mathematics

Weierstrass–Erdmann condition

Weierstrass–Erdmann condition 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 Weierstrass–Erdmann condition rather than just read about it. In short: The Weierstrass–Erdmann condition is a mathematical result from the calculus of variations, which specifies sufficient conditions for broken extremals (that is, an extremal which is constrained to be smooth except at a finite number of "corners"). Conditions The Weierstrass-Erdmann corner conditions stipulate that a broken extremal y ( x ) {\displaystyle y(x)} of a functional J = ∫ a b f ( x , y , y ′ ) d x {\displa…

Key takeaways

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

Reference excerpt

The Weierstrass–Erdmann condition is a mathematical result from the calculus of variations, which specifies sufficient conditions for broken extremals (that is, an extremal which is constrained to be smooth except at a finite number of "corners").

Conditions The Weierstrass-Erdmann corner conditions stipulate that a broken extremal y ( x ) {\displaystyle y(x)} of a functional J = ∫ a b f ( x , y , y ′ ) d x {\displaystyle J=\int \limits _{a}^{b}f(x,y,y')\,dx} satisfies the following two continuity relations at each corner c ∈ [ a , b ] {\displaystyle c\in [a,b]} :

Applications The condition allows one to prove that a corner exists along a given extremal. As a result, there are many applications to differential geometry. In calculations of the Weierstrass excess function, it is often helpful to find where corners exist along the curves. Similarly, the condition allows for one to find a minimizing curve for a given integral.

References

Worked examples

Example 1 — a first encounter with Weierstrass–Erdmann condition

Start with the simplest possible case. Write down what Weierstrass–Erdmann condition 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 Weierstrass–Erdmann condition 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 Weierstrass–Erdmann condition 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 Weierstrass–Erdmann condition

In research: Weierstrass–Erdmann condition 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 Weierstrass–Erdmann condition 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: Weierstrass–Erdmann condition is common in secondary-school and first-year university syllabi. It links to neighbouring topics Calculus of variations, so understanding it makes those chapters shorter.
In everyday life: Look for Weierstrass–Erdmann condition 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 Weierstrass–Erdmann condition in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Weierstrass–Erdmann condition 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 Weierstrass–Erdmann condition out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Weierstrass–Erdmann condition in simple terms?

Why does Weierstrass–Erdmann condition 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 Weierstrass–Erdmann condition?

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 Weierstrass–Erdmann condition.