Reading Sean Carroll's spacetime and geometry he says

If $x^{\sigma*}$ is the coordinate which ${\mu\nu}$ is independent of, let us consider the vector $\partial_{\sigma*}$ which we label as $$K=\partial_{\sigma*}\, , \tag{3.169}$$ which is equivalent in component notation to $$K^\mu=(\partial_{\sigma*})^\mu = \delta^\mu_{\sigma*}. \tag{3.170} $$

Now I ask you, why call a vector that for example looks like this $(1,0,0,0)$ as $\partial_0$?

Is there a point and a reason behind this notation which I always found pretty confusing? For example before reading Sean Carrol the times I read a phrase like the Killing vector is $\partial/\partial t$ I thought it was something like $(\partial_t,0,0,0)$ and beside that I was never sure if it was really $(\partial_t,0,0,0)$ or if it was a spacelike vector with a component equal to $\partial_t$

Reading Sean Carroll's spacetime and geometry he says

If $x^{\sigma_*}$ is the coordinate which ${\mu\nu}$ is independent of, let us consider the vector $\partial_{\sigma_*}$ which we label as $$K=\partial_{\sigma_*}\, , \tag{3.169}$$ which is equivalent in component notation to $$K^\mu=(\partial_{\sigma_*})^\mu = \delta^\mu_{\sigma_*}. \tag{3.170} $$

Now I ask you, why call a vector that for example looks like this $(1,0,0,0)$ as $\partial_0$?

Is there a point and a reason behind this notation which I always found pretty confusing? For example before reading Sean Carroll the times I read a phrase like the Killing vector is $\partial/\partial t$ I thought it was something like $(\partial_t,0,0,0)$ and beside that I was never sure if it was really $(\partial_t,0,0,0)$ or if it was a spacelike vector with a component equal to $\partial_t$