What is the Name for the Linear Mapping f* and How is it Proven?

[yifli]

Is there a name for the linear mapping $f^*:\wedge^k(R^{m}_{f(p)}) \rightarrow \wedge^k(R^{n}_{p})$ where f is a differentiable mapping from $R^n \rightarrow R^m$.

When k is 1, f* is called the adjoint of f. But what about k > 1?

Also can someone show me a proof of $f^*(d\omega)=d(f^*\omega)$ where $\omega$ is a 0-form.

Thanks

 

[mathwonk]

i think that second one is called the chain rule.

 

[yifli]

i think that second one is called the chain rule.

Thanks.

Still, can anyone show me a proof of the f*(dw)=d(f*w)