Matemàtiques / Mathematics
Índex
Mathematics Stack
Exchange
MathWorld
(Mathematica - Wolfram Research)
PlanetMath
The Geometry Center
Basic Research
Institute in the Mathematical Sciences
Eric Weisstein's World
of Mathematics
3.14159265358979323846264338327950288419716939937510582097.org
HyperStat
Online
The MacTutor
History of Mathematics archive
Vídeos
Nature
by
numbers (video) (Eterea Estudios)
Fibonacci
and
the Golden Mean
Dimensions
3Blue1Brown
T1.
Essence of linear algebra
T1.E1. Vectors, què són?
T1.E2. Linear combinations, span, and basis vectors
T1.E3. Linear transformations and matrices
T1.E4. Matrix multiplication as composition
T1.E5. Three-dimensional linear transformation
T1.E6. The determinant
T1.E7. Inverse matrices, column space and null space
T1.E8. Inverse matrices as transformations between
dimensions
T1.E9. Dot products and duality
T1.E10. Cross products
T1.E11. Cross products in the light of linear
transformations
T1.E12. Cramer's rule, explained geometrically
T1.E13. Change of basis
T1.E14. Eigenvectors and eigenvalues
T1.E15. A quick trick for computing eigenvalues
T1.E16. Abstract vector spaces
T2.
Esence of calculus
T2.E1.
The essence of calculus
T2.E2. The paradox of derivative
T2.E3. Derivative formulas through geometry
T2.E4. Visualizing the chain rule and product rule
T2.E5. What's so special about Euler's number e?
T2.E6. Implicit differentiation, what's going on here?
T2.E7. Limits, L'Hôpital's rule, and epsilon delta
definitions
T2.E8. Integration and the fundamental theorem of
calculus
T2.E9. What does area have to do with slope?
T2.E10. Higher order derivatives
T2.E11. Taylor series
T2.E12. The other way to visualize derivatives
T3.
Neural networks
T4.
Differential equations
T4.E1. Differential equations, a tourist's guide
T4.E2. But what is a partial differential equation?
T4.E3. Solving the heat equation
T4.E4. But what is a Fourier series? From heat flow to
drawing with circles
T4.E5. e^(iπ) in 3.14 minutes, using dynamics
T4.E6. How (and why) to raise e to a power of a matrix
Divergence
and curl: the language of Maxwell's equations, fluid
flow, and more
Zach Star
Parth G
Ron
Knott's web pages on Mathematics
The golden
proportion
Monotonic
functions (Connexions)
Publicacions / Publications
Demostracions / Demonstrations
...
Debian
science-mathematics
Open
Source
Mathematical Software: A White Paper
GeoGebra
KDE edu
SageMath
MathML
Mathematica (Wolfram )
Matlab (Mathworks )
SOCR Statistics
Online Computational Resource
Mathtools
Matlab
Central
File Exchange
Octave
Gràfiques / Plots
Wikipedia:How
to
create charts for Wikipedia articles
Geogebra
Octave
Wolfram Alpha
Fooplot (online: 2D, 3D)
(export: svg, pdf, eps...)
Online
3-D
function grapher
kmplot (KDE)
kAlgebra (KDE )
(2D, 3D)
Online
3-D
function grapher
Gnuplot
Documentation
(4.2
HTML (old))
I Gnuplot
II Plotting styles
general style: set style data
{lines,dots,boxes}
(same as with
...
inside plot)
bars:
set boxwidth 0.9 relative
set style fill solid 1.0
plot 'file.dat' with boxes
III Commands
plot
data
column 0 is index
# data.txt: x y
plot 'data.txt' using 1:2 with lines
# data.txt: x
plot 'data.txt' using 1:(log10($1))
with dots
Can
gnuplot compute and plot the delta
between consecutive data points
delta_v(x) = ( vD = x -
old_v, old_v = x, vD)
old_v = 0
set title "Compute Deltas"
set boxwidth 1
set style fill solid 1.0
set style data boxes
plot 'data.dat' using
1:(delta_v($1)) title 'Delta'
splot
IV Terminal types
svg
Wxt (default)
zoom
u: restore zoom
drag right button and click: delineate
region
ctrl+wheel: zoom in, zoom out
p,n: navigate to previous/next zoom
history
scroll
wheel: scroll up/down
shift+wheel: scroll left/right
V Bugs
Gnuplotting
Gnuplot
tips
(not so Frequently Asked Questions)
Xgnuplot
Examples
Demo
scripts
ffprobe
YUV
DSO
CSV files
atop
Can
gnuplot
split data? (sample style)
plot
splot
(2-d projection of 3-d surfaces)
Isometric
view
is distorted
Real
perspective
to splot
gnuplot
patch to add POV-Ray and VRML terminals
Parametric functions
exemple 1
dades.txt
grafica.plot
## some
variables
kzoom=1.2
phi=30.
theta=60.
## Ranges example (10 ist the default
range for x,y here):
set xrange [-10:10]
set yrange [-10:10]
set zrange [0:10]
## the relevant gnuplot commands
set xyplane 0 # removes the offset of
the xy plane
set view equal xyz # force equal units
to all three axes
set view theta,phi,kzoom
set iso 100, 100
splot 'dades.dat' using 1:2:3 with
impulses, "" using 1:2:3 with points
exemple 2: gràfica 3D
## some variables
kzoom=1.2
phi=30.
theta=60.
## Ranges example (10 ist the default
range for x,y here):
set xrange [0:1]
set yrange [0:1]
set zrange [0:1]
# etiquetes
set xlabel "X"
set ylabel "Y"
set zlabel "f(x,y)"
## the relevant gnuplot commands
set xyplane 0 # removes the offset of the
xy plane
#set view equal xyz # force equal units to
all three axes
set view theta,phi,kzoom
# nombre de divisions
set iso 50, 50
splot 1.5*(x*x+y*y), (3*x*x+1)/2.0
gnuplot -p < grafica.plot
Visualització / Visualisation
Rlab
DrGeocaml (dynamic
geometry software)
Mathomatic
The three central problems of calculus
The forward problem: given a curve, fins its slope
everywhere
The backward problem: given a curve's slope everywhere, find
the curve
The area problem: given a curve, find the area under it
\sqrt[n]{x} = \cos{\left( k \frac{2\pi}{n} \right)} + i
\sin{\left( k \frac{2\pi}{n} \right)}
x=2^n
x=4
n=2
\sqrt[2]{4}
\sqrt[2]{4}
n=3
\sqrt[3]{8}
Potències
de les arrels cúbiques de 8 (svg)
\sqrt[3]{4}
n=4
\sqrt[4]{16}
\sqrt[4]{4}
n=5
\sqrt[5]{32}
\sqrt[5]{4}
...
n \to \infty
\lim_{n \to \infty} \sqrt[n]{x}
= 1
\sqrt[\infty]{4}
...
exp
log
demostració
aplicació
multiplicació
a^b a^c = a^{b+c}
\log_a(m·n) = \log_a(m) + \log_a(n)
m = a^b \log_a(m)
= b n = a^c \log_a(n) = c \log_a(m·n)
= \log_a(a^b·a^c) = \log_a(a^{b+c}) = b + c = \log_a(m) +
\log_a(n)
exponenciació
(a^b)^c = a^{bc}
\log_a(m^n) = n·\log_a(m)
\left(a^{\frac{1}{n}}\right)^n =
a^{\frac{1}{n}n} = a ^1 = a \;\Rightarrow\; \boxed{
a^{\frac{1}{n}} = \sqrt[n]{a} }
...
limit
E_n(x) =
\left(1+\frac{x}{n}\right)^n
\mathrm e^x = \lim_{n \to
\infty} E_n(x) = \lim_{n \to \infty}
\left(1+\frac{x}{n}\right)^n
\ln{(x)} = \ln{(x)} + i 2\pik
\quad \Rightarrow \quad \boxed{\mathrm e^x = \mathrm e^x
\mathrm e^{i2\pi k}}
\begin{aligned} E_n(y) &=
\left( 1 + \frac{y}{n}\right)^n = x \\ L_n(x) &= y = n
\left( \sqrt[n]{x} - 1 \right) \end{aligned}
\ln{(x)} = \lim_{n \to \infty}
L_n(x) = \lim_{n \to \infty} n \left( \sqrt[n]{x} - 1
\right)
\left. \begin{array}{r} \ln{(x)}
= \lim_{n \to \infty} n \left( \sqrt[n]{x} - 1 \right) \\
\href{#arrels}{\sqrt[n]{x}} = \sqrt[n]{x} \; \cos{\left(k
\frac{2\pi}{n}\right)} + i \sqrt[n]{x} \; \sin{\left(k
\frac{2\pi}{n}\right)} \end{array} \right\} \quad \ln{(x)}
= \lim_{n \to \infty} n \left[ \sqrt[n]{x} \cos{\left(k
\frac{2\pi}{n}\right)} - 1 + i \sqrt[n]{x} \; \sin{\left(k
\frac{2\pi}{n}\right)} \right] = \left\{ \begin{array}{l}
\left. \begin{array}{r} \Re : \quad \lim_{n \to \infty} n
\left( \sqrt[n]{x} \cos{\left( \frac{2\pi k}{n} \right)} -
1\right) \\ \lim_{n \to \infty} \cos{\left( \frac{2\pi
k}{n} \right)} = 1 \end{array} \right\} = \lim_{n \to
\infty} n \left( \sqrt[n]{x} - 1\right) \triangleq
\ln{(x)} \\ \left. \begin{array}{r} \Im : \quad \lim_{n
\to \infty} n \sqrt[n]{x} \sin{\left( \frac{2\pi k}{n}
\right)} \\ \lim_{n \to \infty} \sqrt[n]{x} = 1
\end{array} \right\} = \lim_{n \to \infty} \frac{\sin{
\left(\frac{2\pi k}{n}\right) }}{\frac{1}{n}}
\stackrel{\text{(H)}}{=} \lim_{n \to \infty} \frac{
\cos{\left( \frac{2\pik}{n} \right)}
\left(\frac{-2\pik}{n^2}\right) }{-\frac{1}{n^2}} = 2\pik
\lim_{n \to \infty} \cos{\left( \frac{2\pik}{n} \right)} =
2\pik \end{array} \right\} \Rightarrow \boxed{ \ln{(x)} =
\ln{(x)} + i 2\pik }
Teorema del sinus (wp )
\frac{a}{\sin
\hat{A}}=\frac{b}{\sin \hat{B}}=\frac{c}{\sin \hat{C}}
Teorema del cosinus (wp )
a^2 = b^2 + c^2 - 2bc\, \cos
\hat{A} b^2 = a^2 + c^2 -
2ac\, \cos \hat{B} c^2 =
a^2 + b^2 - 2ab\, \cos \hat{C}
Derivades de funcions elementals:
funció f(x)
derivada f'(x)
demostració
k
0
kx
k
x^{n}
nx^{n-1}
a^x
a^x · \ln{a}
e^x
e^x
\log_{a}x
\frac{1}{x · \ln{a}}
\ln{x}
\frac{1}{x}
\sin{x}
\cos{x}
\cos{x}
-\sin{x}
\tan{x}
\frac{1}{\cos^2{x}}
\arcsin{x}
\frac{1}{\sqrt{1-x^2}}
\arccos{x}
\frac{-1}{\sqrt{1-x^2}}
\arctan{x}
\frac{1}{1+x^2}
Propietats:
derivada
suma
f(x) + g(x)
f'(x) + g'(x)
producte
f(x) · g(x)
f'(x)·g(x) + f(x)·g'(x)
divisió
\frac{f(x)}{g(x)}
\frac{f'(x)·g(x) -
f(x)·g'(x)}{g^2(x)}
regla de la cadena
f(x) \circ g(x) = f(g(x))
f'(g(x)) · g'(x)
...
...
Càlcul vectorial (wp )
Divergence
and curl: the language of Maxwell's equations, fluid flow, and
more (3blue1brown )
des de:
operador
nabla : \nabla = \left(
\frac{\partial}{\partial x}, \frac{\partial}{\partial
y}, \frac{\partial}{\partial z} \right)
a:
expressa:
exemples
camp escalar
gradient :
\nabla f = \left( \frac{\partial
f}{\partial x}, \frac{\partial f}{\partial y},
\frac{\partial f}{\partial z} \right)
camp vectorial
direcció i magnitud de la màxima variació en un punt
camp vectorial
divergència / divergence: \nabla \cdot \mathbf{F} =
\frac{\partial F_x}{\partial x} + \frac{\partial
F_y}{\partial y} + \frac{\partial F_z}{\partial z}
camp escalar
flux net d'entrada (<0) i sortida (>0) donada
una regió
Maxwell :
\nabla \cdot \mathbf{E} =
\frac{\rho}{\epsilon_0} \nabla
\cdot \mathbf{B} = 0
camp vectorial
rotacional
/ curl:\nabla \times \mathbf{F} =
\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\
\frac{\partial}{\partial x} &
\frac{\partial}{\partial y} &
\frac{\partial}{\partial z} \\ F_x & F_y & F_z
\end{vmatrix}
camp vectorial
efecte de rotació en un punt (com si hi hagués un
molinet), provocat pel camp vectorial
Maxwell :
\nabla \times \mathbf{E} = -
\frac{\partial \mathbf{B}}{\partial t} \nabla \times \mathbf{B} = \mu_0
\left(\mathbf{J} + \epsilon_0 \frac{\partial
\mathbf{E}}{\partial t} \right)
operador
laplacià (divergència del gradient): \Delta
= \nabla \cdot \nabla
camp escalar
\Delta f = (\nabla \cdot
\nabla) f = \nabla^2 f = \frac{\partial^2 f}{\partial
x^2} + \frac{\partial^2 f}{\partial y^2} +
\frac{\partial^2 f}{\partial x^2}
...
...
Latex
Info
continu
Laplace :\mathcal{L}\{f\}(s)
= \int_{0}^{\infty} f(t) e^{-st} dt on:
s = \sigma + i \omega
Two-sided
Laplace (Fourier transform of: f(t)
e^{-\alpha t} ):\mathcal{L}\{f\}(s)
= \int_{-\infty}^{\infty} f(t) e^{-st} dt =
\int_{-\infty}^{\infty} f(t) e^{-\alpha t} e^{-j
\omega t} dt s =
\alpha + j \omega
Fourier
(Laplace
transform at plane: \sigma = 0 ): \mathcal{L}\{f\}(s=j\omega) =
\mathcal{F}\{f\} = \int_{-\infty}^{\infty} f(t)
e^{-j\omegat} dt= \int_{-\infty}^{\infty} f(t)
e^{-j2\pi Ft} dt
discret
Z-Transform :\mathcal{Z}\{f\} =
interpretació geomètrica
càlcul matemàtic
càlcul computacional
info
matriu que multiplica un vector
transformació lineal del vector cap a un nou vector
producte de matrius
composició de transformacions
determinant d'una matriu
com queda afectada l'àrea (2D) o el volum (3D) un cop
aplicada la transformació
rang d'una matriu
dimensió de l'espai de sortida
...
Producte:
resultat
exemples i aplicacions
producte d'un vector per un escalar
k \vec{A}
un vector amb la mateixa direcció i mòdul multiplicat
per k
producte escalar / dot
product
\vec{A}\cdot\vec{B} = |\vec{A}|
|\vec{B}| \cos{\theta} = a_x b_x + a_y b_y
un escalar:
si és 0: els vectors són perpendiculars
si és 1: p.ex. el producte escalar d'un vector
unitari per ell mateix
angle que formen dos vectors:\cos{\theta}
= \frac{a_x b_x + a_y b_y}{|\vec{A}||\vec{B}|}
treball :
W = \vec{F} \cdot \Delta \vec{r} =
\int_{\vec{r_a}}^{\vec{r_b}} \vec{F} \cdot
d\vec{r}
flux
d'un camp magnètic
producte vectorial / cross
product
\vec{A}\times\vec{B} =
\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\
A_x & A_y & A_z \\ B_x & B_y & B_z
\end{vmatrix}
un altre vector, ortogonal a tots dos vectors
direcció segons la regla de la mà dreta, amb els
dits des d'A cap a B
mòdul (àrea del paral·lelogram)
|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}|
\sin{\alpha}
http://www.francescpinyol.cat/matematiques.html
Primera versió: / First version: 27.XI.2021
Darrera modificació: 15 d'agost de 2023 / Last update: 15th August
2023
Cap a casa / Back home .