I'm reading the book "Semi-Riemannian Geometry with Applications to Relativity", which defines a tangent vector at a point $p$ in a manifold $M$ to be a function $v : \mathcal{F}(M) \to \mathbb{R}$ (where $\mathcal{F}(M)$ is all smooth functions $M \to \mathbb{R}$), which is:
- Linear: $v(af + bg) = av(f) + bv(g)$
- Leibnizian: $v(fg) = v(f)g(p) + f(p)v(g)$
I'm confused by this definition. Why is a vector a function $\mathcal{F}(M) \to \mathbb{R}$? Given a function $f : M \to \mathbb{R}$, what does $v(f)$ represent? The Leibnizian condition also seems weird to me, what is it saying?
You will probably see later that $v$ is a derivative. On a manifold in $\mathbb{R}^n$ you can identify $v(f)$ with $\vec{v}\cdot\nabla f.$ The Leibnizian condition is the product rule of derivatives.