understanding of the first fundamental form The following is an excerpt from do Carmo's Differential Geometry of Curves and Surfaces about the first fundamental form:



I don't understand what "without further references to the ambient space ${\Bbb R}^3$" means. When "treating metric questions on a regular surface", what does it mean by


*

*with references to the ambient space ${\Bbb R}^3$?

*without references to the ambient space ${\Bbb R}^3$?


How does the example of calculating the arc length of a curve illustrate the difference of the two different ways above? 

Questions having to do with "magnitudes", like curve lengths, angle amplitudes, and surfaces areas, are said to be metric because they all involve the use of a metric, or a scalar product between vectors. 
In $\mathbb{R}^3$ one usually takes the standard Euclidean metric, for which the scalar product of vectors $X=(x,y,z)$ and $A=(a,b,c)$ is simply:
$$
X\cdot A = xa + yb + zc.
$$
In the first expression you scanned, the quantity under the integral sign is the length of the velocity vector $\alpha'(t)$, given by (the square root of) $\alpha'(t)\cdot \alpha'(t)$, using the metric above.
So we are using the fact that the velocity is a vector in $\mathbb{R}^3$, and we are calculating its length with the metric of $\mathbb{R}^3$. 
The first fundamental form, instead is a quadratic form that gives the scalar product of two tangent vectors without considering them in $\mathbb{R}^3$. But its value is the same as with the $\mathbb{R}^3$ metric. 
As you see, in fact, the second formula does not use the length of any three-dimensional vector, it only uses the quantity $u',v'$, and of course, the first fundamental form.
