I'm studying 'A Most Incomprehensible Thing - Notes towards a very gentle introduction to the mathematics of relativity' by Collier, specifically the section 'More detail - contravariant vectors'.
To give some background, I'm aware that basis vectors in tangent space are given by $\big\{\frac{\partial}{\partial x^i}\big\}$. I'm also aware that if we act these operators on the coordinate functions $x^i$, then we get a specific basis $\{\vec e_i\}$ whose elements are tangent to the coordinate curves that that point. This specific basis is commonly used as far as I understand.
Then in the 'More detail - contravariant vectors' section:
We can now state that a contravariant vector is a tangent vector to a parameterised curve. Let's see how this works. If the parameter of the curve is $\lambda$, and using a coordinate system $x^i$ the components of the tangent vector $\vec V$ are given by $$V^i=\frac{dx^i}{d\lambda}$$
I'm a little confused by this paragraph. Down the line there's another related paragraph:
In more advanced texts you may see the vector $\vec V$ written as $$\vec V=V^i\frac{\partial}{\partial x^i}$$ where $V^i$ are the vector's components and the partial derivative operators $\frac{\partial}{\partial x^i}$ are the coordinate basis vectors. In order to make sense of this formulation, consider an infinitesimal displacement $df$ at a point $p$ on the manifold, where $f$ is a function of some coordinate system $x^i$. If the displacement is along a curve parameterised by $\lambda$ we can drop $f$ into the above to get $$\vec V=V^i\frac{\partial f}{\partial x^i}=\frac{dx^i}{d\lambda}\frac{\partial f}{\partial x^i}=\frac{df}{d\lambda}$$ where $\frac{\partial f}{\partial x^i}$ are the coordinate basis vectors at $p$.
Again, I'm not able to make much sense of the whole parameterisation w.r.t. $\lambda$ thing.
First question: Why does he say that a contravariant vector is tangent to a parameterised curve? I can understand that a contravariant vector, being a member of the tangent space at some point $p$, must be tangent to some curve passing through $p$. But what makes us say that it should be parameterised?
Second question: Why does he claim that $\partial f/\partial x^i$ are basis vectors? I don't see a reason why, for an arbitrary function $f$, $\partial f/\partial x^i$ should form basis vectors. And why does he claim that $V^i=dx^i/d\lambda$ are the components w.r.t. this specific basis?
Would appreciate some guidance!
