Finding the metric of a surface embedded in $\mathbb{R}^3$

Question

I have a problem about finding the metric of a surface defined by $x=\rho\cos\varphi,\ y=\rho\sin\varphi,\ z=\rho^2$, embedded into $\mathbb{R}^3$, where $ds^2=dx^2+dy^2+dz^2$.

I have literally no idea how to do this. Worse perhaps, is I find the expression $ds^2=dx^2+dy^2+dz^2$ hopelessly meaningless, because I can't understand any precise meaning of $ds, dx$ etc. I understand it's supposed to capture the nature of the pythagorean theorem as used to calculate distances in 3D and generalise it, but what exactly $dx$ is precisely escapes me. "A small distance" or "A small change" don't do it for me; it's apparently being used to define distances so calling it a small distance is clearly circular, and what exactly makes a small change? How do you calculate with such quantities? How do you know how the expression for $ds^2$ is going to change when you change coordinates? (These are questions I feel I really shouldn't have to ask, but my lecturer is very sporadic, unclear, and provides very few resources, and feel quite stuck.)

I am looking for an explanation or some resources to enlighten me as to what I should understand by these quantities/expressions, and some hopefully some help with how I would go about finding the metric of the paraboloid above.

I would also appreciate a recommendation of a good text for a first course on differential geometry of curves and surfaces to self-study from.

Thanks in advance.

Answer 1

$dx$ is a basis differential 1-form, or the differential of the coordinate function $x(p)$ -- it "measures" the change of the coordinate along a given tangent.

$ds^2 = dx^2+dy^2+dz^2 = dx\otimes dx+dy\otimes dy+dz\otimes dz$ is indeed a metric tensor (covariant tensor field) corresponding to the Pythagorean metric in $\mathbb R^3$. The metric is induced on the surface embedded, and the corresponding tensor we find by pull-back, or "restriction", $$(d\rho\cos\phi)^2+(d\rho\sin\phi)^2+(d\rho^2)^2=(1+4\rho^2)d\rho^2+\rho^2d\phi^2$$

A good book to start with is Frankel's "Geometry of Physics" complemented by other texts.

Answer 2

The slickest possible reference that I can give you which also builds a lot of intuition is Chapter 7 in Arnold's Mathematical methods of classical mechanics titled "Differential forms". I feel its best explained by the master himself !