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## Main Question or Discussion Point

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:

Would appreciate some guidance!

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:

I'm a little confused by this paragraph. Down the line there's another related paragraph: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 $$V^i=\frac{dx^i}{d\lambda}$$

Again, I'm not able to make much sense of the whole parameterisation w.r.t. ##\lambda## thing.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##.

**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(x^i)##, ##\partial f/\partial x^i## should form basis vectors.**Third question**: Why does he claim that ##V^i=dx^i/d\lambda## are the components w.r.t. this specific basis? What am I missing here?Would appreciate some guidance!