x ) be a coordinate chart about p . The (directional) derivative of f with respect to X at p (or along X , or in the direction of X ) is defined by X p ( f ) = D X ( f ) = n X j = 1 f x j ! ( p ) X j . Theorem 1.3.1 D X ( f ) does not depend on the local coordinate system. Proof : 1. Chain rule. The intrinsic properties are invariant under a change of coordinate system....
View Full Document

This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

Page1 / 2

differential geometry w notes from teacher_Part_8 - 1.3....

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online