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i have book "wstęp do współczesnej geometrii różniczkowej" written by Konstanty Radziszewski, this title in english mean something like... "basics of modern differential geometry", and here are many formulas which look similar to one another, but I never know what it means and sometimes its difficult to understand this, examples:

[tex]A^{i^\prime}_iA^j_{i^\prime}=\delta^i_j[/tex]

what this delta means? here is written that its 1 when i=j and 0 when [tex]i\neq j[/tex], but I dont know what it means, this multiplication value is just 1 or 0 depending on this value?

next problem:) I have formula for skalar multiplication of two vectors: [tex]\partial\mbox{x}(u)=\mbox{x}_i(u)\partial u_x^i,\quad\delta\mbox{x}(u)=\mbox{x}_i(u)\delta u_x^i[/tex]

and the formula:

[tex]\partial\mbox{x}(u)\delta\mbox{x}(u)=\mbox{x}_i(u)\mbox{x}_j(u)\partial u_x^i\delta u_x^j=g_{ij}(u)\partial u_x^i\delta u_x^j[/tex]

why in upper equation there is "i" in upper index, and then suddenly "j" in next formula? what happened after last equation? why there is suddenly [tex]g_{ij}(u)[/tex] instead of [tex]\mbox{x}_i(u)\mbox{x}_j(u)[/tex]? what it means?

thanks for answer mathematicians;]

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# Differential geometry - strange formula

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