Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 1
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Theorem
Let the function $\map h x$ satisfy the equation:
- $-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$
Let $\map h x$ have the boundary conditions:
- $\map h a = \map h b = 0$
Then:
- $\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x = 0$
This article, or a section of it, needs explaining. In particular: Define the domain and codomain of $h$. Presumably real function, but it needs to be made clear. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Proof
\(\ds 0\) | \(=\) | \(\ds \int_a^b \paren 0 h \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \paren {-\map {\frac \d {\d x} } {P h'} + Q h} h \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b Q h^2 \rd x - \int_a^b \map {\frac \d {\d x} } {P h'} h \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b Q h^2 \rd x - \bigintlimits {P h' h} a b + \int_a^b P h' \rd h\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b Q h^2 \rd x + \int_a^b P h'^2 \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x\) |
$\blacksquare$
Sources
1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.26$: Analysis of the Quadratic Functional $ \int_a^b \paren {P h'^2 + Q h^2} \rd x$