
#19
Oct2611, 05:37 PM

Emeritus
Sci Advisor
PF Gold
P: 8,990

In the simplest nontrivial case [itex]V=\mathbb R^2[/itex], [itex]\phi^i(e_j)=\delta^i_j[/itex] are the four equalities [tex]\begin{align}
\phi^1(e_1) &=1\\ \phi^1(e_2) &=0\\ \phi^2(e_1) &=0\\ \phi^2(e_2) &=1 \end{align}[/tex] Since the [itex]\phi^i[/itex] are assumed to be linear, we have [itex]\phi^i(v)=\phi^i(v^je_j)=v^j\phi^i(e_j)=v^j\delta^i_j=v^i[/itex], and in particular [itex]\phi^1(ae_1+be_2)=a\phi^1(e_1)+b\phi^1(e_2)=a[/itex]. (f+g)(v)=f(v)+g(v) (af)(v)=a(f(v)) These definitions give V* the structure of a vector space. Let f in V* and v in V be arbitrary. We have [itex]f(v)=f(v^i e_i)=v^if(e_i)=\phi^i(v)f(e_i)[/itex]. Since v was arbitrary, this means that [itex]f=f(e_i)\phi^i[/itex]. Since f was arbitrary, this means that [itex]\{\phi^i\}[/itex] spans V*. 



#20
Oct2611, 06:13 PM

P: 320

great. now I have almost understood the idea behind defining dual spaces and also the idea behind covariant and contravariant vectors. your post about co/contravariant vectors was a great one.
now the only thing remained unclear for me is that I want to see some mathematical/physical examples of tensors. and I want to understand how we can interpret different physical situations with the new language I've learned. I have almost 0 level of knowledge about differential geometry, so don't go into very advanced topics from physics please (like GR). Is there any other thing that I need to know about tensors? 


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