Question on functional derivative

In summary, the conversation discussed the functional derivatives of a function defined as an integral and how to compute its functional derivative. The result was shown to be the Dirac delta function, indicating that a small change in the function at a specific position will affect the value of the integral. Functional derivatives are useful in understanding how a functional depends on its underlying functions.
  • #1
Einj
470
59
Hi everyone! I have a question on functional derivatives. I have a function defined as:
$$
F[\{u\}]=\int d^3r \sum_{i=1}^3 \frac{\partial u_i}{\partial r_i},
$$ where [itex]u_i(\vec r)[/itex] is a function of the position. I need to compute its functional derivative. To do that I did the following:
$$
F[\{u_k+\delta u_k\}]=F[\{u_k\}]+\int d^3r\sum_{i=1}^3\delta_{ik}\frac{\partial \delta u_k}{\partial r_i}=F[\{u_k\}]+\int d^3r\frac{\partial \delta u_k}{\partial r_k}.
$$
Now take, for example, [itex]k=1[/itex]. Then we have:
$$
\int d^3r\frac{\partial \delta u_x}{dx}=\int dydz \delta u_x(x,y,z)=\int dx'dy'dz'\delta(x-x')\delta u_x(x',y',z'),
$$
is then correct to say that:
$$
\frac{\delta F[\{u\}]}{\delta u_k(r')}=\delta(r_k-r_k')
$$??

Thanks
 
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  • #2
for your question! To answer your question, yes, your derivation is correct. The functional derivative of F with respect to u_k(r') is equal to the Dirac delta function of r_k-r_k'. This means that a small change in the function u_k at position r_k' will result in a change in the value of F. This is a useful tool in functional analysis for understanding how a functional depends on its underlying functions. Let me know if you have any further questions.
 

1. What is a functional derivative?

A functional derivative is a mathematical concept used in functional analysis to calculate the rate of change of a functional with respect to its input function. It is similar to the traditional derivative used in calculus, but instead of operating on a function of a variable, it operates on a functional of a function.

2. How is functional derivative different from a traditional derivative?

The main difference between functional derivative and traditional derivative is that functional derivative operates on functions of a function, while traditional derivative operates on functions of a variable. This means that the functional derivative evaluates the change in a functional caused by a change in the input function, while traditional derivative evaluates the change in a function caused by a change in its independent variable.

3. What is the significance of functional derivative in physics?

In physics, functional derivatives are used to find the equations of motion for a system. This is because the equations of motion can be expressed as a functional of the system's variables, and the functional derivative can be used to determine how these variables change over time. Functional derivatives are also used in field theory, where the fields themselves are functions of space and time.

4. How is functional derivative used in optimization problems?

Functional derivatives are commonly used in optimization problems to find the extrema of a functional. By setting the functional derivative to zero, the optimal function that minimizes or maximizes the functional can be found. This is often used in variational calculus and in finding the path of least action in physics.

5. Are there any applications of functional derivative outside of mathematics and physics?

Yes, functional derivatives have applications in various fields such as economics, chemistry, and biology. In economics, functional derivatives are used in the study of optimal economic growth. In chemistry, they are used in the study of chemical reactions and kinetics. In biology, they are used in the study of population dynamics and evolution.

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