Definition of coordinate basis vectors for manifolds

I am currently learning about manifolds from Hobson's General Relativity: An Introduction for Physicists.

In Chapter 3 on 'Vector Calculus on Manifolds', pg. 57, the author defined coordinate basis vectors for a point $$P$$ on a $$N$$-dimensional manifold with coordinate system $$x^a$$ as $$\vec{e_a}=\lim_{\delta x^a \rightarrow 0}\frac{\delta \vec{s}}{\delta x^a}$$ where $$\delta \vec{s}$$ is the infinitesimal vector displacement between point $$P$$ and a nearby point $$Q$$ whose coordinate separation from $$P$$ is $$\delta x^a$$ along the $$x^a$$ coordinate curve.

Say if $$P$$ and $$Q$$ are points on the $$x^1$$ coordinate cuve, the coordinate separation $$\delta x^1$$ between them will be $$0$$ since on the coordinate curve the $$x^1$$ coordinate stays constant. If this is correct, the definition for $$\vec{e}_a$$ will not make sense since it is dividing by $$0$$. What am I getting wrong?

• Isn't your question answered by: (1) equation 3.4 that follows on page 57 of Hobson text; (2) basic understanding of limits and differentials from Calculus? Commented Aug 27, 2020 at 15:43
• here you are wrong -> "...on the $x^1$ coordinate curve the $x^1$ coordinate stays constant". Commented Aug 27, 2020 at 15:53
• @Umaxo Wikipedia says "In two dimensions, if one of the coordinates in a point coordinate system is held constant and the other coordinate is allowed to vary, then the resulting curve is called a coordinate curve." Does that not mean the $x^1$ coordinate curve has the $x^1$ coordinate held constant? Commented Aug 27, 2020 at 15:56
• @K7PEH I can understand equation 3.4 but not this. Commented Aug 27, 2020 at 15:57
• @TaeNyFan This works only for two dimensions. For example fixing one coordinate in 3 dimensions would produce surface and not curve. You need to fix all coordinates but the one which has the same name. Commented Aug 27, 2020 at 15:58

$$\mathbf e_r = \left(\frac{\partial x}{\partial r} , \frac{\partial y}{\partial r}\right) = (\cos(\theta) , \sin(\theta))$$
$$\mathbf e_{\theta} = \left(\frac{\partial x}{\partial \theta} , \frac{\partial y}{\partial \theta}\right) = (-r\sin(\theta) , r\cos(\theta))$$