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Ciro Santilli  40 Updated 2025-07-16

Like a heat equation but for functions without time dependence, space-only.

TODO confirm: does the solution of the heat equation always converge to the solution of the Laplace equation as time tends to infinity?

In one dimension, the Laplace equation is boring as it is just a straight line since the second derivative must be 0. That also matches our intuition of the limit solution of the heat equation.

Uniqueness: Uniqueness theorem for Poisson's equation.

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Heat equation by

Ciro Santilli  40 Updated 2025-07-16

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Besides being useful in engineering, it was very important historically from a "development of mathematics point of view", e.g. it was the initial motivation for the Fourier series.

Some interesting properties:

TODO confirm: for a fixed boundary condition that does not depend on time, the solutions always approaches one specific equilibrium function.
This is in contrast notably with the wave equation, which can oscillate forever.
TODO: for a given point, can the temperature go down and then up, or is it always monotonic with time?
information propagates instantly to infinitely far. Again in contrast to the wave equation, where information propagates at wave speed.

Sample numerical solutions:

with FreeFem:
- heat-dirichlet.1d.freefem
- heat-dirichlet-2d-freefem

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Wave equation by

Ciro Santilli  40 Updated 2025-07-16

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Describes perfect lossless waves on the surface of a string, or on a water surface.

Uniqueness: math.stackexchange.com/questions/1113622/uniqueness-of-solutions-to-the-wave-equation

As mentioned at: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from solving partial differential equations with the Fourier series citing courses.maths.ox.ac.uk/node/view_material/1720, analogously to the heat equation, the wave linear equation can be be solved nicely with separation of variables.

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Matrix Lie group by

Ciro Santilli  40 Updated 2025-07-16

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This important and common simple case has easy properties.

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Classical group by

Ciro Santilli  40 Updated 2025-07-16

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General linear group by

Ciro Santilli  40 Updated 2025-07-16

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Invertible matrices. Or if you think a bit more generally, an invertible linear map.

When the field

F

is not given, it defaults to the real numbers.

Non-invertible are excluded "because" otherwise it would not form a group (every element must have an inverse). This is therefore the largest possible group under matrix multiplication, other matrix multiplication groups being subgroups of it.

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Lie algebra of

G L (n)

Ciro Santilli  40 Updated 2025-07-16

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Is the set of all n-by-y square matrices.

Because

G L (n)

is a Lie group we can use Section "Lie algebra of a matrix Lie group".

For every matrix

x

in the set of all n-by-y square matrices

M_{n}

e^{x}

has inverse

e^{-} x

Note that this works even if

x

is not invertible, and therefore not in

G L (n)

Therefore, the Lie algebra of

G L (n)

is the entire

M_{n}

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Special linear group by

Ciro Santilli  40 Updated 2025-07-16

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Specials sub case of the general linear group when the determinant equals exactly 1.

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Isometry group by

Ciro Santilli  40 Updated 2025-07-16

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The group of all transformations that preserve some bilinear form, notable examples:

orthogonal group preserves the inner product
unitary group preserves a Hermitian form
Lorentz group preserves the Minkowski inner product

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Orthogonal group by

Ciro Santilli  40 Updated 2025-07-16

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Projective linear group by

Ciro Santilli  40 Updated 2025-07-16

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TODO motivation. Motivation. Motivation. Motivation. The definitin with quotient group is easy to understand.

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Poincaré group by

Ciro Santilli  40 Updated 2025-07-16

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Full set of all possible special relativity symmetries:

In simple and concrete terms. Suppose you observe N particles following different trajectories in Spacetime.

There are two observers traveling at constant speed relative to each other, and so they see different trajectories for those particles:

space and time shifts, because their space origin and time origin (time they consider 0, i.e. when they started their timers) are not synchronized. This can be modelled with a 4-vector addition.
their space axes are rotated relative to one another. This can be modelled with a 4x4 matrix multiplication.
and they are moving relative to each other, which leads to the usual spacetime interactions of special relativity. Also modelled with a 4x4 matrix multiplication.

Note that the first two types of transformation are exactly the non-relativistic Galilean transformations.

The Poincare group is the set of all matrices such that such a relationship like this exists between two frames of reference.

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Cyclic group by

Ciro Santilli  40 Updated 2025-07-16

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The direct product of two cyclic groups of coprime order is another cyclic group by

Ciro Santilli  40 Updated 2025-07-16

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You just map the value (1, 1)

C_{m} \times C_{n}

to the value 1 of

C_{mn}

, and it works out. E.g. for

C_{2} \times C_{3}

, the group generated by of (1, 1) is:

0 = (0, 0)
1 = (1, 1)
2 = (0, 2)
3 = (1, 0)
4 = (0, 1)
5 = (1, 2)
6 = (0, 0) = 0

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Permutation by

Ciro Santilli  40 Updated 2025-07-16

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Dihedral group by

Ciro Santilli  40 Updated 2025-07-16

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Our notation:

D_{n}

, called "dihedral group of degree n", means the dihedral group of the regular polygon with

n

sides, and therefore has order

2 n

(all rotations + flips), called the "dihedral group of order 2n".

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Wallpaper group by

Ciro Santilli  40 Updated 2025-07-16

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17 of them.

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Space group by

Ciro Santilli  40 Updated 2025-07-16

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All possible repetitive crystal structures!

219 of them.

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Klein four-group by

Ciro Santilli  40 Updated 2025-07-16

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C_{2} \times C_{2}

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 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!

Video 1.

Intro to OurBigBook

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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.
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OurBigBook Web topics demo
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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 2.
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Video 3.
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. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
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