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I'm having some troubles with a very basic definition of the metric tensor.

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?

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?

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