Weierstrass's Necessary Condition
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this in more detail, feel free to use the talk page.
Once you have done so, you can remove this instance of {{MissingLinks}} from the code.
Theorem
Let $\mathbf y: \R \to \R^n$ be an $n$-dimensional vector-valued function such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$.
Let $J$ be a functional such that:
- $\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let $\mathbf w$ be an $n$-dimensional vector such that $\mathbf w \in \R^n$.
Let $\gamma$ be a strong minimum of $J$.
exactly what a strong minimum is, by means of a link to a definition page
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Then along $\gamma$ and for every $\mathbf w$:
- $\map E {x, \mathbf y, \mathbf y', \mathbf w} \ge 0$
exactly what along $\gamma$ means, by means of a link to a definition page
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
where $E$ stands for the Weierstrass E-Function.
Proof
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Source of Name
This entry was named for Karl Theodor Wilhelm Weierstrass.
Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 6.34$: The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum