(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I don't have great expectation that this will get a reply but here goes, cuz this is bugging me.

I will assume that you are familiar with the notation used by Spivak.

In the last section of chapter 4, he shows how to integrate a k-form on R^m over a singular k-cube in R^m, namely,

[tex]\int_c \omega = \int_{[0,1]^k}c^{*}\omega[/tex]

He then goes on to define the integral of a k-form over a k-chain [itex]c=\sum a_i c_i[/itex] in R^m by

[tex]\int_c \omega = \sum a_i\int_{c_i} \omega=\sum a_i\int_{[0,1]^k}c_{i}^{*}\omega[/tex]

But actually, a k-chain in R^misa k-cube in R^m, so in principle, the integral of a k-chain is already defined, namely by

[tex]\int_c \omega =\int_{[0,1]^k}(\sum a_i c_i)^{*}\omega[/tex]

and this will be consistent with the above definition if [itex](\sum a_i c_i)^{*}=\sum a_ic_i^{*}[/itex]. But is this so? Taking the case where w is a 1-form for simplicity,

[tex](\sum a_i c_i)^{*}\omega (p)(v_p)=\omega (\sum a_i c_i(p))((\sum a_i c_i)_{*}(v_p))=\omega (\sum a_i c_i(p))(\sum a_i (Dc_i(p)(v))_{c(p)})=\sum a_i \omega (\sum a_i c_i(p))((Dc_i(p)(v))_{c(p)})[/tex]

On the other hand, supposing we define "+" and multiplication by a scalar in the natural way on the f* operators,

[tex](\sum a_ic_i^{*})\omega(p)(v_p)=\sum a_ic_i^{*}\omega(p)(v_p) = \sum a_i \omega(c_i(p))(c_{i*}(v_p))=\sum a_i \omega(c_i(p))(Dc_{i}(v))_{c_i(p)})[/tex]

They are not at all the same.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: A contradiction in Spivak's Calculus on manifold?

**Physics Forums | Science Articles, Homework Help, Discussion**