Along a trajectory, the chain rule gives
Thus is a first integral, and every trajectory lies on one of its level sets.
Using the Taylor expansion
we obtain
When cooling is negligible, the milk lowers the temperature by one unit at . The hot-water term begins at and raises it at one unit per unit time, so one further unit of time is needed to erase the deficit.
The cups have equal temperatures up to the intervention at . For the later recrossing, subtract the formula for from that for and set the result to zero:
Writing gives . Hence the nontrivial meeting time is
For , the Heaviside step function contributes a constant unit forcing. Solving the resulting first-order linear differential equation from the value at gives
Integrating the equation through gives the jump condition for an impulse
For the equation is again homogeneous, so propagation of the post-jump temperature gives
Therefore
Put . Before either forcing term acts, Newton's law of cooling gives
The identity
shows that is a particular solution. Combining it with the homogeneous solution from part (b) gives
The condition gives , and gives . Hence
The exact initial-value solution is . Since ,
Thus the absolute endpoint error is , consistent with the second-order truncation error of the centred scheme.
The characteristic equation of a linear recurrence is
Its repeated root gives
Since , this can be written , exactly the discrete form of the continuous solution of .
The Taylor expansions about are
Adding them cancels the odd powers of and gives
Thus the centred finite difference has truncation error , so .
Every allocation is determined by the set of children on the red bus, so there are equally likely allocations. Exactly one places every boy on red and every girl on green. Hence
Applying Stirling formula to the numerator and denominator gives the central binomial coefficient asymptotic
and therefore
Since the natural logarithm is increasing, comparison with an integral gives
The lower bound is . Dividing all three expressions by and applying the squeeze theorem gives
Let be right-arm length minus left-arm length. A linear combination of jointly normal random variables is normal, and
while
Thus , and standardization of a normal random variable gives
Direct multiplication and the identities and give
Thus is an orthogonal matrix, while preserves the indefinite quadratic form .

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