Invariance of ds2 and transformation properties of dxiInvariance of ds2 and transformation properties of dxids′2=ds2gij′dx′idx′j=gij′∂x′j∂xkdxk∂x′j∂xldxl=gij′∂x′i∂xk∂x′jdxldxkdxl=gkldxkdxlwhere gij is metric tensor1) I want to ask that idea behind this transformation is because distance remains invariant under transformation to different coordinate system2) How this new variable k,l are introduced the text i am reading is physics pages

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Pamela Webb · Accepted Answer

1. Yes. The metric&amp;nbsp;gij&amp;nbsp;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.&amp;nbsp;The method for changing coordinates is&amp;nbsp;dxi=∑k=1n∂xi∂xkdxk,&amp;nbsp;so k is just a &quot;dummy variable&quot; which iterates through the sum. The Einstein summation convention is to leave out the explicit&amp;nbsp;∑, and have the summation be implied by the repeated use of indices. Thus,&amp;nbsp;dxi=∂xi∂xkdxk,&amp;nbsp;where a summation from&amp;nbsp;k=1&amp;nbsp;to n is implied. So k does not really factor in to the expression. Indeed, if n is known (Im

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

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