Calculating Metric/Metric Tensor for Curved Spaces

[marcusesses]

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

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

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

\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

I can see it in the linear case, like when
r = x+y
since
ds^2 = dx^2 + 2 dxdy + dy^2
for example. But what if there is a non-linear relation, like
f(x,y) = x^2 - y
it will produce terms like dx^4 ...(I think, anyway).

Basically, what I'm asking is how do you calculate the metric tensor components
g_{\alpha\beta}?
Are they just found by making assumptions in the curved space you are in?How do you calculate the metric tensor from the metric?

 

[pmb_phy]

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

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

There is no such thing as the metric tensor. There are many metric tensors. I know of only two myself that I find in common use in geometry and physics and that is the Euclidean metric and the metric of spacetime (does this metric tensor have a name??). The metric is different from the distance function. The distance function induces a topology whereas the metric defines the scalar product of two vectors. They need not be the same and in those two I mentioned above are not. Distance let's one define neighborhoods and neighborhoods allow one open and closed sets which allows one to define the topology of the space.

The quantity you have above doesn't look like anything I recognize. What you have ican be viewed as the equation of a surface, i.e. z = f(x,y). The ds is then the difference in height (delta z) of neighboring points. It does not represent the Euclidean distance between two points as I would recognize it. Can you tell us what this f(x,y) is an how its supposed to fit into the definition of a metric?

Basically, what I'm asking is how do you calculate the metric tensor components
g_{\alpha\beta}?
Are they just found by making assumptions in the curved space you are in?

The space need not be curved to define a metric tensor on it. You can't simply say "Here is Rⁿ. What is the metric for this space?". CAn't be done since there is a lack of information here. Someone has to give you the metric or tell you how the metric tensor maps basis vectors to scalars.


Best wishes

Pete