# Articles

## Part 2 - coordinates, vectors and the summation convention

The basic object in GR is the spacetime. As a mathematical object, formally it is a differential manifold, but for our purposes it is enough to consider it as a set of points called events, which can be described by coordinates. In GR, the spacetime is 4-dimensional, which means that we need 4 coordinates - one temporal and three spatial ones.

The coordinates can be denoted by pretty much anything (like $x$, $y$, $z$, $t$), but since we will refer to all four of them at multiple occasions, it will be convenient to denote them by numbers. It is pretty standard to denote time by 0, and the spatial coordinates by 1, 2 and 3. The coordinate number $\mu$ will be written like this: $x^\mu$ (attention: in this case it is not a power!). $\mu$ here is called an index (here: an upper one). By convention, if we mean one of the 4 coordinates, we use a greek letter as the index; if only the spatial ones are to be considered, we use a letter from the latin alphabet.

## Part 1 - partial derivatives

Let's remember the ordinary derivatives first. We denote a derivative of a function $f(x)$ as $f'(x)$ or $\frac{df}{dx}$. It means, basically, how fast the value of the function changes while we change the argument x. For example, when $f(x) = x^2$, then $\frac{df}{dx} = 2x$.
But what if the function depends on more than one variable? Like if we have a function $f(x,y) = x^2 + y^2$ that assigns to each point of the plane the square of its distance from the origin. How do we even define the derivative of such a function?