As of recent, I've been studying Differential Geometry per the Dover Publication on the subject, and I've ran into a bit of an issue with tangent vectors to a parametric surface $ \mathbf{x}(u^1,u^2) $ at a point $ \mathbf{P}\ $ in $\mathbb{R}^3$ with the standard right handed Cartesian basis.
The book was asserting that the tangent vectors
$$ \mathbf{x_\alpha } \equiv \frac{\partial \mathbf{x} }{\partial u^\alpha }\ $$
are orthogonal at all regular (non-singular) points of the surface. I was working through an example ( I don't have the particular example, but the it's immaterial ), and I found that for the particular example:
$$ g_{\alpha\beta} \neq 0 \: \: , \: \: \alpha \neq \beta $$
Where $ g_{\alpha\beta} $ is the metric tensor, thus implying that the tangent vectors at P are not orthogonal. The surface in question was nothing special, and was regular at all points on the surface. I double checked my calculations, and everything seemed to be ok, but of course to err is human.
The question that I have then--assuming my calculations are incorrect--( I'll have to check more later ), and that the book is correct in it's assertion is: What forces the tangent vectors at a point on a surface to be orthogonal?
I haven't gotten much into proofing, but can follow proofs for the most part, and my best conjecture is that they're orthogonal because each parameter $ \: u^\alpha $ is really $ u^\alpha(x_1,x_2,x_3) $, and since $$ x_\alpha x^\beta= {\delta_{\alpha}}^{\beta} $$ ie all $ \: x_\alpha $ are orthogonal, the same must be true for the parameters. Seeing as I lack a proof in this, I don't want to just take that as unfounded fact. Please do correct any notational errors if any are found. As stated, I've just began learning the course, and am applying what I've learned thus far in asking this question, otherwise can anyone shed a bit of light on the matter?