Invariance of ds^{2} and transformation properties of dx^{i} Invariance of ds^{2}

Fastage6bk

Fastage6bk

Answered question

2022-02-12

Invariance of ds2 and transformation properties of dxi
Invariance of ds2 and transformation properties of dxi
ds2=ds2
gijdxidxj=gijxjxkdxkxjxldxl
=gijxixkxjdxldxkdxl
=gkldxkdxl
where gij is metric tensor
1) I want to ask that idea behind this transformation is because distance remains invariant under transformation to different coordinate system
2) How this new variable k,l are introduced the text i am reading is physics pages

Answer & Explanation

Pamela Webb

Pamela Webb

Beginner2022-02-13Added 10 answers

1. Yes. The metric gij is the relationship between the purely mathematical coordinates and the physical properties of your manifold. When you change the coordinates, you change the metric too, which makes up for the change. 
The method for changing coordinates is 
dxi=k=1nxixkdxk
so k is just a "dummy variable" which iterates through the sum. The Einstein summation convention is to leave out the explicit , and have the summation be implied by the repeated use of indices. Thus, 
dxi=xixkdxk
where a summation from k=1 to n is implied. So k does not really factor in to the expression. Indeed, if n is known (Im

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