Assume and abbreviate
The integral of the nonnegative square expands to for every real . If , completing the square, or choosing , gives . Hence the Cauchy-Schwarz inequality is
If , continuity and nonnegativity imply throughout the interval, so the same inequality is immediate.
When , equality holds exactly when . The integrand is continuous and nonnegative, so it must vanish everywhere, giving . Including the case of a zero function, equality is equivalent to linear dependence of and as elements of the real vector space of continuous functions on the interval.
Evaluate the recurrence boundary terms at zero and . For , both and vanish at both endpoints. Therefore the Wallis integrals obey
The starting value for both is . Thus
Write . Apply integration by parts with and :
Using on the right gives . Rearranging proves
For , the analogous integration by parts uses and , giving . Hence
An arbitrary constant of integration absorbs the choices in the lower-order antiderivatives.
Starting from , successive use of gives
These expressions follow directly from the requested recurrence, rather than requiring a separate trigonometric expansion.
The two prescribed directions have dot product zero. Normalize them to
Their cross product is the perpendicular unit vector . Using these as columns gives
The columns form an orthonormal basis, so and is an orthogonal matrix.
Suppose . The given quadratic form bound then gives
so . Thus the kernel of is trivial. A square matrix with trivial kernel is an invertible matrix, proving the assertion. No symmetry assumption on is needed.
The cube is , , so throughout it. Evaluate the six outward surface integrals directly.
On the normal component is , giving flux ; on it is , giving flux . On the normal component is , giving ; on it is zero. On the normal component is and on it is , so those unit-area faces give and .
Adding all six contributions yields
Only direct surface integral was used, as requested.
The homogeneous solution is . If , the method of undetermined coefficients permits a particular integral . Substitution gives and hence
For , the forcing is itself a homogeneous solution, so this trial fails by resonance in a differential equation. Multiplying the sine trial by gives
Thus the resonant general solution is
The secular term containing describes the increasing amplitude under undamped resonant forcing.

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 5. . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact