Understanding Differential Notation

[Hyperreality]

The four velocity component $$u^\mu$$ with coordinate of $$x^\mu(\lambda)$$ is given by

$$u^\mu = \frac{dx^\mu}{d\lambda}$$

where $$\lambda$$ is the proper time. So, the component of acceleration $$a^\mu$$ is

$$a^\mu = \frac{du^\mu}{d\lambda}$$

Using the chain rule we have

$$a^\mu = \frac{\partial u^\mu}{\partial x^\alpha} \frac{dx^\alpha}{d\lambda} = u^\alpha \partial_{\alpha}u^\mu$$

Everything was straight forward except the last part, I don't understand what the notation of $$\partial_{\alpha}$$ meant.

 

[quantumdude]

$\partial_{\alpha}$ is the covariant 4-gradient. It means nothing other than the following:

$$\partial_{\alpha}=\frac{\partial}{\partial x^{\alpha}}$$