- #1

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^{2}=g

_{u}

_{v}dx

^{u}dx

^{v}

Why is the invariance of ds

^{2}as shown above?

Why can't it be something like

ds

^{2}=g

_{u}

_{u}dx

^{u}dx

^{u}or

ds

^{2}=dx

^{u}dx

^{u}?

Isnt it based on Pythagoras theorem? Why must it have 1 u and 1 v instead of just 2 u or 2 v? Forgive me for such dumb question as I just started.