I'm having some troubles with a very basic definition of the metric tensor.(adsbygoogle = window.adsbygoogle || []).push({});

The metric is defined as

[tex]ds^2 =[f(x + dx, y+dy) - f(x,y)]^2 [/tex]

However, I can't see how this is equal to

[tex]\frac{\partial r} {\partial x} \frac{\partial r} {\partial x}dx^2 + 2 \frac{\partial r} {\partial x} \frac{\partial r} {\partial y}dxdy + \frac{\partial r} {\partial y} \frac{\partial r} {\partial y} dy^2[/tex]

I can see it in the linear case, like when

[tex]r = x+y[/tex]

since

[tex]ds^2 = dx^2 + 2 dxdy + dy^2[/tex]

for example. But what if there is a non-linear relation, like

[tex]f(x,y) = x^2 - y[/tex]

it will produce terms like [tex]dx^4[/tex] ...(I think, anyway).

Basically, what I'm asking is how do you calculate the metric tensor components

[tex] g_{\alpha\beta}[/tex]?

Are they just found by making assumptions in the curved space you are in?How do you calculate the metric tensor from the metric?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Metric/metric tensor?

**Physics Forums | Science Articles, Homework Help, Discussion**