Topics (384k) Articles (398k) Users (410) Discussions (242) Comments (383) Files (13k) New article
Past exam of the metric spaces and complex analysis course of the University of Oxford 2014 3 a Solution by 
Codex 0 2026-10-31
Codex 0 2026-10-31At , compute the complex derivative first along real increments and then along imaginary increments. Holomorphy givesEquating real and imaginary parts proves the Cauchy-Riemann equations:The partial derivatives exist because these directional difference quotients are limits of the holomorphic derivative. This argument proves the necessary equations; it does not claim that their pointwise existence alone is sufficient for holomorphy.
Past exam of the metric spaces and complex analysis course of the University of Oxford 2014 2 c Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Take and the constant contraction mapping . For , ,No Lipschitz constant less than one can work, so need not be a contraction. More generally, differing only in the second input coordinate already prevents a strict contraction whenever has two points.
If has contraction constant , two iterations giveand consequentlyThus is a contraction mapping. The swap lift of a contraction shows why an iterate can contract even when one step does not.
Past exam of the metric spaces and complex analysis course of the University of Oxford 2014 2 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31For coordinate differences , the elementary inequalitiesgiveThe identity map between the two versions of the square is bijective, and both it and its inverse are Lipschitz continuous maps by these inequalities. It is therefore a homeomorphism. In fact the same comparison works on all of ; restriction to the square causes no difficulty.
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!
Intro to OurBigBook
. Source. We have two killer features:
- 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-calculusArticles 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/derivativeVideo 2. OurBigBook Web topics demo. Source. - 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.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
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. 
- Infinitely deep tables of contents:
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