If I have $\alpha,\beta\in\mathbb{N}^n$ and the following formula $f(x) = D^{\alpha}x^{\beta}$ why does that lead to a single value (so not dimensional)?
I know this leads to either 0 or a!(recently learned that is a single number) but why? (I am not familiar with the linear algebra so I might need quite basic information)
On
These $\alpha$ and $\beta$ are multiindices. They may seem scary at first glance, but once you got used to them, they are pretty awesome and also handy as they greatly shorten lengthy expressions and also work basically the same as usual indices (once you adapted all notions).
I refrain from listing all the definitions and properties, since they are neatly collected at their Wikipedia page.
Generally the notation $D^\alpha x^\beta$ is shorthand for $$\frac{\partial^{\alpha_1}}{\partial x_1}\cdots \frac{\partial^{\alpha_n}}{\partial x_n}\left[ x_1^{\beta_1}\cdots x_n^{\beta_n}\right]$$ which will, in general, be a monomial and not just a number.