Functional derivative

In the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>, a field of <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, the functional derivative (or variational derivative) relates a change in a <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a> (a functional in this sense is a function that acts on functions) to a change in a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> on which the functional depends.
In the calculus of variations, functionals are usually expressed in terms of an <a href="/facts/Integral/V7lyV997">integral</a> of functions, their <a href="/facts/Argument_of_a_function/QkuPc28I">arguments</a>, and their <a href="/facts/Derivative/7Ito70Dh">derivatives</a>. In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, the coefficient of δf in the first order term is called the functional derivative.
For example, consider the functional

J
        [
        f
        ]
        =
        
          ∫
          
            a
          
          
            b
          
        
        L
        (
        
        x
        ,
        f
        (
        x
        )
        ,
        
          f
          ′
        
        
          (
          x
          )
        
        
        )
        
        d
        x
        
        ,
      
    
    {\displaystyle J[f]=\int _{a}^{b}L(\,x,f(x),f'{(x)}\,)\,dx\,,}

where f ′(x) ≡ df/dx. If f is varied by adding to it a function δf, and the resulting integrand L(x, f +δf, f ′+δf ′) is expanded in powers of δf, then the change in the value of J to first order in δf can be expressed as follows:

δ
                J
              
              
                
                =
                
                  ∫
                  
                    a
                  
                  
                    b
                  
                
                
                  (
                  
                    
                      
                        
                          ∂
                          L
                        
                        
                          ∂
                          f
                        
                      
                    
                    δ
                    f
                    (
                    x
                    )
                    +
                    
                      
                        
                          ∂
                          L
                        
                        
                          ∂
                          
                            f
                            ′
                          
                        
                      
                    
                    
                      
                        d
                        
                          d
                          x
                        
                      
                    
                    δ
                    f
                    (
                    x
                    )
                  
                  )
                
                
                d
                x
                
              
            
            
              
              
                
                =
                
                  ∫
                  
                    a
                  
                  
                    b
                  
                
                
                  (
                  
                    
                      
                        
                          ∂
                          L
                        
                        
                          ∂
                          f
                        
                      
                    
                    −
                    
                      
                        d
                        
                          d
                          x
                        
                      
                    
                    
                      
                        
                          ∂
                          L
                        
                        
                          ∂
                          
                            f
                            ′
                          
                        
                      
                    
                  
                  )
                
                δ
                f
                (
                x
                )
                
                d
                x
                
                +
                
                
                  
                    
                      ∂
                      L
                    
                    
                      ∂
                      
                        f
                        ′
                      
                    
                  
                
                (
                b
                )
                δ
                f
                (
                b
                )
                
                −
                
                
                  
                    
                      ∂
                      L
                    
                    
                      ∂
                      
                        f
                        ′
                      
                    
                  
                
                (
                a
                )
                δ
                f
                (
                a
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\delta J&=\int _{a}^{b}\left({\frac {\partial L}{\partial f}}\delta f(x)+{\frac {\partial L}{\partial f'}}{\frac {d}{dx}}\delta f(x)\right)\,dx\,\\[1ex]&=\int _{a}^{b}\left({\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\right)\delta f(x)\,dx\,+\,{\frac {\partial L}{\partial f'}}(b)\delta f(b)\,-\,{\frac {\partial L}{\partial f'}}(a)\delta f(a)\end{aligned}}}

where the variation in the derivative, δf ′ was rewritten as the derivative of the variation (δf) ′, and <a href="/facts/Integration_by_parts/aV6JgBNd">integration by parts</a> was used in these derivatives.

Functional derivative open-in-new

Functional derivative