How to do functional derivatives

In summary, the speaker is asking for an explanation on a concept involving delta functions in their homework. They mention struggling to find resources online and ask for someone to explain it step by step. They also mention posting a picture of their homework in the appropriate section.
  • #1
leroyjenkens
616
49
Here's an example from my homework. I already turned it in, though. I basically just copied what I could from my notes, but I have no idea how this is done.
Could someone explain this to me? I can't find anything intelligible (at least to me) of this stuff on any website. My notes contain parts where the instructor changed certain terms into delta functions, but I'm not sure why, and I don't know when to do that.
So if anyone has the time, could you explain this step by step to me of what this question is looking for? I'll post the picture from my homework.
Thanks.
 

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  • #2
Please post in the homework section.
 

1. What is a functional derivative?

A functional derivative is a mathematical concept used in functional analysis that represents the rate of change of a functional with respect to a function. It is similar to a regular derivative, which represents the rate of change of a function with respect to a variable, but in this case, the variable is also a function.

2. Why are functional derivatives important?

Functional derivatives are important because they allow us to find the extremum (minimum or maximum) of a functional. This is useful in many fields of science, including physics, chemistry, and mathematics, where finding the minimum or maximum of a functional can provide valuable insights into the underlying system or process.

3. How do you calculate a functional derivative?

To calculate a functional derivative, you use the Euler-Lagrange equation, which is a generalization of the fundamental theorem of calculus for functionals. This equation involves taking the derivative of the functional with respect to the function and setting it equal to 0. The resulting equation can then be solved to find the function that minimizes or maximizes the functional.

4. What are some applications of functional derivatives?

Functional derivatives have many applications in science and engineering. Some examples include finding the path of least action in classical mechanics, optimizing the shape of an airplane wing for maximum lift, and predicting the electron density of a molecule in quantum chemistry.

5. Are there any limitations to using functional derivatives?

One limitation of using functional derivatives is that they can only be applied to continuous functions. Additionally, the functional must be well-defined and differentiable. In some cases, the Euler-Lagrange equation may also have multiple solutions, making it difficult to determine the correct function that minimizes or maximizes the functional.

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