math_for_programmers_by_paul_orland_index

Table of Contents

Math for Programmers by Paul Orland Index
Symbols
Numerics
B
C
D
E
F
G
H
I
J
L
M
N
O
P
Q
R
S
T
U
V
W
Z
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Math for Programmers by Paul Orland Index

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Symbols

__call__ method - Page 231, 235, 251

__dict__ method - Page 630

__eq__ method - Page 631, 634

__mul__ method - Page 631

__repr__ method - Page 208, 371, 373, 631

__rmul__ method - Page 631

_repr_latex_ method - Page 373

- operator - Page 365, 445, 607

@abstractmethod decorator - Page 211

operator - Page 208, 236, 365, 607

** operator - Page 608, 621

/ operator - Page 607

% (modulus operator) - Page 607, 620

+ operator - Page 208, 365, 445, 478, 607, 611

== operator - Page 208, 631

≈ (approximate equality) - Page 54

π(pi) symbol - Page 56

Numerics

2D plane - Page 22

2D vectors. See vectors

3D animation - Page 166-168

3D objects - Page 635-644

change perspective - Page 639-641

exercises - Page 643-644

octahedron - Page 638

rendering in 2D - Page 114-119

defining 3D object with vectors - Page 114-116

exercises - Page 119

orienting faces and shading - Page 116-118

projecting to 2D - Page 116

transformations of - Page 123

composing - Page 126-129

drawing transformed objects - Page 124-126

exercises - Page 134-137

inventing geometric transformations - Page 131-133

rotating objects about an axis - Page 129-131

translating in 4D world - Page 196-199

Utah teapot - Page 641-643

3D space - Page 77

3D vectors. See vectors ==A== * [[ABC (abstract base class) - Page 211, 633

abc module - Page 211

abstract methods - Page 633

abstractions, building with functions - Page 17

acceleration - Page 342-344

activations - Page 560

calculating - Page 572-574

calculating activations of each subsequent layer based on activations in preceding layer - Page 572

each activation in next layer as function of all of activations in input layer - Page 570-571

in matrix notation - Page 574-576

return vector whose entries are activations of output layer - Page 572

setting input layer activations to entries of input vector - Page 570

addition * [[of 2D vectors - Page 32-35

of 3D vectors - Page 83-84

proving projection preserves vector sums - Page 182-183

amplitude - Page 469

Anaconda - Page 596-597

angles * [[converting from components to - Page 57-59

converting to components - Page 52-55

measuring with dot products - Page 97-100

of 3D vectors - Page 87-88

antiderivatives, integrals as - Page 387-388

append method - Page 615

Apply class - Page 369, 384

approximate equality (≈) - Page 54

approximate gradient - Page 450

arange function - Page 311

arctangent function - Page 66

arguments - Page 619

arithmetic * [[2D vectors * [[addition - Page 32-35

components and lengths - Page 35-37

exercises - Page 51

multiplying by numbers - Page 37-38

subtraction - Page 39-41

3D vectors - Page 83-92

addition - Page 83-84

angles and directions - Page 87-88

exercises - Page 89-92

lengths and distances - Page 86-87

scalar multiplication - Page 85

scalar subtraction - Page 85

subtraction - Page 86

arithmetic operators - Page 365, 607

Arrow class - Page 26, 72

Arrow3D class - Page 80-81, 89-90

artificial neural networks - Page 559

AssertionError - Page 214

asteroid arcade game - Page 258

accelerating spaceship - Page 343-344

constant velocity motion * [[adding velocities to asteroids - Page 339

adding velocities to objects - Page 339

keeping asteroids on screen - Page 340-342

updating game engine to move asteroids - Page 339-340

deciding whether laser hits an asteroid - Page 270

modeling game - Page 259

rendering game - Page 260-261

shooting laser - Page 261-262

astype method - Page 476

average_flow_rate function - Page 305-306

axes - Page 24

axn expression - Page 382

B

backpropagation - Page 562-592

calculating partial derivatives for last layer weights using chain rule - Page 590-591

exercises - Page 591-592

finding cost in terms of last layer weights - Page 590

basis - Page 243

bias - Page 573

bit depth - Page 467

BlackHole class - Page 396, 399

bounds of integration - Page 334

C

calculus - Page 9, 301, 303

callable classes - Page 225

Cartesian coordinates - Page 51-52, 54-55, 57, 68, 88, 129

chain rule - Page 383-384, 590-591

classes - Page 632-634

classification - Page 561

closed subspace - Page 239

coefficients - Page 511

column vectors - Page 160, 178

combinators - Page 359

combining terms - Page 370

commutative operations - Page 170

components, of 2D vectors - Page 35-37

converting angles to - Page 52-55

converting to angles - Page 57-59

composition of functions - Page 127

computer algebra - Page 371

constant velocity motion - Page 338-342

adding velocities to objects - Page 339

exercises - Page 342

keeping objects on screen - Page 340-342

updating game engine to move objects - Page 339-340

constructive interference - Page 480

constructors - Page 629

converging values - Page 333

CoordinateVector class - Page 216-217

cost functions - Page 501

cross products - Page 92, 103-113

computing - Page 109-110

exercises - Page 110-113

finding direction of - Page 106-108

finding length of - Page 108-109

overview - Page 103-106

currying - Page 129-130

D

Darcy’s law - Page 10

data classification * [[with logistic regression - Page 526-558

exploring possible logistic functions - Page 544-557

finding best logistic function - Page 551-557

framing classification as regression problem - Page 536-544

picturing decision boundary - Page 532-535

testing classification functions - Page 528-530

with neural networks - Page 561-568

building random digit classifier - Page 565-566

exercises - Page 567-568

measuring performance of digit classifier - Page 566

data collections - Page 610

lists - Page 610

indexing and slicing - Page 611-612

iterating over - Page 612

decision boundaries - Page 532-535

drawing better decision boundary - Page 533

exercises - Page 535

implementing classification function - Page 534-535

definite integral - Page 325

degrees of freedom - Page 285

denominator - Page 363

derivative method - Page 384

derivative of a function at the point - Page 318

derivative operators - Page 321

derivatives - Page 303, 315, 320

finding - Page 374-381

exercises - Page 379-381

of powers - Page 374-375

of products and compositions - Page 378-379

of special functions - Page 377-378

of transformed functions - Page 375-376

finding with computer algebra system - Page 355-356

taking automatically - Page 381-386

chain rule - Page 383-384

derivative method for expressions - Page 382-383

exercises - Page 386

power rule - Page 384-386

product rule - Page 383-384

destructive interference - Page 480

df/dx notation - Page 412

Difference combinator - Page 363

differential equations - Page 9

dimensions - Page 206, 243

directions * [[between 3D vectors - Page 87-88

between cross products - Page 106-108

displacement - Page 40, 85

distance * [[of 2D vectors - Page 39-41

of 3D vectors - Page 86-87

distinct_variables function - Page 366

distributive property - Page 369

does_collide function - Page 277

does_intersect function - Page 263, 270-271

dot products - Page 92-103

computing - Page 95-97

examples - Page 97

exercises - Page 100-103

measuring angles with - Page 97-100

picturing - Page 93-95

drag - Page 448

drawing 2D - Page 24

dt value - Page 329, 331, 349

E

Euler’s method - Page 344-353

building projectile simulation with - Page 426-427

carrying out by hand - Page 344-346

exercises - Page 349-353

implementing algorithm in Python - Page 346-347

running with smaller time steps - Page 348-353

evaluate function - Page 367-368

expand function - Page 370-371

exponential decay - Page 520

Expression class - Page 365, 367, 370, 373, 382

expression trees * [[building - Page 359-360

translating to Python - Page 360-362

expressions.py file - Page 360

F

faces, of 3D objects, orienting - Page 116-118

false positive - Page 530

feedforward method - Page 581-582

first-class values - Page 621

flow rate * [[approximating change in volume - Page 323-328

breaking up time into smaller intervals - Page 324-325

exercises - Page 328

for short time interval - Page 323-324

picturing on flow rate graph - Page 325-327

approximating instantaneous - Page 315-322

building function - Page 318-319

currying and plotting function - Page 320-321

exercises - Page 322

finding slope of small secant lines - Page 315-318

flow rate (continued) * [[calculating average - Page 305-309

average_flow_rate function - Page 305-306

negative rates of change - Page 308-309

picturing with secant line - Page 306-308

plotting average over time - Page 310-314

exercises - Page 313

finding average in different time intervals - Page 310-311

plotting interval flow rates - Page 311-313

force field - Page 392

four-dimensional vector - Page 5

Fourier series, decomposing sound waves into - Page 465-466, 486-495

defining inner products for periodic functions - Page 488-490

exercises - Page 494-495

finding vector components with inner products - Page 487-488

Fourier coefficients for other waveforms - Page 492-494

Fourier coefficients for square wave - Page 491-492

Fourier coefficients, writing function to find - Page 490-491

frequency - Page 464

function application - Page 359

Function class - Page 233

Function(Vector) class - Page 231

Function2(Vector) class - Page 233

functional programming paradigm - Page 621

functional programming techniques - Page 128

functions * [[building abstractions with - Page 17

derivatives of * [[finding - Page 374-381

taking automatically - Page 381-386

integrating symbolically - Page 387-389

exercises - Page 389

integrals as antiderivatives - Page 387-388

SymPy library - Page 388-389

treating as vectors - Page 223-226

viewing square and non-square matrices as - Page 180-181

G

gcf (get current figure) method - Page 628

global maximum value - Page 438

global minimum value - Page 438

gradient ascent - Page 424, 451, 453

gradient operator - Page 11

gradients - Page 393, 414

calculating with backpropagation - Page 588-592

calculating partial derivatives for last layer weights using chain rule - Page 590-591

exercises - Page 591-592

finding cost in terms of last layer weights - Page 589-590

optimizing projectile range using gradient ascent - Page 449-461

exercises - Page 457-461

finding uphill direction with gradient - Page 451-453

gradient of range function - Page 450-451

implementing gradient ascent - Page 453-457

plotting range vs. launch parameters - Page 449-450

training neural networks using gradient descent - Page 582-588

automatic training with scikit-learn - Page 585-586

calculating gradients with backpropagation - Page 584

exercises - Page 586-588

framing training as minimization problem - Page 582-584

gravitational fields - Page 393

H

hertz (Hz) - Page 471

hidden layers - Page 572

hypotenuse - 36, 112

hypothesis - Page 500

I

identity matrices - Page 171

identity transformation - Page 150, 170

images * [[manipulating with vector operations - Page 227-230

vector subspaces of - Page 245-248

ImageVector class - Page 228-229, 236, 254

immutable tuples - Page 615

independent system - Page 285

infinite dimensional - Page 226

inheritance - Page 206

inner products * [[defining for periodic functions - Page 488-490

finding vector components with - Page 487-488

input vectors - Page 190

instantaneous speed - Page 313

instantaneous value - Page 313

integral of the flow rate - Page 333

integration - Page 323

choosing right formula for line - Page 263-264

finding standard form equation for line - Page 265-267

linear equations in matrix notation - Page 267-268

solving linear equations with NumPy - Page 268-270

inverse trigonometric function - Page 58

items method - Page 618

J

Jupyter notebooks - Page 600-606

L

length * [[of 2D vectors - Page 35-37

of 3D vectors - Page 86-87

of cross products - Page 108-109

line of best fit - Page 501

linear algebra - Page 5, 199, 205

linear combination - Page 142

linear equations, systems of - Page 257

arcade game - Page 258

intersection points of lines - Page 270

modeling game - Page 259-260

rendering game - Page 260-261

shooting laser - Page 261-262

choosing right formula for line - Page 263-264

finding standard form equation for line - Page 265-267

linear equations in matrix notation - Page 267-268

solving linear equations with NumPy - Page 268-270

linear functions - Page 163, 181, 244, 501

linear maps * [[composing - Page 184-185

projection as - Page 181-184

proving projection preserves scalar multiples - Page 183-184

proving projection preserves vector sums - Page 182-183

linear regression - Page 3, 501

linear transformation - Page 138

linear transformations - Page 155, 163, 380

computing - Page 146-149

exercises - Page 149-155

picturing - Page 140-142

preserving vector arithmetic - Page 138-140

reasons for using - Page 142-146

representing with matrices - Page 159-170

3D animation with matrix transformations - Page 166-168

composing linear transformations by matrix multiplication - Page 163-166

exercises - Page 169-170

implementing matrix multiplication - Page 166

multiplying matrices with vector - Page 161, 163

writing vectors and linear transformations as matrices - Page 159-160

LinearFunction class - Page 244, 251-252

linearly dependent - Page 243

linearly independent - Page 242-243

lists - Page 610

indexing and slicing - Page 611-612

iterating over - Page 612

local maximum value - Page 438

local minimum value - Page 438

logistic functions - Page 527, 540

logistic regression - Page 526, 540-558

exploring possible logistic functions - Page 544-557

exercises - Page 549-551

measuring quality of fit - Page 546-547

parameterizing logistic functions - Page 545-546

testing - Page 548

finding best logistic function - Page 551-557

exercises - Page 555-557

gradient descent in 3D - Page 551-552

testing an understanding best logistic classifier - Page 554-555

using gradient descent - Page 552-553

framing classification as regression problem - Page 536-544

exercises - Page 543-544

measuring likelihood - Page 538-539

scaling raw data - Page 536-538

sigmoid function - Page 540-542

picturing decision boundary - Page 532-535

drawing better decision boundary - Page 533-534

exercises - Page 535

implementing classification function - Page 534-535

testing classification functions - Page 528-530

exercises - Page 530

loading car data - Page 529

testing classification function - Page 529-530

M

m-by-n matrix - Page 181, 188, 578

m -dimensional space - Page 227

m/s/s (meters per second per second) - Page 343

math anxiety - Page 2

math module - Page 16, 608-609

math, learning through software and code - Page 1-17

abstractions, building with functions - Page 17

real-world use cases - Page 2

comparing deals - Page 5

Mathematica - Page 355

Matplotlib * [[drawing - Page 73

drawing 2D vectors in - Page 72

matrices - Page 158-204

interpreting matrices of different shapes - Page 175-191

column vectors as matrices - Page 176-178

composing linear maps - Page 184-185

determining which pairs of matrices can be multiplied - Page 178-180

exercises - Page 186-191

projection as linear maps - Page 181-184

viewing square and non-square matrices as vector functions - Page 180-181

linear equations in matrix notation - Page 267-268

representing linear transformations with - Page 159-170

3D animation with matrix transformations - Page 166-168

composing linear transformations by matrix multiplication - Page 163-166

exercises - Page 169-170

implementing matrix multiplication - Page 166

multiplying matrices with vector - Page 161-163

writing vectors and linear transformations as matrices - Page 159-160

translating vectors with - Page 191-203

combining translation with other linear transformations - Page 195-196

exercises - Page 199-203

finding 3D matrix for 2D translation - Page 194-195

making plane translations linear - Page 191-194

translating 3D objects in 4D world - Page 196-199

treating as vectors - Page 226-227

Matrix class - Page 227, 234

matrix dimensions - Page 179

matrix multiplication - Page 161

matrix notation - Page 159

maxima, identifying - Page 437-438

measuring functions - Page 511

meters per second per second (m/s/s) - Page 343

minima, identifying - Page 437-438

MLP (multilayer perceptron) - Page 568

evaluating - Page 580-581

implementing MLP class - Page 578-580

testing performance of - Page 581-582

MLP class - Page 577, 585, 588

MLPClassifier class - Page 588

modulus operator (%) - Page 607, 620

motion simulation - Page 337-353

acceleration - Page 342-344

constant velocity motion - Page 338-342

adding velocities to objects - Page 339

exercises - Page 342

updating game engine to move objects - Page 339-340

Euler’s method - Page 344-353

carrying out by hand - Page 344-346

exercises - Page 349-353

implementing algorithm in Python - Page 346-347

running with smaller time steps - Page 348-353

move method - Page 340-341, 344, 401, 404

Mul constructor - Page 388

multilayer perceptron. See MLP (multilayer perceptron) * [[multiplication * [[matrix * [[composing linear transformations by - Page 163-166

implementing - Page 166

when to use - Page 178-180

of 2D vectors by numbers - Page 37-38

scalar * [[of 3D vectors - Page 85

proving projection preserves scalar multiples - Page 183-184

N

n -dimensional space - Page 227

n -dimensional vectors - Page 181

Negative combinator - Page 364

negative gradient - Page 415

negative slope - Page 308

neural networks - Page 559-593

building in Python - Page 577-582

evaluating MLP - Page 580-581

exercises - Page 582

implementing MLP class - Page 578-580

testing performance of MLP - Page 581-582

calculating gradients with backpropagation - Page 588-592

calculating partial derivatives for last layer weights using chain rule - Page 590-591

exercises - Page 591-592

finding cost in terms of last layer weights - Page 589-590

classifying images of handwritten digits - Page 561-568

building 64-dimensional image vectors - Page 563-565

building random digit classifier - Page 565-566

exercises - Page 567-568

measuring performance of digit classifier - Page 566

designing - Page 568-577

calculating activations - Page 572, 576

data flow - Page 569-572

exercises - Page 576-577

organizing neurons and connections - Page 568-569

training using gradient descent - Page 582-588

automatic training with scikit-learn - Page 585-586

calculating gradients with backpropagation - Page 584

exercises - Page 586-588

framing training as minimization problem - Page 582-584

neurons - Page 560

noise - Page 467

Number class - Page 386

numerator - Page 363

NumPy library - Page 268-270

O

octahedrons - Page 114-115, 118, 635-638

OOP (object-oriented programming) - Page 206-634

class methods - Page 632

inheritance and abstract classes - Page 632-634

OpenGL - Page 635-644

change perspective - Page 639-641

exercises - Page 643-644

octahedron - Page 635-638

Utah teapot - Page 641-643

operator overloading - Page 208

origin - Page 7, 22

outliers - Page 500

output layer - Page 572

P

parabola - Page 440

periodic functions - Page 471

pi (π) symbol - Page 56

pitch - Page 471

pixels - Page 227

planes - Page 22

play() method - Page 470

Points class - Page 26, 72

Points3D class - Page 80

polar coordinates - Page 51-52, 68, 129

Polygon class - Page 26, 72

PolygonModel class - Page 259, 262, 270, 276, 340

Polynomial class - Page 233

Power class - Page 385

power rule - Page 384-386

preserve vector sums - Page 139

Product constructor - Page 388

product rule - Page 378, 383-384

projectile simulation - Page 422-462

building with Euler’s method - Page 426-427

calculating optimal range - Page 432-440

as function of launch angle - Page 432-435

exercises - Page 439-440

identifying maxima and minima - Page 437-438

solving for maximum range - Page 435-437

enhancing - Page 440-447

adding another dimension - Page 441-442

exercises - Page 447

modeling terrain around cannon - Page 442-443

solving for range of projectile in 3D - Page 443-446

optimizing using gradient ascent - Page 449-461

exercises - Page 457-461

finding uphill direction with gradient - Page 451-453

gradient of range function - Page 450-451

implementing gradient ascent - Page 453-457

plotting range vs. launch parameters - Page 449-450

testing - Page 425-426

exercises - Page 429-432

exploring different launch angles - Page 428-429

measuring properties of trajectory - Page 427-428

proofs, writing - Page 214

properties - Page 629

property-based testing - Page 152

pure functions - Page 16

PyGame - Page 635-644

change perspective - Page 639-641

exercises - Page 643-644

octahedron - Page 635-638

Utah teapot - Page 641-643

pyplot - Page 624

Pythagorean theorem - Page 36, 57, 86

Python - Page 595-634

building neural networks in - Page 577-582

evaluating MLP - Page 580-581

exercises - Page 582

implementing MLP class - Page 578-580

testing classification performance of MLP - Page 581-582

checking for existing installation - Page 595-596

data collections - Page 610

Python (continued) * [[downloading and installing Anaconda - Page 596-597

drawing 2D vectors in - Page 26-29

drawing 3D vectors in - Page 80-81

numbers and math - Page 607-610

math module - Page 608-609

random numbers - Page 609-610

object-oriented programming - Page 634

class methods - Page 632

inheritance and abstract classes - Page 632-634

symbolic algebra in - Page 356-358

translating expression trees to - Page 360-362

trigonometry in - Page 56-57

using in interactive mode - Page 597-606

creating and running script file - Page 598-600

using Jupyter notebooks - Page 600-606

Q

quadratic formula - Page 434

quadratic functions - Page 245

QuadraticFunction(Vector) class - Page 252

quants - Page 2

Quotient combinator - Page 363

R

radians - Page 56-57

random module - Page 609

random numbers - Page 609-610

range of projectiles * [[calculating optimal - Page 432-440

as function of launch angle - Page 432-435

exercises - Page 439-440

identifying maxima and minima - Page 437-438

solving for maximum range - Page 435-437

optimizing using gradient ascent - Page 449-461

exercises - Page 457-461

finding uphill direction with gradient - Page 451-453

gradient of range function - Page 450-451

implementing gradient ascent - Page 453-457

plotting range vs. launch parameters - Page 449-450

solving for in 3D - Page 443-446

ranges - Page 613

rates of change - Page 303-337

approximating change in volume - Page 323-328

breaking up time into smaller intervals - Page 324-325

exercises - Page 328

for short time interval - Page 323-324

picturing on flow rate graph - Page 325-327

approximating instantaneous flow rates - Page 315-322

building function - Page 318-319

currying and plotting function - Page 320-321

exercises - Page 322

finding slope of small secant lines - Page 315-318

calculating average flow rate - Page 305-309

average_flow_rate function - Page 305-306

exercises - Page 309

negative rates of change - Page 308-309

picturing with secant line - Page 306-308

plotting average flow rate over time - Page 310-314

exercises - Page 313-314

finding average in different time intervals - Page 310-311

plotting interval flow rates - Page 311-313

plotting volume over time - Page 328-335

definite and indefinite integrals - Page 334-335

finding volume over time - Page 328-329

improving approximation - Page 332-333

picturing Riemann sums for volume function - Page 329-331

Rectangle class - Page 629-630, 632-634

reflection - Page 152

regression - Page 499

REPL (read-evaluate-print loop) - Page 597

reverse() method - Page 619

RGB (red, green, and blue) - Page 227

Riemann sum - Page 327, 488

right-hand rule - Page 106

rotating objects - Page 129-131

row vector - Page 177

S

saddle point - Page 451

sampling - Page 465, 473

scalar field - Page 394

scalar multiplication - Page 37

scalars - Page 37

scale function - Page 211

scaling - Page 37

scikit-learn library - Page 562

secant lines * [[finding slope of small - Page 315, 318

picturing average flow rate - Page 306, 308

second derivatives - Page 344

Segment class - Page 26, 28, 72

segments() method - Page 277

set_size_inches method - Page 628

shading, of 3D objects - Page 116-118

Shape class - Page 633-634

Ship class - Page 261

sigmoid function - Page 540-542

sinusoidal functions - Page 466

sinusoidal waves - Page 471-477

building familiar function with - Page 483-486

building linear combination of - Page 481-482

changing frequency of - Page 473-475

exercises - Page 477

making audio from - Page 471-473

sampling and playing sound waves - Page 475-476

sort() method - Page 619

sound wave analysis - Page 463-496

combining sound waves - Page 465-466, 478-486

adding sampled sound waves to build chords - Page 478-479

building familiar function with sinusoids - Page 483-486

building linear combination of sinusoids - Page 481-482

exercises - Page 486

picturing sum of two sound waves - Page 479-481

decomposing sound waves into Fourier series - Page 465-466, 486-495

defining inner products for periodic functions - Page 488-490

exercises - Page 494-495

finding vector components with inner products - Page 487-488

Fourier coefficients for other waveforms - Page 492-494

Fourier coefficients for square wave - Page 491-492

Fourier coefficients, writing function to find - Page 490-491

playing sound waves in Python - Page 466, 471

exercises - Page 471

playing musical notes - Page 469-471

producing sound - Page 467-469

turning sinusoidal wave into sound - Page 471-477

changing frequency of sinusoid - Page 473-475

exercises - Page 477

making audio from sinusoidal functions - Page 471-473

sampling and playing sound wave - Page 475-476

spacetime - Page 197

spans - Page 240-243

spherical coordinates - Page 88

Square class - Page 632

square wave - Page 470

standard form - Page 264

steepest ascent - Page 415

steepest descent - Page 415

steepness - Page 408

straight-line path - Page 26

subscripts - Page 569

subtraction * [[of 2D vectors - Page 39-41

of 3D vectors - Page 85-86

Sum constructor - Page 388

summation symbol - Page 282

superscripts - Page 569

Symbol constructor - Page 388

symbolic expressions - Page 354-391

evaluating expressions - Page 366-369

exercises - Page 372-374

expanding expressions - Page 369-371

finding all variables in expressions - Page 365-366

finding derivatives - Page 374-381

exercises - Page 379-381

of powers - Page 374-375

of products and compositions - Page 378-379

of special functions - Page 377-378

of transformed functions - Page 375-376

with computer algebra system - Page 355-356

integrating functions symbolically - Page 387-389

exercises - Page 389

integrals as antiderivatives - Page 387-388

SymPy library - Page 388-389

modeling algebraic expressions - Page 358-365

breaking expressions into pieces - Page 358-359

building expression trees - Page 359-360

exercises - Page 362-365

translating expression trees to Python - Page 360-362

symbolic algebra in Python - Page 356-358

taking derivatives automatically - Page 381-386

chain rule - Page 383-384

derivative method for expressions - Page 382-383

exercises - Page 386

power rule - Page 384-386

product rule - Page 383-384

symbolic programming - Page 356, 371

SymPy library - Page 388-389

systems of linear equations - Page 257

arcade game - Page 258

intersection points of lines - Page 270

modeling game - Page 259-260

rendering game - Page 260-261

shooting laser - Page 261-262

choosing right formula for line - Page 263-264

finding standard form equation for line - Page 265-267

linear equations in matrix notation - Page 267-268

solving linear equations with NumPy - Page 268-270

T

t values - Page 473

t%1 value - Page 491

taking a derivative - Page 320

tangent of the angle - Page 54

theorems - Page 13

tick() method - Page 640

timbre - Page 464, 476

tip-to-tail addition - Page 33, 43

tolerance - Page 453

transformations - Page 67-72,]] - Page 121-157

combining - Page 69-70

exercises - Page 71-72

linear transformations - Page 138-155

computing - Page 146-149

exercises - Page 149-155

picturing - Page 140-142

preserving vector arithmetic - Page 138-140

reasons for using - Page 142-146

representing with matrices - Page 159-170

of 3D objects - Page 123-137

composing vector transformations - Page 126-129

drawing transformed objects - Page 124-126

exercises - Page 134-137

inventing geometric transformations - Page 131-133

rotating objects about an axis - Page 129-131

translating by a vector - Page 125

translating vectors with matrices * [[combining translation with other linear transformations - Page 195-196

exercises - Page 199-203

finding 3D matrix for 2D translation - Page 194-195

making plane translations linear - Page 191-194

translating 3D objects in 4D world - Page 196-199

translations - Page 191-203

transposition - Page 188

Triangle class - Page 634

trigonometric function - Page 54

trigonometry - Page 51-67

converting from angles to components - Page 52-55

converting from components to angles - Page 57-59

exercises - Page 60-67

in Python - Page 56-57

true negative - Page 530

true positive - Page 530

Turing complete - Page 16

U

uniform - Page 609

unit testing - Page 214-215

updating game engine to move objects - Page 339

Utah teapot - Page 641-643

V

variable bindings - Page 367

Variable class - Page 368

Variable constructor - Page 388

Vec0 class - Page 220

Vec1 class - Page 219-220

Vec2 class - Page 207-208, 210-213, 215, 218, 251

creating - Page 207-208

improving - Page 208-209

Vec3 class - Page 209-213, 218

vector addition - Page 32

Vector base class - Page 210-212

Vector class - Page 211-212, 214-215, 219, 221, 226, 228, 230, 234-235, 244, 367

vector mathematics - Page 22

vector operations, image manipulation with - Page 227-230

vector products - Page 88

vector spaces - Page 206, 213, 219-237

defined - Page 212-213

enumerating all coodinate vector space - Page 219-221

exercises - Page 230-237

identifying - Page 221-223

manipulating images with vector operations - Page 227-230

treating functions as vectors - Page 223-226

treating matrices as vectors - Page 226-227

unit testing vector space classes - Page 214-215

vector subspaces - Page 237-250

dimensions - Page 243-244

exercises - Page 248-250

identifying - Page 238-240

of images - Page 245-248

of vector space of functions - Page 244-245

spans - Page 240-243

starting with single vector - Page 240

vector sums - Page 33, 154

vector_drawing module - Page 27

vectors - Page 5, 21, 24-120

arithmetic for 2D * [[addition - Page 32-35

components and lengths - Page 35-37

displacement - Page 39-41

distance - Page 39-41

exercises - Page 42-51

multiplying by numbers - Page 38

subtraction - Page 39-41

arithmetic for 3D - Page 83-92

addition - Page 83-84

angles and directions - Page 87-88

exercises - Page 89-92

lengths and distances - Page 86-87

scalar multiplication - Page 85

subtraction - Page 85-86

column vectors as matrices - Page 176-178

defining 3D objects with - Page 114-116

multiplying matrices with - Page 161-163

overview - Page 22-24

transformations - Page 67-72

combining - Page 69-70

exercises - Page 71-72

treating functions as - Page 223-226

treating matrices as - Page 226-227

viewing square and non-square matrices as vector functions - Page 180-181

writing as matrices - Page 159-160

volume * [[approximating change in - Page 323-328

W

weights - Page 573

white noise - Page 469

Z

zero method - Page 217-218

zero vector - Page 230

zero-dimensional subspace - Page 240

zero-dimensional vector - Page 220

zero-indexed - Page 610

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