mathematics

Functional derivative

Functional derivative is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Functional derivative rather than just read about it. In short: In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends. In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives.

Key takeaways

Functional derivative belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Functional derivative to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Functional derivative from memory before moving on to harder problems.

Reference excerpt

In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends. In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. 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 integration by parts was used in these derivatives.

Definition In this section, the functional differential (or variation or first variation) is defined. Then the functional derivative is defined in terms of the functional differential.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Functional derivative

Start with the simplest possible case. Write down what Functional derivative claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Functional derivative before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Functional derivative ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Functional derivative

In research: Functional derivative appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Functional derivative in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Functional derivative is common in secondary-school and first-year university syllabi. It links to neighbouring topics Calculus of variations, Differential calculus, Differential operators, so understanding it makes those chapters shorter.
In everyday life: Look for Functional derivative outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Functional derivative in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Functional derivative means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Functional derivative out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Functional derivative in simple terms?

Why does Functional derivative matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Functional derivative?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Functional derivative.