Skip to content

Commit 3595ab9

Browse files
leiosleios
committed
Merge branch 'beta'
2 parents 40e2e9d + 17c3c77 commit 3595ab9

File tree

7 files changed

+163
-2
lines changed

7 files changed

+163
-2
lines changed

‎chapters/computational_mathematics/cooley_tukey.md

Lines changed: 24 additions & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -1,3 +1,26 @@
1+
<script>
2+
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
3+
</script>
4+
$$
5+
\newcommand{\d}{\mathrm{d}}
6+
\newcommand{\bff}{\boldsymbol{f}}
7+
\newcommand{\bfg}{\boldsymbol{g}}
8+
\newcommand{\bfp}{\boldsymbol{p}}
9+
\newcommand{\bfq}{\boldsymbol{q}}
10+
\newcommand{\bfx}{\boldsymbol{x}}
11+
\newcommand{\bfu}{\boldsymbol{u}}
12+
\newcommand{\bfv}{\boldsymbol{v}}
13+
\newcommand{\bfA}{\boldsymbol{A}}
14+
\newcommand{\bfB}{\boldsymbol{B}}
15+
\newcommand{\bfC}{\boldsymbol{C}}
16+
\newcommand{\bfM}{\boldsymbol{M}}
17+
\newcommand{\bfJ}{\boldsymbol{J}}
18+
\newcommand{\bfR}{\boldsymbol{R}}
19+
\newcommand{\bfT}{\boldsymbol{T}}
20+
\newcommand{\bfomega}{\boldsymbol{\omega}}
21+
\newcommand{\bftau}{\boldsymbol{\tau}}
22+
$$
23+
124
## What Makes a Fourier Transform Fast?
225

326
If there were ever an algorithm to radically change the landscape of computer science and engineering by making seemingly impossible possible, it would be the Fast Fourier Transform (FFT). On the surface, the algorithm seems like a simple application of recursion, and in principle, that is exactly what it is; however, the Fourier Transform is no ordinary transform -- it allows researchers and engineers to easily bounce back and forth between real space and frequency space and is the heart of many physics and engineering applications.
@@ -10,7 +33,7 @@ Simply put, the Fourier Transform is a beautiful application of complex number s
1033

1134
To an outsider, the Fourier Transform looks like a mathematical mess -- certainly a far cry from the heroic portal between two domains I have depicted it to be; however, like most things, it's not as bad as it initially appears to be. So, here it is in all it's glory!
1235

13-
$$F(\xi) = \int_{-\infty} ^\infty f(x) e^{-2 \pi i x \xi} dx$$
36+
$F(\xi) = \int_{-\infty} ^\infty f(x) e^{-2 \pi i x \xi} dx$
1437

1538
and
1639

‎chapters/computational_mathematics/matrix_methods/gaussian_elimination.md

Lines changed: 23 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -1,3 +1,26 @@
1+
<script>
2+
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
3+
</script>
4+
$$
5+
\newcommand{\d}{\mathrm{d}}
6+
\newcommand{\bff}{\boldsymbol{f}}
7+
\newcommand{\bfg}{\boldsymbol{g}}
8+
\newcommand{\bfp}{\boldsymbol{p}}
9+
\newcommand{\bfq}{\boldsymbol{q}}
10+
\newcommand{\bfx}{\boldsymbol{x}}
11+
\newcommand{\bfu}{\boldsymbol{u}}
12+
\newcommand{\bfv}{\boldsymbol{v}}
13+
\newcommand{\bfA}{\boldsymbol{A}}
14+
\newcommand{\bfB}{\boldsymbol{B}}
15+
\newcommand{\bfC}{\boldsymbol{C}}
16+
\newcommand{\bfM}{\boldsymbol{M}}
17+
\newcommand{\bfJ}{\boldsymbol{J}}
18+
\newcommand{\bfR}{\boldsymbol{R}}
19+
\newcommand{\bfT}{\boldsymbol{T}}
20+
\newcommand{\bfomega}{\boldsymbol{\omega}}
21+
\newcommand{\bftau}{\boldsymbol{\tau}}
22+
$$
23+
124
# Gaussian Elimination
225

326
So, how exactly do we go about solving a system of linear equations? Well, one way is _Gaussian Elimination_, which you may have encountered before in a math class or two. The basic idea is that we take a system of equations,

‎chapters/computational_mathematics/matrix_methods/matrix_methods.md

Lines changed: 24 additions & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -1,3 +1,26 @@
1+
<script>
2+
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
3+
</script>
4+
$$
5+
\newcommand{\d}{\mathrm{d}}
6+
\newcommand{\bff}{\boldsymbol{f}}
7+
\newcommand{\bfg}{\boldsymbol{g}}
8+
\newcommand{\bfp}{\boldsymbol{p}}
9+
\newcommand{\bfq}{\boldsymbol{q}}
10+
\newcommand{\bfx}{\boldsymbol{x}}
11+
\newcommand{\bfu}{\boldsymbol{u}}
12+
\newcommand{\bfv}{\boldsymbol{v}}
13+
\newcommand{\bfA}{\boldsymbol{A}}
14+
\newcommand{\bfB}{\boldsymbol{B}}
15+
\newcommand{\bfC}{\boldsymbol{C}}
16+
\newcommand{\bfM}{\boldsymbol{M}}
17+
\newcommand{\bfJ}{\boldsymbol{J}}
18+
\newcommand{\bfR}{\boldsymbol{R}}
19+
\newcommand{\bfT}{\boldsymbol{T}}
20+
\newcommand{\bfomega}{\boldsymbol{\omega}}
21+
\newcommand{\bftau}{\boldsymbol{\tau}}
22+
$$
23+
124
# Matrix Methods
225

326
Mathematics is the foundation for many areas of science and engineering, and it is not uncommon for these applications to require the manipulation of large matrices. Though these applications are not mathematics, they use mathematics in everything they do and the methods to manipulate matrices are certainly mathematical. For this reason, I feel we should tour many methods used to manipulate matrices into any usable form. For the most part, we can divide the mathematical matrix methods into the following categories:
@@ -6,4 +29,4 @@ Mathematics is the foundation for many areas of science and engineering, and it
629
2. Creating upper and lower diagonal matrices
730
3. Solving systems of Equations
831

9-
There are certainly more methods out there, so don't be offended if I have failed to list one of your favorite types of algorithms. We'll be filling more in as we go along!
32+
There are certainly more methods out there, so don't be offended if I have failed to list one of your favorite types of algorithms. We'll be filling more in as we go along!

‎chapters/computational_mathematics/matrix_methods/thomas.md

Lines changed: 23 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -1,3 +1,26 @@
1+
<script>
2+
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
3+
</script>
4+
$$
5+
\newcommand{\d}{\mathrm{d}}
6+
\newcommand{\bff}{\boldsymbol{f}}
7+
\newcommand{\bfg}{\boldsymbol{g}}
8+
\newcommand{\bfp}{\boldsymbol{p}}
9+
\newcommand{\bfq}{\boldsymbol{q}}
10+
\newcommand{\bfx}{\boldsymbol{x}}
11+
\newcommand{\bfu}{\boldsymbol{u}}
12+
\newcommand{\bfv}{\boldsymbol{v}}
13+
\newcommand{\bfA}{\boldsymbol{A}}
14+
\newcommand{\bfB}{\boldsymbol{B}}
15+
\newcommand{\bfC}{\boldsymbol{C}}
16+
\newcommand{\bfM}{\boldsymbol{M}}
17+
\newcommand{\bfJ}{\boldsymbol{J}}
18+
\newcommand{\bfR}{\boldsymbol{R}}
19+
\newcommand{\bfT}{\boldsymbol{T}}
20+
\newcommand{\bfomega}{\boldsymbol{\omega}}
21+
\newcommand{\bftau}{\boldsymbol{\tau}}
22+
$$
23+
124
##### Dependencies
225
* [Gaussian Elimination](gaussian_elimination.md)
326

‎chapters/computational_physics/euler.md

Lines changed: 23 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -1,3 +1,26 @@
1+
<script>
2+
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
3+
</script>
4+
$$
5+
\newcommand{\d}{\mathrm{d}}
6+
\newcommand{\bff}{\boldsymbol{f}}
7+
\newcommand{\bfg}{\boldsymbol{g}}
8+
\newcommand{\bfp}{\boldsymbol{p}}
9+
\newcommand{\bfq}{\boldsymbol{q}}
10+
\newcommand{\bfx}{\boldsymbol{x}}
11+
\newcommand{\bfu}{\boldsymbol{u}}
12+
\newcommand{\bfv}{\boldsymbol{v}}
13+
\newcommand{\bfA}{\boldsymbol{A}}
14+
\newcommand{\bfB}{\boldsymbol{B}}
15+
\newcommand{\bfC}{\boldsymbol{C}}
16+
\newcommand{\bfM}{\boldsymbol{M}}
17+
\newcommand{\bfJ}{\boldsymbol{J}}
18+
\newcommand{\bfR}{\boldsymbol{R}}
19+
\newcommand{\bfT}{\boldsymbol{T}}
20+
\newcommand{\bfomega}{\boldsymbol{\omega}}
21+
\newcommand{\bftau}{\boldsymbol{\tau}}
22+
$$
23+
124
##### Dependencies
225

326
* [Taylor Series Expansions](../mathematical_background/taylor_series.md)

‎chapters/math.md

Lines changed: 23 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,23 @@
1+
<script>
2+
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
3+
</script>
4+
$$
5+
\newcommand{\d}{\mathrm{d}}
6+
\newcommand{\bff}{\boldsymbol{f}}
7+
\newcommand{\bfg}{\boldsymbol{g}}
8+
\newcommand{\bfp}{\boldsymbol{p}}
9+
\newcommand{\bfq}{\boldsymbol{q}}
10+
\newcommand{\bfx}{\boldsymbol{x}}
11+
\newcommand{\bfu}{\boldsymbol{u}}
12+
\newcommand{\bfv}{\boldsymbol{v}}
13+
\newcommand{\bfA}{\boldsymbol{A}}
14+
\newcommand{\bfB}{\boldsymbol{B}}
15+
\newcommand{\bfC}{\boldsymbol{C}}
16+
\newcommand{\bfM}{\boldsymbol{M}}
17+
\newcommand{\bfJ}{\boldsymbol{J}}
18+
\newcommand{\bfR}{\boldsymbol{R}}
19+
\newcommand{\bfT}{\boldsymbol{T}}
20+
\newcommand{\bfomega}{\boldsymbol{\omega}}
21+
\newcommand{\bftau}{\boldsymbol{\tau}}
22+
$$
23+

‎chapters/mathematical_background/taylor_series.md

Lines changed: 23 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -1,3 +1,26 @@
1+
<script>
2+
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
3+
</script>
4+
$$
5+
\newcommand{\d}{\mathrm{d}}
6+
\newcommand{\bff}{\boldsymbol{f}}
7+
\newcommand{\bfg}{\boldsymbol{g}}
8+
\newcommand{\bfp}{\boldsymbol{p}}
9+
\newcommand{\bfq}{\boldsymbol{q}}
10+
\newcommand{\bfx}{\boldsymbol{x}}
11+
\newcommand{\bfu}{\boldsymbol{u}}
12+
\newcommand{\bfv}{\boldsymbol{v}}
13+
\newcommand{\bfA}{\boldsymbol{A}}
14+
\newcommand{\bfB}{\boldsymbol{B}}
15+
\newcommand{\bfC}{\boldsymbol{C}}
16+
\newcommand{\bfM}{\boldsymbol{M}}
17+
\newcommand{\bfJ}{\boldsymbol{J}}
18+
\newcommand{\bfR}{\boldsymbol{R}}
19+
\newcommand{\bfT}{\boldsymbol{T}}
20+
\newcommand{\bfomega}{\boldsymbol{\omega}}
21+
\newcommand{\bftau}{\boldsymbol{\tau}}
22+
$$
23+
124
# Taylor Series Expansion
225

326
I have been formally trained as a physicist. In my mind, there are several mathematical topics that blur the boundary between mathematics and physics. Taylor Series Expansions are one of those topics.

0 commit comments

Comments
 (0)