Math for Programmers

692 Pages·2020·36.474 MB·English

by  Paul Orland

3D graphics, machine learning,
and simulations with Python
Paul Orland
M A N N I N G
Mathematical Notation Reference 3
Mathematical Notation Reference
Notation example Name Defined in section
(x,y) Coordinate(cid:1)vector(cid:1)in(cid:1)2D 2.1.1
sin(x) Trigonometric(cid:1)sine(cid:1)function 2.3.1
cos(x) Trigonometric(cid:1)cosine(cid:1)function 2.3.1
θ Greek(cid:1)letter(cid:1)theta,(cid:1)commonly(cid:1)representing(cid:1)angle(cid:1)measure(cid:1) 2.3.1
π Greek(cid:1)letter(cid:1)pi,(cid:1)representing(cid:1)the(cid:1)number(cid:1)3.14159(cid:1).(cid:1).(cid:1). 2.3.2
(x,y,z) Coordinate(cid:1)vector(cid:1)in(cid:1)3D 3.1.1
u·v Dot(cid:1)product(cid:1)of(cid:1)two(cid:1)vectors(cid:1)u(cid:1)and(cid:1)v 3.3
u×v Cross(cid:1)product(cid:1)of(cid:1)two(cid:1)vectors(cid:1)u(cid:1)and(cid:1)v 3.4
f(g(x)) Composition(cid:1)of(cid:1)functions 4.1.2
f(x,y)versusf(x)(y) See(cid:1)discussion(cid:1)of(cid:1)currying. 4.1.2
au+bv Linear(cid:1)combination(cid:1)(cid:80)(cid:71)(cid:1)two(cid:1)vectors(cid:1)u(cid:1)and(cid:1)v 4.2.3
e1,e2,e3,... Standard(cid:1)basis(cid:1)vectors 4.2.4
⎛ ⎞
1
v=⎝2⎠ Column(cid:1)vector 5.1.1
3
⎛ ⎞
1 2 3
Matrix 5.1.1
A=⎝4 5 6⎠
7 8 9
Rn Real(cid:1)coordinate(cid:1)vector(cid:1)space(cid:1)of(cid:1)dimension(cid:1)n 6.2.1
(f+g)(x)orf(x)+g(x) Adding(cid:1)two(cid:1)functions 6.2.3
c·f(x) Scalar(cid:1)multiplication(cid:1)of(cid:1)a(cid:1)function 6.2.3
span({u,v,w}) The(cid:1)span(cid:1)of(cid:1)a(cid:1)set(cid:1)of(cid:1)vectors 6.3.3
f(x)=ax+b Linear(cid:1)function 6.3.5
f(x)=a0+a1x+···+anxn Polynomial(cid:1)function 6.3.5
(continues on inside back cover)
Math for Programmers
3D graphics, machine learning and simulations with Python
ii
Math for
Programmers
3D GRAPHICS, MACHINE LEARNING AND
SIMULATIONS WITH PYTHON
PAUL ORLAND
MANNING
SHELTER ISLAND
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To my first math teacher and
my first programming teacher—Dad.
vi
brief contents
1 ■ Learning math with code 1
PART 1 VECTORS AND GRAPHICS ............................................... 19
2 ■ Drawing with 2D vectors 21
3 ■ Ascending to the 3D world 75
4 ■ Transforming vectors and graphics 121
5 ■ Computing transformations with matrices 158
6 ■ Generalizing to higher dimensions 205
7 ■ Solving systems of linear equations 257
PART 2 CALCULUS AND PHYSICAL SIMULATION ........................ 301
8 ■ Understanding rates of change 303
9 ■ Simulating moving objects 337
10 ■ Working with symbolic expressions 354
11 ■ Simulating force fields 392
12 ■ Optimizing a physical system 422
13 ■ Analyzing sound waves with a Fourier series 463
PART 3 MACHINE LEARNING APPLICATIONS ............................. 497
14 ■ Fitting functions to data 499
15 ■ Classifying data with logistic regression 526
16 ■ Training neural networks 559
vii
viii BRIEF CONTENTS