What is the Inverse of a Differential Operator?

[laguna]

Hello everybody,
If I define $$z_\mu = \frac{\partial{\phi}}{\partial{x^{\mu}}}, \, \mu = 0,1,...,n$$, (for some scalar function phi of x=(x_0,...,x_n)) how is then $$\frac{\partial{}}{\partial{z_{\mu}}}$$ defined or rather what is it equal to? How would you call this expression? the inverse of a differential?

Thank you.

 

[mathman]

It is not an inverse of anything. It is a partial derivative.

 

[laguna]

It is not an inverse of anything. It is a partial derivative.

What is it equal to or how can i calculate it please?

 

[Stephen Tashi]

How would you call this expression?

Thank you.

The expression is a differential operator. Unless you state a specific function of $z_\mu$ for it to operate upon, there is nothing to calculate.

 

[Mark44]

Hello everybody,
how is then $$\frac{\partial{}}{\partial{z_{\mu}}}$$ defined or rather what is it equal to?

What is it equal to or how can i calculate it please?

As already explained, it's an operator. By itself, without a function for it to operate on, it has no meaning. Your question is similar to "What is $\sqrt{}$ equal to?