A Givens rotation is the identity except in rows and columns , where it has the block
For and , choose
when . Then
If both entries vanish, any angle works.
For the given matrix, first use
It gives
Then use
Thus is
All leading row entries are positive. The resulting QR decomposition by Givens rotations is
where
and is the matrix above. Since it is a product of transposed rotations, is orthogonal.
Let point downslope and point normally away from the plane, with . Write the parallel velocity as and take to be the magnitude of the air's upslope stress. The steady equations are
with boundary conditions
The free-surface condition makes , and integration gives
The surface velocity, downslope shear exerted by the fluid on the plane, and volume flux per unit width are
Thus the inclined viscous film with opposing surface shear reverses in the three senses when
The order of increasing required air stress is therefore
Differentiate under the integral and use :
The omitted boundary term vanishes if the current is localized sufficiently rapidly. A steady current obeys charge conservation , hence
For much larger than the source size,
Localization and imply . They also imply
by integrating . Thus the first nonzero moment is antisymmetric and can be written using the magnetic dipole moment
Consequently the Coulomb-gauge vector potential of a localized steady current has far field
The dimensions are
Continuity at is automatic in the proposed expression. Integrating the differential equation through gives the derivative jump
Thus
In terms of the Wronskian
one has
The boundary conditions hold because and .
The Abel identity gives . If , the Wronskian and hence are constant. The two branches of the displayed formula are then interchanged by , proving
This is the symmetry of the Neumann Green function for a second-order ordinary differential equation in the self-adjoint case.
For , choose
Their Wronskian is
Writing and gives
The solution of the inhomogeneous problem is . Equivalently, solving directly gives
The two Neumann conditions yield and
Therefore

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