- #1
marcusesses
- 24
- 1
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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