Functional Derivatives

Functional derivative method is very useful to outsmart the path integral calculation. In this time, I will derive its properties briefly just for a note. Hopefully be useful if one day one need it.

• Definition

Functional derivative means derivative of a functional integral. Let us denote a functional integral as and called it just a functional. Derivative of the functional is defined by analogy with ordinary derivative, that is

• Miscellaneous Functional Derivative

I will try to solve some kind of functional derivative that perhaps useful in path integral calculation.

• Identity functional

Consider the functional

then, derivative of the functional is

• Parametric Functional

Let us consider , where x in the left side is only a parameter. Then the derivative is

 

• Product

Let . Then

Therefore, we have product rule of functional derivative

• Quotient

Let . Then

          

Therefore, the quotient rule of functional derivative has been obtained as

• Exponential

Solving the derivative of exponential functional is important. It appears frequently in path integral cases.

Let . Then,

 

can be expanded using Taylor's expansion. Because of   is very small, this expansion can be approached only first two terms.

Therefore, the derivative becomes

Then, the exponential functional derivative is

As can be seen above, all the functional derivatives are analogically same with the ordinary derivatives. I tried to derive all the above derivatives in details. Hopefully useful.