"
493 | ]
494 | },
495 | "execution_count": 17,
496 | "metadata": {},
497 | "output_type": "execute_result"
498 | }
499 | ],
500 | "source": [
501 | "from IPython.core.display import HTML\n",
502 | "def css_styling():\n",
503 | " styles = open('./styles/custom_barba.css', 'r').read()\n",
504 | " return HTML(styles)\n",
505 | "css_styling()"
506 | ]
507 | },
508 | {
509 | "cell_type": "code",
510 | "execution_count": null,
511 | "metadata": {},
512 | "outputs": [],
513 | "source": []
514 | }
515 | ],
516 | "metadata": {
517 | "kernelspec": {
518 | "display_name": "Python 3",
519 | "language": "python",
520 | "name": "python3"
521 | },
522 | "language_info": {
523 | "codemirror_mode": {
524 | "name": "ipython",
525 | "version": 3
526 | },
527 | "file_extension": ".py",
528 | "mimetype": "text/x-python",
529 | "name": "python",
530 | "nbconvert_exporter": "python",
531 | "pygments_lexer": "ipython3",
532 | "version": "3.6.7"
533 | },
534 | "varInspector": {
535 | "cols": {
536 | "lenName": 16,
537 | "lenType": 16,
538 | "lenVar": 40
539 | },
540 | "kernels_config": {
541 | "python": {
542 | "delete_cmd_postfix": "",
543 | "delete_cmd_prefix": "del ",
544 | "library": "var_list.py",
545 | "varRefreshCmd": "print(var_dic_list())"
546 | },
547 | "r": {
548 | "delete_cmd_postfix": ") ",
549 | "delete_cmd_prefix": "rm(",
550 | "library": "var_list.r",
551 | "varRefreshCmd": "cat(var_dic_list()) "
552 | }
553 | },
554 | "types_to_exclude": [
555 | "module",
556 | "function",
557 | "builtin_function_or_method",
558 | "instance",
559 | "_Feature"
560 | ],
561 | "window_display": false
562 | }
563 | },
564 | "nbformat": 4,
565 | "nbformat_minor": 4
566 | }
567 |
--------------------------------------------------------------------------------
/program.md:
--------------------------------------------------------------------------------
1 | # Advanced mathematics for engineers
2 |
3 | ## Description
4 |
5 | This is a course of mathematical methods for beginning graduate and senior
6 | engineering students. The topics presented in this course pretend to be
7 | and appetizer for the student, allowing her to self-study engineering
8 | systems that involve mathematical models not covered in basic engineering
9 | subjects.
10 |
11 |
12 | ## Objectives
13 |
14 | At the end of the course the student should be able to use mathematical
15 | models to solve engineering problems. In particular, the students
16 | should
17 |
18 | - Understand different vector operators in several coordinate
19 | systems.
20 |
21 | - Solve second order ordinary differential equations.
22 |
23 | - Apply mathematical methods to solve important boundary value problems:
24 | Laplace, Poisson, Wave and Heat equations.
25 |
26 | - Identify types of equations and decide the method to solve it.
27 |
28 | - Identify the origin of some special functions and understand some of
29 | their property.
30 |
31 |
32 | ## Methodology
33 |
34 | Lectures, Examples, recommended reading. The course is divided in 5 units,
35 | emphasizing the concepts. For this reason is suggested the use of a [Computer
36 | Algebra System (CAS)](https://en.wikipedia.org/wiki/Computer_algebra_system)
37 | such as Maple, [Maxima](http://maxima.sourceforge.net/) or
38 | [SymPy](http://www.sympy.org/en/index.html). The instructor will show examples
39 | using SymPy. There will be assignments for each unit that will help to
40 | reinforce the understanding of the topics.
41 |
42 | ## Contents
43 |
44 | ### Linear Algebra Review (1 week)
45 |
46 | It is suggested to watch the series of videos
47 | ["Essence of Linear Algebra"](http://www.3blue1brown.com/essence-of-linear-algebra/).
48 |
49 | 1. Linear transformations
50 |
51 | 2. Vector spaces and bases
52 |
53 | 3. Eigenvalue and Eigenvector problems
54 |
55 | ### Vector calculus and coordinates (2 weeks)
56 |
57 | 1. Vectors and tensors
58 |
59 | 2. Coordinate Systems
60 |
61 | 3. Line, surface and volume differentials
62 |
63 | 4. Differential operators
64 |
65 | 5. Vector identities
66 |
67 | 6. Integral theorems
68 |
69 | ### Ordinary differential equations (5 weeks)
70 |
71 | 1. First order differential equations
72 |
73 | 2. Systems of differential equations
74 |
75 | 3. Power series solutions
76 |
77 | 4. Frobenius method
78 |
79 | 5. Laplace transform method
80 |
81 | 6. Qualitative methods for non-linear systems
82 |
83 | ### Orthogonal bases and Fourier analysis (3 weeks)
84 |
85 | 1. Discrete bases
86 |
87 | 2. Continuous bases
88 |
89 | 3. Fourier Series
90 |
91 | 4. Fourier Integrals
92 |
93 | ### Partial differential equations (5 weeks)
94 |
95 | 1. Classification of partial differential equations
96 |
97 | 2. Common equations
98 |
99 | 1. Poisson equation
100 |
101 | 2. Diffusion equation
102 |
103 | 3. Wave equation
104 |
105 | 4. Navier-Cauchy equation
106 |
107 | 3. Separation of variables
108 |
109 | 1. Sturm-Liouville problems
110 | 2. Bessel functions
111 |
112 | 4. Ritz method
113 |
114 | 5. Weighted residual methods
115 |
116 | ## Textbook
117 |
118 | The textbookd for the course are “Física Matemática” by Alonso Sepúlveda, and
119 | “Advanced Engineering Mathematics” by Erwin Kreyszig.
120 |
121 |
122 | ## Evaluation
123 |
124 | - Assignments 30%
125 |
126 | - 2 Midterms 40%
127 |
128 | - Final project 30%
129 |
130 | ## Pre-requisites
131 |
132 | - Linear Algebra.
133 |
134 | - Differential equations.
135 |
136 | - Vector calculus.
137 |
138 | ## References
139 |
140 | 1. SymPy Development Team. [Sympy’s documentation.](http://docs.sympy.org/latest/index.html), 2016.
141 |
142 | 2. Grant Sanderson. [Essence of linear algebra.](http://www.3blue1brown.com/essence-of-linear-algebra/), 2016.
143 |
144 | 3. Erwin Kreyszig. Advanced engineering mathematics. John Wiley & Sons, 2010.
145 |
146 | 4. Antonio Velasco and Ruben Sánchez. Curso Básico de Álgebra Lineal (Spanish).
147 | Comex, 1980.
148 |
149 | 5. Alonso Sepulveda Soto. Fı́sica matemática (Spanish). Ciencia y Tecnologı́a.
150 | Universidad de Antioquia, 2009.
151 |
152 | 6. FWJ Olver, DW Lozier, RF Boisvert, and CW Clark. [NIST digital library of mathematical functions.](http://dlmf.nist.gov). NIST, 2010.
153 |
154 | 7. Louis Leithold. The calculus. New York, USA: Harper and Row Publishers,
155 | 7 edition, 1995.
156 |
157 | 8. H. Hochstadt. Differential equations: a modern approach. Courier Dover
158 | Publications, 1975.
159 |
160 | 9. Stanley J Farlow. Partial differential equations for scientists and
161 | engineers. Courier Corporation, 2012.
162 |
--------------------------------------------------------------------------------
/references.md:
--------------------------------------------------------------------------------
1 | # References
2 |
3 | This is a list of references related to Advanced Mathematics,
4 | [Mathematical Physics](https://en.wikipedia.org/wiki/Mathematical_physics),
5 | and other topics of interest.
6 |
7 |
8 | - Alonso Sepulveda Soto. Fı́sica matemática (Spanish). Ciencia y Tecnologı́a.
9 | Universidad de Antioquia, 2009.
10 |
11 | This is a nice little book about Mathematical Physics. I think that it covers
12 | most of the relevant topics and it is short enough. The only caveat is the
13 | jargon, that it might be a little bit too specific for physicists.
14 |
15 | - Erwin Kreyszig. Advanced engineering mathematics. John Wiley & Sons, 2011.
16 |
17 | A good book in Advanced Mathematics (for engineers). The only but is its
18 | extension, that would not make it suitable for a one-semester course. But,
19 | definitely a good book to have in one's bookshelf.
20 |
21 | - Stanley J Farlow. Partial differential equations for scientists and
22 | engineers. Courier Corporation, 2012.
23 |
24 | This is a [PDE](https://en.wikipedia.org/wiki/Partial_differential_equation)
25 | book intended for students in areas other than mathematics who are studying
26 | partial differential equations. It presents the content in 47 _independent_
27 | lessons instead of presenting it by chapters.
28 |
29 | - H. Hochstadt. Differential equations: a modern approach. Courier Dover
30 | Publications, 1975.
31 |
32 | My favorite book on [ODE](https://en.wikipedia.org/wiki/Ordinary_differential_equation),
33 | it does not describe all the common methods for second order equations as is
34 | common in most ODE books. The emphasis is on concepts and in matrix methods
35 | that are more algorithmic, in my opinion.
36 |
37 | - Antonio Velasco and Ruben Sánchez. Curso Básico de Álgebra Lineal (Spanish).
38 | Comex, 1980.
39 |
40 | Yet another little book that I like. It does not give any special treatment
41 | to the topics of linear algebra, but it is short (208 page), so you can read
42 | it pretty fast. If I had to choose a book with a different approach it would
43 | be [_Coding the Matrix_](http://codingthematrix.com/) by Philip N. Klein,
44 | that gives an intertwined presentation between theory, concepts and
45 | (Python) programming.
46 |
47 | - Louis Leithold. The calculus. New York, USA: Harper and Row Publishers,
48 | 7 edition, 1995.
49 |
50 | I like this calculus book, but it is probably because I studied in my
51 | undergrad with it. Regarding vector calculus, I find more useful the book
52 | by Stewart.
53 |
54 |
55 | ## Freely-available references
56 |
57 | - Grant Sanderson. [Essence of linear algebra.](http://www.3blue1brown.com/essence-of-linear-algebra/), 2016.
58 |
59 | A series of videos that clearly explain the concepts behind the most Common
60 | topics in linear algebra.
61 |
62 | - Lloyd Trefethen and Kristine Embree (Editors). [The (Unfinished) PDE Coffee Table Book.](https://people.maths.ox.ac.uk/trefethen/pdectb.html), 2011.
63 |
64 | This is a collection of 2-pages spreads talking about relevant information
65 | for different partial differential equations.
66 |
67 | - FWJ Olver, DW Lozier, RF Boisvert, and CW Clark. [NIST digital library of mathematical functions.](http://dlmf.nist.gov). NIST, 2010.
68 |
69 | The updated version of the classical _Handbook of Mathematical Functions_.
70 | It is an online version, so you can access it wherever you are (with an
71 | internet connection, of course).
72 |
73 | - Ondřej Čertík. [Theoretical Physics Reference](http://www.theoretical-physics.net/dev/index.html), 2011.
74 |
75 | This is an e-book generated using [Sphinx](http://www.sphinx-doc.org/en/stable/)
76 | with source code stored in [GitHub](https://github.com/certik/theoretical-physics).
77 | The book contains notes related to theoretical and mathematical physics and
78 | snippets of code in SymPy.
79 |
80 | - Hans Petter Langtangen, Svein Linge ["Finite Difference Computing with PDEs."](https://link.springer.com/book/10.1007/978-3-319-55456-3),
81 | Springer, 2017.
82 |
83 | This is a book on Finite Difference methods for PDEs. It has a stronger
84 | emphasis on computer implementation and verification, key aspects of
85 | scientific computing.
86 |
87 | - Hans Petter Langtangen, Geir K. Pedersen ["Scaling of Differential Equations."](https://link.springer.com/book/10.1007/978-3-319-32726-6),
88 | Springer, 2017.
89 |
90 | This book is centered on scaling of differential equations. Rewriting
91 | differential equations in dimensionless form has several advantages.
92 | These advantages are also present in numerical solutions.
93 |
--------------------------------------------------------------------------------
/slides/course_presentation.html:
--------------------------------------------------------------------------------
1 |
2 |
4 |
5 |
6 | Advanced Mathematics for Engineers
13 |
19 |
28 |
Advanced Mathematics for Engineers
29 |
30 | Nicolás Guarín-Zapata
32 |
July 2019
33 |
35 |
Objectives
36 |
At the end of the course the student should be able to use mathematical methods to solve engineering problems. Particularly, the student should
37 |
38 | Understand different vector operators in several coordinate systems.
39 | Solve ordinary differential equation of second order.
40 | Apply mathematical methods to solve common boundary value problems: Laplace, Poisson, Wave and Heat equations.
41 | Identify types of equations and decide the method to solve it.
42 | Identify the origin of some special functions and understand some of their properties.
43 |
44 |
46 |
Evaluation
47 |
48 | Homework 30%
49 | 2 Midterms 40%
50 | Project 30%
51 |
52 |
54 |
Evaluation: Description
55 |
56 | First midterm: Written examination (Week 8)
57 | Second midterm: Written examination (Week 16)
58 | Final exam: “Small” project. Final report in paper format.
59 |
60 | Topic selection: Week 4
61 | Partial report: Week 10
62 | Deadline: Week 17
63 |
64 | Homework is due on the following weeks
65 |
66 | Week 2
67 | Week 5
68 | Week 9
69 | Week 12
70 | Week 16
71 |
72 |
73 |
75 |
Textbook
76 |
The main books for the course are “Física Matemática” by Alonso Sepúlveda and “Advanced Engineering Mathematics” by Kreyszig (there are several copies in the library).
77 |
79 |
GitHub repository
80 |
I will create contents such as Jupyter Notebooks and Slides (and maybe some other types) regularly into the following repository
81 |
https://github.com/nicoguaro/AdvancedMath
82 |
84 |
Python Distribution
85 |
I recommend to use Anaconda as Python distribution, it includes SymPy and Jupyter already.
86 |
A good Python tutorial for scientific purpose is
87 |
90 |
And I wrote a short tutorial for SymPy. You can find it in the course GitHub’s repository, or follow this link .
91 |
93 |
References
94 |
Main books:
95 |
96 | Alonso Sepúlveda Soto. Física matemática. Ciencia y Tecnología. Universidad de Antioquia, 2009.
97 | Erwin Kreyszig. Advanced engineering mathematics. John Wiley & Sons, 2010.
98 |
99 |
For further reference
100 |
101 | SymPy Development Team. Sympy’s documentation. , 2016.
102 | Grant Sanderson. Essence of linear algebra. , 2016.
103 | Antonio Velasco and Ruben Sánchez. Curso Básico de Álgebra Lineal. Comex, 1980.
104 | FWJ Olver, DW Lozier, RF Boisvert, and CW Clark. NIST digital library of mathematical functions. NIST, 2010.
105 | Louis Leithold. The calculus. New York, USA: Harper and Row Publishers, 7 edition, 1995.
106 | H. Hochstadt. Differential equations: a modern approach. Courier Dover Publications, 1975.
107 | Stanley J Farlow. Partial differential equations for scientists and engineers. Courier Corporation, 2012.
108 |
109 |
Orthogonal bases and Fourier analysis
13 |
19 |
29 |
Orthogonal bases and Fourier analysis
30 |
31 | Nicolás Guarín-Zapata
33 |
March, 2019
34 |
36 |
Inner products
37 |
Inner products let us extend geometrical notions such as length of a vector or angle between vectors for vector spaces that are more abstract than \(\mathbb{R}^2\) or \(\mathbb{R}^3\) . It also let us define the orthogonality between vectors. Inner product spaces generalize the notion of Euclidean spaces to any dimension.
38 |
40 |
Orthogonal basis
41 |
An orthogonal basis for an inner product spaces \(V\) , is a basis for \(V\) whose vectors are mutually orthogonal. The angle between vectors (\(\theta\) ) is defined using the inner product as
42 |
\[\theta = \arccos\left(\frac{\langle x, y\rangle}{\Vert x\Vert\, \Vert y\Vert}\right) \, .\]
43 |
If they have magnitude 1, then the base is called orthonormal .
44 |
46 |
Examples of (discrete) orthogonal basis: Fourier basis
47 |
\[\left\lbrace \frac{1}{\sqrt{\pi}} \sin(nx),
48 | \frac{1}{\sqrt{\pi}} \cos(nx),
49 | \frac{1}{\sqrt{2\pi}}\middle|
50 | \forall n \in \mathbb{N}, \forall x\in [-\pi, \pi]\right\rbrace\]
51 |
54 |
56 |
Examples of (discrete) orthogonal basis: Hermite polynomials
57 |
\[\left\lbrace (-1)^n e^{x^2}\frac{d^n}{dx^n} e^{-x^2},\middle|
58 | \forall n \in \mathbb{N}, \forall x\in [-\infty, \infty]\right\rbrace\]
59 |
with orthogonality as
60 |
\[\int\limits_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} dx = \sqrt{\pi}s^n n! \delta_{mn}\]
61 |
64 |
66 |
Examples of (discrete) orthogonal basis: Chebyshev polynomials
67 |
They are defined by the recursion relation
68 |
\[T_0(x) = 1,\, T_1(x) = x,\, T_{n+1} = 2x T_n(x) - T_{n-1}(x),\quad \forall x\in [-1,1]\]
69 |
with orthogonality as
70 |
\[\int\limits_{-1}^{1} T_m(x) T_n(x) \frac{dx}{\sqrt{1 - x^2}} =
71 | \begin{cases}
72 | 0 &n\neq m \\
73 | \pi &n=m=0\\
74 | \pi/2 &n=m\neq 0\end{cases}\]
75 |
78 |
80 |
Fourier analysis: definition
81 |
From Wikipedia
82 |
83 | In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
84 |
85 |
87 |
Fourier analysis: scientific applications
88 |
Fourier analysis has many scientific applications:
89 |
90 | Signal Processing. It may be the best application of Fourier analysis.
Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial.
Control Theory. The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation. They are useful to find out the dynamics of the solution.
Partial Differential equation. We use it to solve higher order partial differential equations by the method of separation of variables.
95 |
97 |
Fourier analysis: applications
98 |
Some examples include:
99 |
105 |
107 |
Fourier series
108 |
A Fourier series allow us to represent a (periodic) function as the sum of sine and cosine functions.
109 |
For a function \(f(x)\) defined over \([x_0, x_0 + P]\) , that is continuous or piecewise continuous, we write
110 |
\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[
111 | a_n \cos\left(\frac{2\pi nx}{P}\right) +
112 | b_n \sin\left(\frac{2\pi nx}{P}\right) \right]\]
113 |
where the coefficients are obtained computing the inner product with the elements of the base, i.e.
114 |
\[a_0 = \frac{2}{P}\int_{x_0}^{x_0+P} f(x)\, dx\\
115 | a_n = \frac{2}{P}\int_{x_0}^{x_0+P} \cos\left(\frac{2\pi nx}{P}\right) f(x)\, dx\\
116 | b_n = \frac{2}{P}\int_{x_0}^{x_0+P} \sin\left(\frac{2\pi nx}{P}\right) f(x)\, dx\]
117 |
119 |
Fourier series visualisation
120 |
122 |
123 |
124 |
Orthogonal basis: continuum case
125 |
A set \(\lbrace \phi(k, x)\rbrace\) with \(x\) and \(k\) defined over \((a, b)\) , and \((c, d)\) are orthogonal with weight \(w(x)\) (\(w(x)\) real) if:
126 |
\[\int\limits_{a}^{b} w(x) \phi^* (k, x) \phi(k', x)\, dx = \delta(k - k')\, ,
127 | \quad x\in(a, b),\, k\in(c, d)\, .\]
128 |
130 |
Orthogonal basis: continuum case
131 |
If the basis is complete we can write a function \(f(x)\) as
132 |
\[f(x) = \int\limits_{c}^{d} C(k) \phi(k, x)\, dk\, ,\]
133 |
with
134 |
\[C(k) = \int\limits_{a}^{b} f(x) w(x)\phi(k, x)\, dx\, .\]
135 |
\(C(k)\) is known as the tranform of \(f(x)\) .
136 |
170 |
References
171 |
180 |
7 | Produced by GNUPLOT 4.2 patchlevel 2
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92 | 3
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101 | H_n (x)
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104 | x
105 |
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107 | Hermite (physicists') Polynomials
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116 | n = 0
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182 | n = 3
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219 | n = 5
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/slides/img/PenduloTmg.gif:
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/slides/img/airy_functions.py:
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1 | #!/usr/bin/env python2
2 | # -*- coding: utf-8 -*-
3 | """
4 | Plots of the Airy functions
5 | """
6 | from __future__ import division
7 | import numpy as np
8 | from scipy.special import airy
9 | import matplotlib.pyplot as plt
10 |
11 | plt.rcParams["mathtext.fontset"] = "cm"
12 |
13 |
14 | z = np.linspace(-10, 5, 500)
15 | Ai, _, Bi, _ = airy(z)
16 |
17 |
18 | plt.figure(figsize=(4, 3))
19 | plt.plot(z, Ai)
20 | plt.plot(z, Bi)
21 | plt.ylim(-0.5, 1)
22 | plt.xlabel(r"$z$")
23 | plt.ylabel(r"$y$")
24 | plt.legend([r"$\mathrm{Ai}(z)$", r"$\mathrm{Bi}(z)$"])
25 | plt.tight_layout()
26 | plt.savefig("airy_functions.svg", transparent=True)
27 |
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/slides/img/bessel_functions.py:
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1 | #!/usr/bin/env python2
2 | # -*- coding: utf-8 -*-
3 | """
4 | Plots of the Bessel functions of the first kind
5 | """
6 | from __future__ import division
7 | import numpy as np
8 | from scipy.special import jn
9 | import matplotlib.pyplot as plt
10 |
11 | plt.rcParams["mathtext.fontset"] = "cm"
12 |
13 |
14 | z = np.linspace(0, 10, 500)
15 |
16 | plt.figure(figsize=(4, 3))
17 | for nu in [0, 1, 2, 3]:
18 | J = jn(nu,z)
19 | plt.plot(z, J, label=r"$J_{}$".format(nu))
20 |
21 |
22 | plt.xlabel(r"$z$")
23 | plt.ylabel(r"$y$")
24 | plt.legend()
25 | plt.tight_layout()
26 | plt.savefig("bessel_functions.svg", transparent=True)
27 |
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/slides/img/drag_fall.mp4:
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https://raw.githubusercontent.com/nicoguaro/AdvancedMath/2749068de442f67b89d3f57827367193ce61a09c/slides/img/drag_fall.mp4
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/slides/img/drag_fall.py:
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1 | #!/usr/bin/env python2
2 | # -*- coding: utf-8 -*-
3 | """
4 | Animate the fall of a person with parachute
5 | """
6 | from __future__ import division, print_function
7 | import numpy as np
8 | from scipy.integrate import odeint
9 | import matplotlib.pyplot as plt
10 | import matplotlib.animation as animation
11 |
12 |
13 | plt.rcParams["font.family"] = "serif"
14 | plt.rcParams["font.size"] = 18
15 | plt.rcParams["mathtext.fontset"] = "cm"
16 |
17 |
18 | def drag_fall(x, t, gravity, drag, mass):
19 | y, v = x
20 | return [v, -gravity + drag/mass*v**2]
21 |
22 |
23 | def update(num, ax, t, y):
24 | ax.cla()
25 | plot = ax.plot(0, y[num], "ko")
26 | ax.set_title(r"$t={:.2f}$".format(t[num]))
27 | ax.set_xlim(-1, 1)
28 | ax.set_xticks([])
29 | return plot
30 |
31 |
32 | dist = 500
33 | y0 = [dist, 0]
34 | g = 9.81
35 | mass = 70
36 | drag = 1.2
37 | tmax = np.sqrt(2*dist/g)
38 | t = np.linspace(0, tmax, 101)
39 | sol = odeint(drag_fall, y0, t, args=(g, drag, mass))
40 |
41 | fig, (ax1, ax2) = plt.subplots(1, 2, sharey=True, **{"figsize": (8, 5)})
42 | ax1.plot(t, sol[:, 0])
43 | ax1.set_xlabel(r"Time (s)")
44 | ax1.set_ylabel(r"Height (m)")
45 |
46 | ani = animation.FuncAnimation(fig, update, fargs=(ax2, t, sol[:, 0]),
47 | interval=50, blit=True)
48 | ani.save("drag_fall.mp4", dpi=600)
49 | #plt.show()
50 |
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/slides/img/free_fall.mp4:
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https://raw.githubusercontent.com/nicoguaro/AdvancedMath/2749068de442f67b89d3f57827367193ce61a09c/slides/img/free_fall.mp4
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/slides/img/free_fall.py:
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1 | #!/usr/bin/env python2
2 | # -*- coding: utf-8 -*-
3 | """
4 | Animate the free fall of a body
5 | """
6 | from __future__ import division, print_function
7 | import numpy as np
8 | from scipy.integrate import odeint
9 | import matplotlib.pyplot as plt
10 | import matplotlib.animation as animation
11 |
12 |
13 | plt.rcParams["font.family"] = "serif"
14 | plt.rcParams["font.size"] = 18
15 | plt.rcParams["mathtext.fontset"] = "cm"
16 |
17 | def free_fall(x, t, gravity):
18 | y, v = x
19 | return [v, -gravity]
20 |
21 |
22 | def update(num, ax, t, y):
23 | ax.cla()
24 | plot = ax.plot(0, y[num], "ko")
25 | ax.set_title(r"$t={:.2f}$".format(t[num]))
26 | ax.set_xlim(-1, 1)
27 | ax.set_xticks([])
28 | return plot
29 |
30 |
31 | y0 = [10, 0]
32 | g = 9.81
33 | dist = 10
34 | tmax = np.sqrt(2*dist/g)
35 | t = np.linspace(0, tmax, 101)
36 | sol = odeint(free_fall, y0, t, args=(g,))
37 |
38 | fig, (ax1, ax2) = plt.subplots(1, 2, sharey=True, **{"figsize": (8, 5)})
39 | ax1.plot(t, sol[:, 0])
40 | ax1.set_xlabel(r"Time (s)")
41 | ax1.set_ylabel(r"Height (m)")
42 |
43 | ani = animation.FuncAnimation(fig, update, fargs=(ax2, t, sol[:, 0]),
44 | interval=50, blit=True)
45 | ani.save("free_fall.mp4", dpi=600)
46 | #plt.show()
47 |
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/slides/img/gamma_function.py:
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1 | #!/usr/bin/env python2
2 | # -*- coding: utf-8 -*-
3 | """
4 | Plot of the gamma function
5 | """
6 | from __future__ import division
7 | import numpy as np
8 | from scipy.special import gamma
9 | import matplotlib.pyplot as plt
10 |
11 | plt.rcParams["mathtext.fontset"] = "cm"
12 |
13 |
14 | z = np.linspace(-5, 5, 5000)
15 |
16 | plt.figure(figsize=(4, 3))
17 | y = gamma(z)
18 | y[np.abs(y)>5.1] = np.nan
19 | plt.plot(z, y)
20 |
21 |
22 | plt.xlabel(r"$z$")
23 | plt.ylabel(r"$y$")
24 | plt.ylim(-4, 4)
25 | plt.tight_layout()
26 | plt.savefig("gamma_function.svg", transparent=True)
27 |
28 |
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/slides/linear_transformations.html:
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1 |
2 |
4 |
5 |
6 | Linear Tranformations and Eigenvalues/Eigenvectors
13 |
19 |
29 |
Linear Tranformations and Eigenvalues/Eigenvectors
30 |
31 | Nicolás Guarín-Zapata
33 |
July 2019
34 |
45 |
Video lecture: Linear transformations
46 | VIDEO
48 |
49 |
50 |
Applet: Linear transformations
51 |
53 |
54 |
71 |
72 |
Eigenvalues and Eigenvectors
73 |
An eigenvector or characteristic vector of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it. More formally, if \(T\) is a linear transformation from a vector space \(V\) over a field \(F\) into itself and \(v\) is a vector in \(V\) that is not the zero vector, then \(v\) is an eigenvector of \(T\) if \(T(v)\) is a scalar multiple of \(v\) . This condition can be written as the equation
74 |
\[T ( v ) = λ v ,\]
75 |
where \(λ\) is a scalar in the field \(F\) , known as the eigenvalue, characteristic value, or characteristic root associated with the eigenvector \(v\) .
76 |
78 |
79 |
82 |
In this shear mapping the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it doesn’t change direction, and since its length is unchanged, its eigenvalue is 1.
83 |
85 |
References
86 |
91 |
VIDEO
29 |
30 | ------------------
31 |
32 | # Applet: Linear transformations
33 |
34 |
40 |
41 | ------------------
42 |
43 | # Examples of linear transformation matrices
44 |
45 | In two-dimensional space $\mathbb{R}^2$ linear maps are described by 2 × 2 real matrices. These are some examples:
46 |
47 | - rotation by 90 degrees counterclockwise:
48 |
49 | $$\mathbf{A}=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$$
50 |
51 | - rotation by angle ''θ'' counterclockwise: $$\mathbf{A}=\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
52 | - reflection against the ''x'' axis: $$\mathbf{A}=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$$
53 | - reflection against the ''y'' axis: $$\mathbf{A}=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}$$
54 | - scaling by 2 in all directions: $$\mathbf{A}=\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}$$
55 | - horizontal shear mapping: $$\mathbf{A}=\begin{pmatrix}1 & m\\ 0 & 1\end{pmatrix}$$
56 | - squeeze mapping: $$\mathbf{A}=\begin{pmatrix}k & 0\\ 0 & 1/k\end{pmatrix}$$
57 | - projection onto the ''y'' axis: $$\mathbf{A}=\begin{pmatrix}0 & 0\\ 0 & 1\end{pmatrix}.$$
58 |
59 | ------------------
60 |
61 | # Eigenvalues and Eigenvectors
62 |
63 | An eigenvector or characteristic vector of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it. More formally, if $T$ is a linear transformation from a vector space $V$ over a field $F$ into itself and $v$ is a vector in $V$ that is not the zero vector, then $v$ is an eigenvector of $T$ if $T(v)$ is a scalar multiple of $v$. This condition can be written as the equation
64 |
65 | $$T ( v ) = λ v ,$$
66 |
67 | where $λ$ is a scalar in the field $F$, known as the eigenvalue, characteristic value, or characteristic root associated with the eigenvector $v$.
68 |
69 | ------------------
70 |
71 |
91 |
92 |
93 | In this case, gravity is the only force acting upon an object. The equation
94 | reads
95 | $$y'' =g = \text{const}$$
96 |
97 |
98 |
99 |
100 | Your browser does not support the video tag.
101 |
102 |
103 |
104 |
105 |
106 | ------------------
107 |
108 | # Examples: Parachute falling
109 |
110 |
111 |
112 |
113 | This case is similar to the previous one, but there is air resistance (drag).
114 | The drag force is normally as a constant multiplied by the square of the speed
115 | $$m\frac{d^2 y}{dt^2} = mg - b\left(\frac{dy}{dt}\right)^2$$
116 |
117 |
118 |
119 |
120 | Your browser does not support the video tag.
121 |
122 |
123 |
124 |
125 |
126 | ------------------
127 |
128 | # Examples: Pendulum
129 |
130 |
131 |
132 |
133 |
137 |
138 | A pendulum is a weight suspended from a pivot that can swing freely. The
139 | equation for a simple pendulum is
140 |
141 | $$L\theta'' + g\sin\theta = 0$$
142 |
143 |
144 |
145 |
146 | ------------------
147 |
148 | # Equations of first order
149 |
150 | Equations of first order are of the form
151 |
152 | $$\frac{\mathrm{d} y}{\mathrm{d} z} = f(z, y)\, .$$
153 |
154 | For an arbitrary function $f$ there is not method for solving this equation
155 | in terms of elementary functions.
156 |
157 | ------------------
158 |
159 | # Linear equations of first order
160 |
161 | They are of the form
162 |
163 | $$a_0(z) y' + a_1(z) y = f(z)\, .$$
164 |
165 | We are assuming that $a_0(z)$, $a_1(z)$, and $f(z)$ are continuous.
166 |
167 | ------------------
168 |
169 | # Linear equations of first order: homogeneous case
170 |
171 | We can write it as
172 |
173 | $$\frac{\mathrm{d}} {\mathrm{d}z}\ln y + \frac{a_1(z)}{a_0(z)} = 0\, ,$$
174 |
175 | after integration
176 |
177 | $$y = y_0 \exp\left[-\int\limits_{z_0}^z \frac{a_1(t)}{a_0(t)}\mathrm{d}t\right]\, .$$
178 |
179 | ------------------
180 |
181 | # Linear equations of first order: inhomogeneous case
182 |
183 | To solve the equation
184 |
185 | $$y' + \frac{a_1(z)}{a_0(z)} y = \frac{f(z)}{a_0(z)}\, ,$$
186 |
187 | with $y(0)=y_0$, we use the adjoint equation
188 |
189 | $$x' - \frac{a_1(z)}{a_0(z)} x = 0\, ,$$
190 |
191 | with $x(0) = 1$.
192 |
193 | ------------------
194 |
195 | # Linear equations of first order: inhomogeneous case
196 |
197 | And we solve the differential equation
198 |
199 | $$(xy)' = xy' + x' y = x\frac{f(z)}{a_0(z)}\, ,$$
200 |
201 | and, after integration
202 |
203 | $$y = \frac{y_0}{x} + \frac{1}{x}\int\limits_{0}^{z}\frac{x f(t)}{a_0(t)}\mathrm{d}t\, .$$
204 |
205 | ------------------
206 |
207 | # Systems of differential equations
208 |
209 | Sometimes we have more than one differential equations. For example,
210 |
211 | $$
212 | \begin{align}
213 | &F_1(z, y_1, y_1', y_1'', y_2, y_2') = 0\, ,\\
214 | &F_2(z, y_1, y_1', y_2, y_2', y_2'', y_2''') = 0\, .
215 | \end{align}
216 | $$
217 |
218 |
219 | ------------------
220 |
221 | # Systems of differential equations
222 |
223 | Using the new variables
224 |
225 | $$
226 | \begin{align}
227 | &x_1 = y_1\, , & x_4 = y_2'\, ,\\
228 | &x_2 = y_1'\, , & x_5 = y_2''\, ,\\
229 | &x_3 = y_2\, , &
230 | \end{align}
231 | $$
232 |
233 | we can rewrite the following system
234 |
235 | $$
236 | \begin{align}
237 | &x_1' = G_1(z, x_1, x_2, x_3, x_4) = x_2\, ,\\
238 | &x_2' = G_2(z, x_1, x_2, x_3, x_4)\, ,\\
239 | &x_3' = G_3(z, x_1, x_2, x_3, x_4) = x_4\, ,\\
240 | &x_4' = G_4(z, x_1, x_2, x_3, x_4) = x_5\, ,\\
241 | &x_5' = G_5(z, x_1, x_2, x_3, x_4)\, .\\
242 | \end{align}
243 | $$
244 |
245 |
246 | ------------------
247 |
248 | # Power series solutions
249 |
250 | Consider the second-order linear differential equation
251 |
252 | $$a_0(z)f''(z)+a_1(z)f'(z)+a_2(z)f(z)=0\, ,$$
253 |
254 | where $a_0$ is nonzero for all $z$, and $a_1/a_0$ and $a_2/a_0$ are analytic
255 | functions.
256 |
257 | The power series method calls for the construction of a power series solution
258 | $$f=\sum_{k=0}^\infty A_kz^k\, ,$$
259 |
260 | to find the form of the coefficients $A_k$.
261 |
262 | ------------------
263 |
264 | # Power series solutions: Example 1
265 |
266 | Consider the equation
267 |
268 | $$y' = y\, .$$
269 |
270 | Assuming $y=\sum_{k=0}^\infty A_k z^k$, leads to
271 |
272 | $$\sum_{k=1}^\infty[A_{k-1} - k A_k] z^k = 0\, ,$$
273 |
274 | or
275 |
276 | $$A_k = \frac{A_{k-1}}{k}\, .$$
277 |
278 | ------------------
279 |
280 | # Power series solutions: Example 2
281 |
282 | Consider the (not so simple) equation
283 |
284 | $$y'' - z y = 0\, .$$
285 |
286 | Assuming $y=\sum_{k=0}^\infty A_k z^k$, leads to
287 |
288 | $$2A_2 + \sum_{k=1}^\infty[(k + 2)(k + 1)A_{k+2} - A_{k-1}] z^k = 0\, .$$
289 |
290 | ------------------
291 |
292 | # Power series solutions: Example 2
293 |
294 | We obtain the general solution
295 |
296 | \begin{align}
297 | y(z) =&
298 | C_0\left[1 +
299 | \sum_{k=1}^{\infty} \frac{z^{3k}}{(2\cdot 3) (5\cdot 6) \cdots ((3k-1) \cdot (3k))}\right]\\
300 | &+ C_1\left[z +
301 | \sum_{k=1}^{\infty} \frac{z^{3k+1}}{(3\cdot 4) (6\cdot 7) \cdots ((3k) \cdot (3k+1))}\right]
302 | \end{align}
303 |
304 | ------------------
305 |
306 | # Power series solutions: Example 2
307 |
308 |
309 |
310 |
311 |
312 | These series represent a group of _special functions_ named Airy functions
313 |
314 | $$y(z) = \mathrm{Ai}(z) + \mathrm{Bi}(z)\, ,$$
315 |
316 | due to the British astronomer George Biddel Airy.
317 |
318 | They appear in the solution of Scrödinger equation for a particle
319 | confined within a triangular potential. They are also important
320 | in microscopy and astronomy.
321 |
322 |
323 |
327 |
328 |
329 |
330 | ------------------
331 |
332 | # Power series solutions: A nonlinear example
333 |
334 | The equation $x' = 1 + x^2$, with initial condition $x(0) = 0$, has the
335 | solution $x = \tan(t)$. We could use the power series method, assuming
336 | $x = \sum_{k=0}^\infty A_k t^k$, to obtain
337 |
338 | $$\sum_{k=0}^{\infty} (k + 1) A_{k+1} t^k =
339 | 1 + \left(\sum_{k=0}^{\infty} A_k t^k\right)^2 =
340 | 1 + A_0^2 + \sum_{k=1}^\infty\left[\sum_{j=0}^{k} A_j A_{k-j}\right]t^k$$
341 |
342 | and the coefficients are
343 |
344 | \begin{align}
345 | A_0 &= 0\\
346 | A_1 &= 1 + A_0^2\\
347 | A_{k + 1} &= \frac{1}{k + 1}\sum_{j=0}^{k} A_{k} A_{j - k}\quad \forall k\in \mathbb{N}
348 | \end{align}
349 |
350 | ------------------
351 |
352 | # Frobenius method
353 |
354 | This method gives as a way to find infinite series solutions for the
355 | differential equation
356 |
357 | $$z^2 u'' + p(z) z u'' + q(z) u = 0\, ,$$
358 |
359 | in the vinicinity of the regular singular point $z=0$. We could divide by
360 | $z^2$ to obtain
361 |
362 | $$u'' + \frac{p(z)}{z} u' + \frac{q(z)}{z^2} u = 0\, .$$
363 |
364 |
365 | ------------------
366 |
367 | # Frobenius method
368 |
369 | We are looking for a solution of the form
370 |
371 | $$u(z) = \sum_{k=0}^\infty A_k z^{k + r}\quad A_0\neq 0, r\in \mathbb{R}\, ,$$
372 |
373 | that leads us to the equation
374 |
375 | \begin{align}
376 | 0 =&[r (r - 1) + p(z) + q(z)]A_0 z^r \\
377 | &+ \sum_{k=1}^{\infty} [(k + r - 1)(k + r) + p(z)(k + r) + q(z)]A_k z^{k + r}
378 | \end{align}
379 |
380 | ------------------
381 |
382 | # Frobenius method
383 |
384 | The expression
385 |
386 | $$I(r) \equiv r(r - 1) + p(0)r + q(0)$$
387 |
388 | is called the inditial equation, and its roots give us the values for $r$.
389 |
390 | Depending on the values for the roots, we have different ways to obtain
391 | two linearly independent solutions.
392 |
393 |
394 | ------------------
395 |
396 | # Frobenius method: Case 1
397 |
398 |
399 | The roots do not differ by an integer
400 |
401 | \begin{align}
402 | y_1 &= \sum_{k=0}^\infty A_k z^{k + r_1}\quad A_0\neq 0\\
403 | y_2 &= \sum_{k=0}^\infty A_k z^{k + r_2}\quad A_0\neq 0
404 | \end{align}
405 |
406 | ------------------
407 |
408 | # Frobenius method: Case 2
409 |
410 |
411 | The roots do differ by an integer
412 |
413 | \begin{align}
414 | y_1 &= \sum_{k=0}^\infty A_k z^{k + r_1}\quad A_0\neq 0\\
415 | y_2 &= C y_1 \ln(z) + \sum_{k=0}^\infty A_k z^{k + r_2}\quad A_0\neq 0
416 | \end{align}
417 |
418 | where $C$ is a constant that could be zero.
419 |
420 | ------------------
421 |
422 | # Frobenius method: Case 3
423 |
424 |
425 | The roots are the same
426 |
427 | \begin{align}
428 | y_1 &= \sum_{k=0}^\infty A_k z^{k + r_1}\quad A_0\neq 0\\
429 | y_2 &= y_1 \ln(z) + \sum_{k=0}^\infty A_k z^{k + r_2}\quad A_0\neq 0
430 | \end{align}
431 |
432 | ------------------
433 |
434 | # Bessel equation
435 |
436 | The Bessel equation is of the form
437 |
438 | $$z^2 y'' + z y' + (z^2 - \nu^2) y = 0\, .$$
439 |
440 | And it appears frequently when solving problems in cylindrical or spherical
441 | coordinates. Particularly, when solving the temperature distribution of a
442 | circular plate or the vibration or a membrane.
443 |
444 |
445 | ------------------
446 |
447 | # Bessel equation: Sketch of solution
448 |
449 | After pluging our infinite series we obtain
450 |
451 | $$A_0 (r^2 - \nu^2) z^r + z^r\sum_{k=1}^\infty A_k [(n + r)^2 - \nu^2]z^n +
452 | z^r \sum_{k=0}^\infty A_k z^{n + 2} = 0 $$
453 |
454 | And the indicial equation
455 |
456 | $$r^2 - \nu^2 = 0$$
457 |
458 | has as solutions $r_1 = \nu$, $r_2 = -\nu$.
459 |
460 | ------------------
461 |
462 | # Bessel equation: Sketch of solution
463 |
464 |
465 | If we solve the recurrence relations for $r_1 =\nu$, we end up with
466 |
467 |
468 | $$A_{2k} = -\frac{A_{2k - 2}}{2^2 n (n + \nu)}$$
469 |
470 | and the solution renders
471 |
472 | $$y = A_0 \sum_{n=0}^\infty \frac{(-1)^n}{2^n n! (1 +\nu)(2 + \nu)\cdots(n + \nu)}
473 | \left(\frac{z}{2}\right)^{2n + \nu}\, .$$
474 |
475 |
476 | ------------------
477 |
478 | # Bessel equation: Sketch of solution
479 |
480 | We could rewrite this _prettier_ using the Gamma function
481 |
482 | $$y = A_0 \sum_{n=0}^\infty \frac{(-1)^n}{n! \Gamma(1 + \nu + n)}
483 | \left(\frac{z}{2}\right)^{2n + \nu}\, ,$$
484 |
485 | or simply
486 |
487 | $$y = A_0 \mathrm{J}_\nu(z)\, .$$
488 |
489 | ------------------
490 |
491 | # Bessel functions of the first kind
492 |
493 |
494 |
495 |
496 |
497 | They appear as the solution of Bessel equation for integer or positive $\nu$.
498 |
499 | For non-integer $\nu$, the functions $\mathrm{J}_\nu(z)$ and
500 | $\mathrm{J}_{-\nu}(z)$ are linearly independent.
501 |
502 |
503 |
507 |
508 |
509 |
510 |
511 |
512 | ------------------
513 |
514 | # Gamma function
515 |
516 |
517 |
518 |
519 | The gamma function is an extension of the factorial
520 |
521 | $$\Gamma(n) = (n - 10)!\, .$$
522 |
523 | The Gamma function is defined for all complex numbers except for non-positive
524 | integers. For complex numbers with a positive real part, it is defined via a
525 | convergent improper integral:
526 | $$\Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} \mathrm{d}t\, .$$
527 |
528 |
529 |
533 |
534 |
535 |
536 | ------------------
537 |
538 | # References
539 |
540 | - H. Hochstadt. Differential equations: a modern approach.
541 | Courier Dover Publications, 1975.
542 |
543 | - Erwin Kreyszig. Advanced engineering mathematics. John Wiley & Sons, 2010.
544 |
545 | - Dennis G. Zill. Ecuaciones diferenciales: con aplicaciones demodelado.
546 | Iternational Thomson Editores, 1997.
547 |
548 | - Ruryk, 2011. Oscilatting pendulum. Retrieved from:
549 | https://commons.wikimedia.org/wiki/File:PenduloTmg.gif
550 |
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/slides/vector_calculus.html:
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1 |
2 |
4 |
5 |
6 | Curvilinear coordinates and vector calculus
13 |
19 |
29 |
Curvilinear coordinates and vector calculus
30 |
31 | Nicolás Guarín-Zapata
33 |
January, 2019
34 |
36 |
Tensors
37 |
38 |
39 |
40 | Depiction of the Cauchy stress tensor, a second-order tensor. The tensor’s components, in a three-dimensional Cartesian coordinate system, form the matrix
41 | \[\begin{align}
42 | \sigma & = \begin{bmatrix}\mathbf{T}^{(\mathbf{e}_1)} \mathbf{T}^{(\mathbf{e}_2)} \mathbf{T}^{(\mathbf{e}_3)} \\ \end{bmatrix} \\
43 | & = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}\\
44 | \end{align}\]
45 | whose columns are the stresses (forces per unit area) acting on the \(\mathbf{e}_1\) , \(\mathbf{e}_2\) , and \(\mathbf{e}_3\) faces of the cube.
46 |
47 |
48 |
52 |
53 |
54 |
56 |
Kronecker Delta
57 |
The Kronecker delta is defined by
58 |
\[\delta_{ij} = \begin{cases}
59 | 1\quad i=j\\
60 | 0\quad i\neq j
61 | \end{cases}\]
62 |
and it can be thought as the components of the identity tensor.
63 |
65 |
Levi-Civita symbol
66 |
67 |
68 |
69 | It is defined by the cross product between element of an orthonormal basis
70 | \[\hat{\mathbf{e}}_i \times \hat{\mathbf{e}}_j = \sum_{k=1}^{3}\epsilon_{ijk} \hat{\mathbf{e}}_k\]
71 | with
72 | \[\begin{align}
73 | \epsilon_{ijk} &= \frac{1}{2} (i - j)(j - k)(k - i)\\
74 | &= \begin{cases}
75 | 1\quad \text{even permutation}\\
76 | -1\quad \text{odd permutation}\\
77 | 0\quad \text{repeated indices}\\
78 | \end{cases}
79 | \end{align}\]
80 |
81 |
82 |
86 |
87 |
88 |
90 |
Some operations between second-order Tensors
91 |
92 | \(\underline{\underline{T}} \cdot \mathbf{A} = T_{ik} A_k \hat{\mathbf{e}}_i\)
\(\underline{\underline{T}} \times \mathbf{A} = \hat{\mathbf{e}}_i \epsilon_{jkl} \hat{\mathbf{e}}_l T_{ik} A_k \neq \mathbf{A} \times \underline{\underline{T}}\)
\(\underline{\underline{T}} \cdot \underline{\underline{V}} = T_{ij} V_{kl}\delta_{jk} \hat{\mathbf{e}}_i \hat{\mathbf{e}}_l = T_{ik} V_{kl} \hat{\mathbf{e}}_i \hat{\mathbf{e}}_l\)
\(\underline{\underline{T}} : \underline{\underline{V}} = T_{ik} V_{ki}\)
97 |
99 |
Curvilinear coordinates example: Cylindrical Coordinates
100 |
102 |
103 |
104 |
Coordinate systems
105 |
Two common examples of curvilinear coordinates are
106 |
Cylindrical coordinates
107 |
\[\begin{align}
108 | x =& \rho \cos\varphi\\
109 | y =& \rho \sin\varphi\\
110 | z =& z
111 | \end{align}\]
112 |
Spherical coordinates
113 |
\[\begin{align}
114 | x =& r \sin\theta \cos\varphi\\
115 | y =& r \sin\theta \sin\varphi\\
116 | z =& r \cos\theta
117 | \end{align}\]
118 |
130 |
Jacobian
131 |
We can rewrite the transformation, in components, as
132 |
\[d x_i = \sum_j \frac{\partial x_i}{\partial u_j} du_j = \sum_j J_{ij} du_j \, ,\]
133 |
where \(J_{ij} = \partial x_i/\partial u_j\) are the components of the Jacobian matrix .
134 |
And, its determinant represents the (local) change in volume of the transformation
135 |
\[ |J| = h_1 h_2 h_3\, .\]
136 |
138 |
Line, surface and volume differentials
139 |
We can rewrite the transformation as
140 |
\[ d\mathbf{r} = \sum_{i=1} h_i \hat{\mathbf{e}}_i du_i = \sum_{i=1}^3 d\mathbf{l}_i \, ,\]
141 |
where \(d\mathbf{l}_i = h_i \hat{\mathbf{e}}_i du_i\) is the line differential along the coordinate \(u_i\) .
142 |
We can define the surface differentials as the vectors that are perpendicular to the differential areas according to the right hand convention, namely
143 |
\[d\mathbf{S}_i = d\mathbf{l}_j \times d\mathbf{l}_k =
144 | h_j h_k \hat{\mathbf{e}}_i du_j du_k = \hat{\mathbf{e}}_i dS_i\]
145 |
And the volume differential is given by the volume of the curvilinear parallelepiped defined by the line differentials, i.e.
146 |
\[dV = d\mathbf{l}_1 \cdot (d\mathbf{l}_2 \times \mathbf{l}_3) =
147 | h_1 h_2 h_3 du_1 du_2 du_3 \, .\]
148 |
150 |
Differential operators
151 |
Many physical equations are written as differential equations, and these can be interpreted as the action of certain operators over some particular functions knows as fields. There are fields of different nature:
152 |
153 | Scalar fields (temperature, pressure): \(\phi = zy^3 - x^2\)
Vector fields (velocity, displacement): \(\mathbf{A} = yz^2\hat{\imath} - 2zx^3\hat{\jmath} + xy^2\hat{k}\)
Tensor fields (strain, stresses):
157 |
\[\underline{\underline{\sigma}} =
158 | \begin{pmatrix}
159 | x^2 &xy &0\\
160 | xy &y^2 &0\\
161 | 0 &0 &1
162 | \end{pmatrix}\]
163 |
165 |
166 |
Gradient
167 |
The gradient of a scalar function \(\phi\) , denoted by \(\nabla \phi\) is given by
168 |
\[\nabla \phi \equiv \sum_{i=1}^3\frac{\hat{\mathbf{e}}_i}{h_i}
169 | \frac{\partial \phi}{\partial u_i}\]
170 |
In spherical coordinates it reads
171 |
\[\begin{align}
172 | \nabla \phi &= \frac{\hat{\mathbf{e}}_r}{h_r}\frac{\partial \phi}{\partial r} +
173 | \frac{\hat{\mathbf{e}}_\theta}{h_\theta}\frac{\partial \phi}{\partial \theta} +
174 | \frac{\hat{\mathbf{e}}_\varphi}{h_\varphi}\frac{\partial \phi}{\partial \varphi} \\
175 | &= \hat{\mathbf{e}}_r \frac{\partial \phi}{\partial r} +
176 | + \frac{\hat{\mathbf{e}}_\theta}{r}\frac{\partial \phi}{\partial \theta}
177 | + \frac{\hat{\mathbf{e}}_\varphi}{r\sin\theta}\frac{\partial \phi}{\partial \varphi}
178 | \end{align}\]
179 |
181 |
182 |
Gradient: example
183 |
The function \(f(x,y) = -[\cos^2x + \cos^2y]^2\) , has as gradient
184 |
\[ \nabla f(x, y) = 4[\cos^2 x + \cos^2 y](\cos x \sin x, \sin x \cos y )\]
185 |
188 |
190 |
191 |
Divergence
192 |
The divergence of a vector function \(\mathbf{B}\) , denoted by \(\nabla\cdot\mathbf{B}\) or , is given by
193 |
\[\nabla \cdot \mathbf{B} = \operatorname{div} \mathbf{B} = \frac{1}{h}\sum_{i=1}^3
194 | \frac{\partial}{\partial u_i} \left(\frac{B_i h}{h_i}\right)\]
195 |
with \(h = h_1 h_2 h_3\) .
196 |
In spherical coordinates it reads
197 |
\[
198 | \nabla \cdot \mathbf{B} = \operatorname{div} \mathbf{B}
199 | = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 B_r)
200 | + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}(\sin\theta B_\theta)
201 | + \frac{1}{r\sin\theta}\frac{\partial}{\partial \varphi}(B_\varphi)
202 | \]
203 |
205 |
206 |
Divergence and the Gauss theorem
207 |
We can understand the divergence as the flux per unit volume \(\operatorname{div}\mathbf{B} = d\phi/dV\) , and we can extend this calculation to a finite surface
208 |
\[\phi = \oint_S \mathbf{B}\cdot d\mathbf{S} = \int_V \operatorname{div}\mathbf{B} dV\, ,\]
209 |
The second part is the Gauss theorem
210 |
\[\oint_S \mathbf{B}\cdot d\mathbf{S} = \int_V \operatorname{div}\mathbf{B} dV\, .\]
211 |
213 |
214 |
Curl
215 |
The curl describes the infinitesimal rotation of a (3-dimensional) vector field. The curl of a field \(\mathbf{B}\) , denoted by \(\operatorname{curl}\mathbf{B}\) , \(\operatorname{rot}\mathbf{B}\) , or \(\nabla\times\mathbf{B}\) , is defined by
216 |
\[\operatorname{rot} \mathbf{B} = \frac{1}{h}\sum_{i,j,k=1}^1 \hat{\mathbf{e}}_i
217 | \epsilon_{ijk} h_i \frac{\partial}{\partial u_j} (B_k h_k)\, ,\]
218 |
where \(h = h_1 h_2 h_3\) , and \(\epsilon_{ijk}\) is the Levi-Civita symbol.
219 |
The Stokes theorem tranform line integrals to surface integrals, and involves the curl
220 |
\[\oint_c \mathbf{B} \cdot d\mathbf{l} = \int_S \operatorname{rot}\mathbf{B}\cdot d\mathbf{S}\]
221 |
223 |
224 |
Laplacian
225 |
The Laplacian of a scalar function \(f(u_i)\) , denoted by \(\nabla\cdot\nabla f\) , \(\nabla^2 f\) , or \(\Delta f\) , it is defined as \(\nabla\cdot\nabla f\) . It is defined as
226 |
\[\nabla^2 f = \frac{1}{h}\sum_{i=1}^3 \frac{\partial}{\partial u_i}
227 | \left(\frac{h}{h_i^2}\frac{\partial f}{\partial u_i}\right)\, ,\]
228 |
with \(h = h_1 h_2 h_3\) .
229 |
For a vector function, the Laplacian is defined like
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\[\nabla^2\mathbf{A} = \nabla^2A_x\hat{\imath} + \nabla^2A_y\hat{\jmath} + \nabla^2A_y\hat{k}\]
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for Cartesian coordinates, and
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\[\nabla^2\mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times( \nabla \times \mathbf{A})\]
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for other coordinate systems.
234 |
236 |
References
237 |
238 | Alonso Sepúlveda Soto. Física matemática. Ciencia y Tecnología. Universidad de Antioquia, 2009.
Wikipedia contributors. “Tensor.” Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, Retrieved: 2 Aug. 2017.
Wikipedia contributors. “Curvilinear coordinates.” Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, Retrieved: 3 Feb. 2017.
Wikipedia contributors. “Gradient.” Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, Retrieved: Web. 15 Feb. 2017.
Wikipedia contributors. “Divergence.” Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, Retrieved: Web. 15 Feb. 2017.
Wikipedia contributors. “Curl (mathematics).” Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, Retrieved: Web. 15 Feb. 2017.
Wikipedia contributors. “Laplace operator.” Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, Retrieved: Web. 17 Feb. 2017.
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14 |
15 |
16 | Depiction of the Cauchy stress tensor, a second-order tensor.
17 | The tensor's components, in a three-dimensional Cartesian coordinate
18 | system, form the matrix
19 |
20 | $$\begin{align}
21 | \sigma & = \begin{bmatrix}\mathbf{T}^{(\mathbf{e}_1)} \mathbf{T}^{(\mathbf{e}_2)} \mathbf{T}^{(\mathbf{e}_3)} \\ \end{bmatrix} \\
22 | & = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}\\
23 | \end{align}$$
24 |
25 | whose columns are the stresses (forces per unit area) acting on the $\mathbf{e}_1$ , $\mathbf{e}_2$, and $\mathbf{e}_3$ faces of the cube.
26 |
27 |
28 |
32 |
33 |
34 |
35 | ------------------
36 |
37 | # Kronecker Delta
38 |
39 | The Kronecker delta is defined by
40 |
41 | $$\delta_{ij} = \begin{cases}
42 | 1\quad i=j\\
43 | 0\quad i\neq j
44 | \end{cases}$$
45 |
46 | and it can be thought as the components of the identity tensor.
47 |
48 | ------------------
49 |
50 | # Levi-Civita symbol
51 |
52 |
53 |
54 |
55 | It is defined by the cross product between element of an orthonormal basis
56 |
57 | $$\hat{\mathbf{e}}_i \times \hat{\mathbf{e}}_j = \sum_{k=1}^{3}\epsilon_{ijk} \hat{\mathbf{e}}_k$$
58 |
59 | with
60 |
61 | $$\begin{align}
62 | \epsilon_{ijk} &= \frac{1}{2} (i - j)(j - k)(k - i)\\
63 | &= \begin{cases}
64 | 1\quad \text{even permutation}\\
65 | -1\quad \text{odd permutation}\\
66 | 0\quad \text{repeated indices}\\
67 | \end{cases}
68 | \end{align}$$
69 |
70 |
71 |
72 |
76 |
77 |
78 |
79 | ------------------
80 |
81 | # Some operations between second-order Tensors
82 |
83 | - $\underline{\underline{T}} \cdot \mathbf{A} = T_{ik} A_k \hat{\mathbf{e}}_i$
84 |
85 | - $\underline{\underline{T}} \times \mathbf{A} = \hat{\mathbf{e}}_i \epsilon_{jkl} \hat{\mathbf{e}}_l T_{ik} A_k \neq \mathbf{A} \times \underline{\underline{T}}$
86 |
87 | - $\underline{\underline{T}} \cdot \underline{\underline{V}} = T_{ij} V_{kl}\delta_{jk} \hat{\mathbf{e}}_i \hat{\mathbf{e}}_l = T_{ik} V_{kl} \hat{\mathbf{e}}_i \hat{\mathbf{e}}_l$
88 |
89 | - $\underline{\underline{T}} : \underline{\underline{V}} = T_{ik} V_{ki}$
90 |
91 | ------------------
92 |
93 | # Curvilinear coordinates example: Cylindrical Coordinates
94 |
95 |
96 |
102 |
103 | ------------------
104 |
105 | # Coordinate systems
106 |
107 | Two common examples of curvilinear coordinates are
108 |
109 | ### Cylindrical coordinates
110 |
111 | $$\begin{align}
112 | x =& \rho \cos\varphi\\
113 | y =& \rho \sin\varphi\\
114 | z =& z
115 | \end{align}$$
116 |
117 | ### Spherical coordinates
118 |
119 | $$\begin{align}
120 | x =& r \sin\theta \cos\varphi\\
121 | y =& r \sin\theta \sin\varphi\\
122 | z =& r \cos\theta
123 | \end{align}$$
124 |
125 | ------------------
126 |
127 | # Transformations
128 |
129 | We can write the position vector as
130 |
131 | $$d\mathbf{r} = \sum\limits_{i=1}^3 \frac{\partial \mathbf{r}}{\partial u_i} du_i \, .$$
132 |
133 | The factor $\partial \mathbf{r}/\partial u_i$ is a non-unitary vector, we can
134 | introduce a normalized base $\hat{\mathbf{e}}_i$
135 |
136 | $$\frac{\partial \mathbf{r}}{\partial u_i} = h_i \hat{\mathbf{e}}_i $$
137 |
138 | where
139 |
140 | $$\left| \frac{\partial \mathbf{r}}{\partial u_i}\right| = h_i$$
141 |
142 | is the [**scale factor**](https://en.wikipedia.org/wiki/Curvilinear_coordinates#Relation_to_Lam.C3.A9_coefficients).
143 |
144 | ------------------
145 |
146 | # Jacobian
147 |
148 | We can rewrite the transformation, in components, as
149 |
150 | $$d x_i = \sum_j \frac{\partial x_i}{\partial u_j} du_j = \sum_j J_{ij} du_j \, ,$$
151 |
152 | where $J_{ij} = \partial x_i/\partial u_j$ are the components of the [Jacobian
153 | matrix](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant).
154 |
155 | And, its determinant represents the (local) change in volume of the
156 | transformation
157 |
158 | $$ |J| = h_1 h_2 h_3\, .$$
159 |
160 | ------------------
161 |
162 | # Line, surface and volume differentials
163 |
164 | We can rewrite the transformation as
165 |
166 | $$ d\mathbf{r} = \sum_{i=1} h_i \hat{\mathbf{e}}_i du_i = \sum_{i=1}^3 d\mathbf{l}_i \, ,$$
167 |
168 | where $d\mathbf{l}_i = h_i \hat{\mathbf{e}}_i du_i$ is the _line differential_
169 | along the coordinate $u_i$.
170 |
171 | We can define the _surface differentials_ as the vectors that are perpendicular
172 | to the differential areas according to the right hand convention, namely
173 |
174 | $$d\mathbf{S}_i = d\mathbf{l}_j \times d\mathbf{l}_k =
175 | h_j h_k \hat{\mathbf{e}}_i du_j du_k = \hat{\mathbf{e}}_i dS_i$$
176 |
177 | And the volume differential is given by the volume of the curvilinear
178 | parallelepiped defined by the line differentials, i.e.
179 |
180 | $$dV = d\mathbf{l}_1 \cdot (d\mathbf{l}_2 \times \mathbf{l}_3) =
181 | h_1 h_2 h_3 du_1 du_2 du_3 \, .$$
182 |
183 | ------------------
184 |
185 | # Differential operators
186 |
187 | Many physical equations are written as differential equations, and these can be
188 | interpreted as the action of certain operators over some particular functions
189 | knows as fields. There are fields of different nature:
190 |
191 | - Scalar fields (temperature, pressure): $\phi = zy^3 - x^2$
192 |
193 | - Vector fields (velocity, displacement): $\mathbf{A} = yz^2\hat{\imath} - 2zx^3\hat{\jmath} + xy^2\hat{k}$
194 |
195 | - Tensor fields (strain, stresses):
196 |
197 | $$\underline{\underline{\sigma}} =
198 | \begin{pmatrix}
199 | x^2 &xy &0\\
200 | xy &y^2 &0\\
201 | 0 &0 &1
202 | \end{pmatrix}$$
203 |
204 | ------------------
205 |
206 | ## Gradient
207 |
208 | The gradient of a scalar function $\phi$, denoted by $\nabla \phi$ is given by
209 |
210 | $$\nabla \phi \equiv \sum_{i=1}^3\frac{\hat{\mathbf{e}}_i}{h_i}
211 | \frac{\partial \phi}{\partial u_i}$$
212 |
213 | In spherical coordinates it reads
214 |
215 | $$\begin{align}
216 | \nabla \phi &= \frac{\hat{\mathbf{e}}_r}{h_r}\frac{\partial \phi}{\partial r} +
217 | \frac{\hat{\mathbf{e}}_\theta}{h_\theta}\frac{\partial \phi}{\partial \theta} +
218 | \frac{\hat{\mathbf{e}}_\varphi}{h_\varphi}\frac{\partial \phi}{\partial \varphi} \\
219 | &= \hat{\mathbf{e}}_r \frac{\partial \phi}{\partial r} +
220 | + \frac{\hat{\mathbf{e}}_\theta}{r}\frac{\partial \phi}{\partial \theta}
221 | + \frac{\hat{\mathbf{e}}_\varphi}{r\sin\theta}\frac{\partial \phi}{\partial \varphi}
222 | \end{align}$$
223 |
224 | ------------------
225 |
226 | ## Gradient: example
227 |
228 | The function $f(x,y) = -[\cos^2x + \cos^2y]^2$, has as gradient
229 |
230 | $$ \nabla f(x, y) = 4[\cos^2 x + \cos^2 y](\cos x \sin x, \sin x \cos y )$$
231 |
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532 | 
51 | 
85 | 