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I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Appendix A : Introduction to Tensors ...I need help to understand some statements/equations by Fortney concerning rank one tensors ...

Those remarks by Fortney read as follows:

View attachment 8786

View attachment 8787In the above text by Fortney we read the following:

" ... ... Suppose we change the coordinates from \(\displaystyle (x^1, x^2, \ ... \ ... \ , x^n )\) to \(\displaystyle (u^1, u^2, \ ... \ ... \ , u^n )\) using the \(\displaystyle n\) functions

\(\displaystyle u^1 (x^1, x^2, \ ... \ ... \ , x^n ) = u_1 \)

\(\displaystyle u^2 (x^1, x^2, \ ... \ ... \ , x^n ) = u_2\)

... ...

... ...

\(\displaystyle u^n (x^1, x^2, \ ... \ ... \ , x^n ) = u_n\) ... ... "

My question is as follows:

What do the equations \(\displaystyle u^i (x^1, x^2, \ ... \ ... \ , x^n ) = u_i\) mean ... ? ... how do we interpret them ...?

What would it mean for example if we wanted to form the differentials \(\displaystyle du^i\) ... ?Help will be appreciated ...

Peter

EDIT ... Reflecting on the above ... a further question ... are the coordinate functions \(\displaystyle (x^1, x^2, \ ... \ ... \ , x^n )\) essentially a basis for M ... (I am assuming the manifold is a vector space ... hmmm bt not sure it is ...?)

Hope someone can clarify ...

Peter

Those remarks by Fortney read as follows:

View attachment 8786

View attachment 8787In the above text by Fortney we read the following:

" ... ... Suppose we change the coordinates from \(\displaystyle (x^1, x^2, \ ... \ ... \ , x^n )\) to \(\displaystyle (u^1, u^2, \ ... \ ... \ , u^n )\) using the \(\displaystyle n\) functions

\(\displaystyle u^1 (x^1, x^2, \ ... \ ... \ , x^n ) = u_1 \)

\(\displaystyle u^2 (x^1, x^2, \ ... \ ... \ , x^n ) = u_2\)

... ...

... ...

\(\displaystyle u^n (x^1, x^2, \ ... \ ... \ , x^n ) = u_n\) ... ... "

My question is as follows:

What do the equations \(\displaystyle u^i (x^1, x^2, \ ... \ ... \ , x^n ) = u_i\) mean ... ? ... how do we interpret them ...?

What would it mean for example if we wanted to form the differentials \(\displaystyle du^i\) ... ?Help will be appreciated ...

Peter

EDIT ... Reflecting on the above ... a further question ... are the coordinate functions \(\displaystyle (x^1, x^2, \ ... \ ... \ , x^n )\) essentially a basis for M ... (I am assuming the manifold is a vector space ... hmmm bt not sure it is ...?)

Hope someone can clarify ...

Peter

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