enciclopedia matematica - volume 1 - analisi, algebra, geometria
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ENCICLOPEDIA MATEMATICA
di Corrado Brogi
http://spazioweb.libero.it/corradobrogi
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Prefazione
A mio modesto (anzi modestissimo) parere questa enciclopedia in sette volumi è uno dei
lavori più chiari e completi sulla matematica presenti in Rete. L’Enciclopedia può essere utilizzata
sia come riferimento per richiamare concetti noti, sia come fonte di studio introduttiva su pressoché
tutti gli argomenti di matematica di comune interesse. Ho raccolto in sette documenti PDF tutto il
contenuto del sito Web dell’autore (sperando di non aver contravvenuto alle sue volontà) in modo
da rendere il materiale facilmente consultabile ed eventualmente stampabile.
Buona lettura.
Stillbelieveit
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Volume I
La Matematica, La Geometria, L'Analisi per chi voglia ripartire da zero.
INDICE Premessa 1 Posizione del problema 1 La memoria - la comunicabilità - I linguaggi 2 Cosa memorizzare 3 Le quantità 4 Le frazioni 6 Criteri di divisibilità - Numeri primi 10 L'algebra 12 L'equivalenza di due frazioni - Le proporzioni 17 Le regole del tre semplice e del tre composto 19 Ripartizione direttamente proporzionale 24 Ripartizione inversamente proporzionale 25 Applicazioni geometriche 26 La similitudine 29 Gli Angoli 34 I principali teoremi della geometria 45 Alcune applicazioni grafiche del II° teorema di Euclide 48 Altre costruzioni grafiche 51 Confronto di due segmenti 53 Commensurabilità ed incommensurabilità di segmenti 54 Le numerazioni 59 La sezione aurea 63 Il decagono regolare 67 Gli angoli notevoli 68 Come costruire graficamente gli angoli notevoli 69 Dividere il cerchio in un numero qualsiasi di parti uguali 71 Alcuni teoremi dedotti dalla proporzionalità 74 Teorema di Talete 77 La linearizzazione 78 I diagrammi ed il concetto di funzione 79 Il simbolismo 84 Le cinque proprietà delle potenze 96 I logaritmi 101 Le radici, i radicali, le potenze ad esponente frazionario 108 Prodotti notevoli 109 Divisibilità di binomi 109 Razionalizzazione del denominatore 110 Metodi per calcolare le radici 110 I radicali doppi 111 Gli operatori 112
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Le successioni 119 Progressione aritmetica 120 Progressioni geometriche 122 Frazione generatrice di un numero periodico 126 Il fattoriale 127 Numeri reali e numeri immaginari 128 L'operatore immaginario 129 I numeri complessi 133 Tavola dei numeri 141 I numeri - le misure - le dimensioni - l'equivalenza 142 Le aree 146 Integrazione grafica 151 Le capacità - i volumi 152 Principio di Cavalieri 154 Le equazioni 158 Equazioni lineari 161 Equazioni indipendenti 164 Metodo di Cramer o dei determinanti 166 Sistemi lineari in "n" incognite 171 Equazioni di grado superiore al primo 183 Equazioni di secondo grado 184 Equazioni reciproche 187 Equazioni biquadratiche 193 Sui vari modi di considerare un espressione algebrica 194 Luoghi geometrici 196 La traslazione di assi 199 Il cerchio 201 La parabola 204 Regola dei segni di Cartesio 211 I passaggi algebrici e l'assurdo 213 Le disequazioni 215 Disequazioni di primo grado 218 Disequazioni qualsiasi 224 La discussione delle equazioni di 2° grado 229 La trigonometria 235 Correlazioni fra le funzioni trigonometriche 240 Rappresentazioni grafiche 242 Angoli notevoli 245 Le tavole trigonometriche 246 Le funzioni somma e differenza di angoli 248 Formule di duplicazione 250 Formule di bisezione 250 Formule parametriche 252 Formule di Prostaferesi 253 I teoremi per la risoluzione dei triangoli 254 Teorema dei seni 254
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Area del triangolo 254 Teorema di Carnot 255 Teorema di Neper 256 Formule di Brigg 257 Risoluzione di triangoli (tavola) 258 Funzioni trigonometriche inverse 259 Giochetti sulle funzioni trigonometriche 260 La simmetria 262 Il triangolo 268 Punti notevoli di un triangolo 270 Circocentro 270 Ortocentro 271 Incentro 274 Lunghezza delle bisettrici 276 Exincentri 277 Distanze degli exincentri dai vertici 281 Baricentro 282 Complementi sul triangolo 285 Teorema di Ceva 290 Teorema di Menelao 291 Retta di Eulero 292 Retta di Simpson 293 Cerchio dei nove punti 293 I punti di Brocard 294 I poligoni regolari 295 Triangolo equilatero 299 Quadrato 299 Pentagono 300 Esagono 303 Ettagono 304 Ottagono 310 Ennagono 311 Decagono 313 Undecagono 315 Dodecagono 317 Il processo di esaustione 319 Le classi contigue 321 Il calcolo infinitesimale 323 I differenziali 328 Le derivate 330 I limiti 334 Paradosso di Achille e la tartaruga 334 Operazioni di passaggio al limite 337 Limiti notevoli (trigonometrici) 341 Limiti notevoli (derivanti da "e") 343 Tabella dei limiti notevoli 347
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Ordine degli infiniti e degli infinitesimi 348 Metodi di calcolo dei limiti 350 Calcolo di derivate 351 Regole di derivazione di funzioni 355 Derivata logaritmica 359 Derivata esponenziale 360 Derivata di funzioni inverse 361 Calcolo dei massimi e dei minimi di una funzione 364 Le derivate di ordine "n" 365 Flessi 367 I teoremi di Rolle, di Lagrange, di Cauchy 369 Generalizzazione delle formule di Cauchy e di Lagrange 375 Teoremi di De L'Hospital 376 Artifizi per poter applicare i teoremi di De L'Hospital 377 Limiti a destra, limiti a sinistra 380 Lo studio delle funzioni 381 Semplificazioni 381 Valori assoluti 381 campi di esistenza 383 Operazioni preparatorie all'esposizione dati 385 Calcolo di altri punti caratteristici 387 Calcolo di asintoti 388 Il grafico della funzione 390 Esercizio 393 Il calcolo integrale 400 Saper portare sotto il segno di differenziale 408 Metodi di integrazione 412 Integrazione per parti 414 Artifizi di integrazione 416 Integrazione di funzioni razionali fratte 421 Se il polinomio al denominatore ha solo radici immaginarie 424 Se il polinomio al denominatore ha due radici reali 425 Integrazione di irrazionali algebrici 427 Integrali impropri o generalizzati 429 Formula di Wallis 434 Quadro riepilogativo delle figure geometriche elementari 438 Solidi elementari 442 Triangoli pitagorici 451 Triangoli pitagorici aventi un lato comune 451 Triangoli eroniani 453 Costruzione approssimata della circonferenza rettificata 464 Calcolo numerico approssimato 465 Sistema metrico decimale 471 Misure di valore 481 Tavole di ragguaglio di Antiche Misure Italiane ed Estere (precedenti l'adozione del sistema metrico decimale) 485
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misure di superficie 489 misure di capacità 491 misure di peso 494 monete antiche italiane 498 Tavole di confronto per misure lineari 501 Considerazioni di aritmetica Razionale 508 Le operazioni aritmetiche (Somma) 514 Le operazioni aritmetiche (Sottrazione) 515 Le operazioni aritmetiche (Moltiplicazione) 516 Le operazioni aritmetiche (Divisione) 526 Prova del 9 per le quattro operazioni 532 Le lunule e le figure piane delimitate da archi di cerchio 536 L'arbelo 538 Il salinon 539 Drepanoide 540 Pelecoide 543 Prisma Prismatoide Prismoide 546 Centri isogonici; metapoli; punti isodinamici 550
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