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lecture22

# lecture22 - Lecture 22 In R3 we had the operators grad div...

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Lecture 22 In R 3 we had the operators grad, div, and curl. What are the analogues in R n ? Answering this question is the goal of today’s lecture. 4.9 Tangent Spaces and k -forms Let p R n . Definition 4.36. The tangent space of p in R n is T p R n = { ( p, v ) : v R n } . (4.144) R n We identify the tangent space with via the identification T p R n = R n (4.145) ( p, v ) v. (4.146) according to the following diagram: Via this identification, the vector space structure on R n gives a vector space structure on T p R n . Let U be an open set in R n , and let f : U R m be a C 1 map. Also, let p U and define q = f ( p ). We define a linear map df p : T p R n T q R m (4.147) T p R n df p T q R m −−−→ = = (4.148) R n Df ( p ) −−−→ R m . So, df p ( p, v ) = ( q, Df ( p ) v ) . (4.149) Definition 4.37. The cotangent space of R n at p is the space T p R n ( T p R n ) , (4.150) which is the dual of the tangent space of R n at p . Definition 4.38. Let U be an open subset of R n . A k -form on U is a function ω which assigns to every point p U an element ω p of Λ k ( T R n ) (the k th exterior power p of T R n ). p 1

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Let us look at a simple example. Let f ∈ C ( U ), p U , and c = f ( p ). The map df p : T p R n T c R = R (4.151) is a linear map of T p R n into R . That is, df p T R n . So, df is the one-form
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lecture22 - Lecture 22 In R3 we had the operators grad div...

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