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Começamos por ferramentas que certamente usaremos na empreitada. -\subsection{O sistema Hindu-Arábico de numerais} +\subsection{O sistema hindu-arábico de numerais} \newcommand{\al}{al-Khow\={a}rism\={\i}} @@ -390,12 +390,12 @@ que é 1 ---, teríamos marcado um {\tt x} só nos algarismos {\tt 123456}, indicando que ainda consideraremos os algarismos {\tt 789123456789} ao longo da computação do algoritmo. -Por que se chama divisão longa? Pra ser distinguida com divisão -curta. Na divisão longa, escrevemos a multiplicação parcial do -quociente pelo divisor logo abaixo do dividendo parcial; em seguida, -efetuamos a subtração. Na divisão curta, omitimos a subtração tal que -escrevemos abaixo do dividendo apenas o resultado da subtração, -tornando a verificação do trabalho mais difícil. Também exibida pela +Por que se chama divisão longa? Pra ser distinguida da divisão curta. +Na divisão longa, escrevemos a multiplicação parcial do quociente pelo +divisor logo abaixo do dividendo parcial; em seguida, efetuamos a +subtração. Na divisão curta, omitimos a subtração tal que escrevemos +abaixo do dividendo apenas o resultado da subtração, tornando a +verificação do trabalho mais difícil. Também exibida pela Figura~\ref{fg-alkw-long-div} (na página \pageref{fg-alkw-long-div}), desenhamos uma seta indicando o produto do algarismo corrente do quociente (corrente durante a execução do algoritmo) que deu origem ao @@ -450,8 +450,8 @@ página~\pageref{fg-alkw-long-div}). -------- 42042 --> resto \end{verbatim} - \caption{A multiplicação entre 123456789 por 987654321 segundo \al. - O quociente é 1002711021689 e o resto, 42042. Se tivéssemos + \caption{A divisão de 123456789123456789 por 123123 segundo \al. O + quociente é 1002711021689 e o resto, 42042. Se tivéssemos interromido o algoritmo em qualquer momento, teríamos no papel tudo que precisamos pra continuá-lo. Por falta de espaço, exibimos a tabuada de 123123 na @@ -504,7 +504,7 @@ Estudemos agora a divisão entre 322 e 37. A tabuada revela imediatamente que o primeiro algarismo do quociente é 8. O resto de $26 < 37$ revela que terminamos. Desejando a parte fracionada da expansão decimal da divisão, continuamos. O zero que -justamos ao 26 representa o fato de que multiplicamos o resto 26 por +justapomos ao 26 representa o fato de que multiplicamos o resto 26 por 10. Por causa disso, precisaremos corrigir o quociente por via de uma divisão por 10 --- o que é feito imediatamente com a adição do ponto decimal. Encontramos o algarismo 7 e como resto obtemos $1 < @@ -517,7 +517,7 @@ coisa, mas no contexto em que estamos significa uma quantidade menor que 1. O termo ``periódica'' é evidenciado pelo fato de que temos um ciclo, o que representamos pelos colchetes --- ou seja, $322 \div 37 = 8.702[702] = 8.[702]$. A justificativa para esse sistema de -computação em papel é +computação em papel é % \begin{align*} \frac{322}{37} &= 8 + \frac{26}{37}\\ @@ -530,7 +530,11 @@ computação em papel é &= 8 + 0.7 + 0.002 + 0.702 \cdot\frac{1}{1000}\\ &= 8 + 0.702 + 0.000702 + \cdots \end{align*} -% + +Cabe aqui agora uma explicação do porquê essas equações justificam o +algoritmo da divisão---em outras palavras, estamos sem uma explicação +por que o algoritmo funciona. + \begin{figure}[!htb] \centering \begin{verbatim} @@ -699,7 +703,7 @@ transcendentais cujos algarismos jamais poderiam ser obtidos por um programa de computador. A relevância do fato é que programa de computador é apenas uma realização da expressibilidade dos nossos sistemas de cálculo. Logo, a limitação se aplica a nós --- porque os -computadores são apenas uma realização do nossos sistemas de +computadores são apenas uma realização dos nossos sistemas de raciocínio. Temos números ``demais'' na linha real no sentido de que não temos (e não teremos) uma linguagem expressiva o suficiente pra alcançá-los, o que nos leva a questionar nossa concepção intelectual @@ -936,7 +940,7 @@ esse quadrado, fica faltando adicionar $(b/2a)^2$. O lado desse quadrado é, portanto, $x + b/2a$, o que nos dá a identidade % \begin{align*} - (x + \fr{b}{2a})^2 = x^2 + \fr{b}{a} + \fr{b^2}{4a^2}. + \left(x + \fr{b}{2a}\right)^2 = x^2 + \fr{b}{a} + \fr{b^2}{4a^2}. \end{align*} % Assim, podemos resolver a equação por computar @@ -968,7 +972,7 @@ conjugados $(t + 3)(t - 3)$. %% sendo $a, b$ números racionais --- digamos. Essa função é %% tipicamente chamada de conjugação. Ela pode ser vista como uma %% generalização da conjugação complexa restrita a inteiros -%% gaussianos, que não números complexos da forma $a + bi$, sendo $a, +%% gaussianos, que são números complexos da forma $a + bi$, sendo $a, %% b$ inteiros. A generalização, portanto, é relaxar os inteiros %% gaussianos e permitir racionais. Se definirmos $f$ assim, obtemos %% uma contradição quando $n$ é um quadrado perfeito. Por exemplo, @@ -1018,6 +1022,12 @@ $a, b$ estão em $\{3, 9\}$, mas é preciso saber se tentamos $(5x + 3)(x + 9)$ ou $(5x + 9)(x + 3)$. Devemos considerar a segunda opção primeiro já que $9\times 5 = 45$, muito grande pra $24$. +Dito tudo isso, o fato é que estamos sempre resolvendo um sistema +não-linear em que $ab$ é um inteiro e $Lb + a$ é outro. A +investigação acima é meramente uma tentativa de nos clarificar como +pensar no problema---como procurar por soluções sem recorrer à equação +quadrática. + \subsubsection{Fatoração de soma e diferença de cubos} Outros produtos notáveis são a {\em Soma de cubos} e a {\em diferença @@ -1839,7 +1849,7 @@ Parece que Newton usou alguma ideia bem diferente, evitando o sistema de equações lineares, mas não sei o que ele fez. Fica aí como um exercício pra um dia descobrir o que ele fez e catalogar a técnica. -\section{Definitions, limits and continuity} +\section{Limits and continuity} \masciidef{} The absolute value of a number $x$, written $|x|$ is the distance from 0 to $x$ on the real line. Distances are always positive, @@ -6676,8 +6686,8 @@ observações sobre o tema. Uma equação diferencial é chamada de {\em ordinária} se a função que soluciona a equação depende de apenas uma única váriavel independente, ou seja, a função desejada é $y(x)$. Se ela depender de mais de uma -variável, então trata-se de uma equação diferencial parcial, quando a -solução é uma função como, por exemplo, $y(x_1, x_2)$. +variável, então trata-se de uma equação diferencial {\em parcial}, +quando a solução é uma função como, por exemplo, $y(x_1, x_2)$. Uma equação é chamada de {\em separável de primeira ordem} quando ela pode ser escrita como @@ -6828,7 +6838,7 @@ diferencial linear de primeira ordem. \begin{definition}[linear] Uma equação diferencial é linear de $n$-ésima ordem se ela é da forma % - $$f_n(x) y^(n) + f_{n-1}(x)y^(n - 1) + \cdots + f_2(x)y'' + f_(1) y' + f_o(x)y = Q(x).$$ + $$f_n(x) y^{(n)} + f_{n-1}(x)y^{(n - 1)} + \cdots + f_2(x)y'' + f_1(x) y' + f_0(x)y = Q(x).$$ % \end{definition} @@ -6839,25 +6849,30 @@ resolver a linear de primeira ordem, que é da forma % \begin{remark*} Note que uma equação como $q(x)y' + P(x)y = Q(x)$ ainda é linear de -primeira ordem porque basta a gente reescrevê-la pra $y' + 1/q(x) -P(x)y = 1/q(x) Q(x)$. +primeira ordem porque basta a gente reescrevê-la pra +% + $$y' + \fr{1}{q(x)} P(x)y = \fr{1}{q(x)} Q(x).$$ +% \end{remark*} - +% É nesse tipo de equação que usamos o chamado ``método do fator -integrante'', que exibiremos agora. O método nos instruir a procurar +integrante'', que exibiremos agora. O método nos instrui a procurar por $f(x)$ que quando multiplicada em ambos os lados da equação produza a regra do produto $f(x)y' + f'(x)y = (f(x)y)'$. Ocorre que uma tal função de $x$ pode ser sempre $\exp(\int P(x) \dx)$. ``Por -quê?'' é a pergunta interesse aqui. Antes de exibirmos o porquê, +quê?'' é a pergunta de interesse aqui. Antes de exibirmos o porquê, vejamos o método em ação. \begin{exercise*} -Resolva a equação $y' + \fr{1}{x}y = x^2$. \sol. Primeiro computamos -$\exp(\int x^{-1} \dx) = \exp(\ln x) = x$. Observe que queremos uma -função $f(x)$ qualquer, então podemos sempre tomar $c = 0$ na -integração de $P(x)$. Agora multiplique $x$ em ambos lados da equação, -obtendo $xy' + x x^{-1}y = x^3$. Observe que o lado esquerdo é -exatamente $(xy)'$ e, portanto, temos +Resolva a equação +% + $$y' + \fr{1}{x}y = x^2.$$ +% +\sol. Primeiro computamos $\exp(\int x^{-1} \dx) = \exp(\ln x) = +x$. Observe que queremos uma função $f(x)$ qualquer, então podemos +sempre tomar $c = 0$ na integração de $P(x)$. Agora multiplique $x$ em +ambos lados da equação, obtendo $xy' + x x^{-1}y = x^3$. Observe que o +lado esquerdo é exatamente $(xy)'$ e, portanto, temos % \begin{align*} xy' + x x^{-1}y &= x^3\\ @@ -6874,52 +6889,35 @@ O método ilustra o fato de que a gente sempre recorre às regras de derivação pra conseguir uma integração. Uma das perguntas importantes aqui é por que o fator integrante é -$\int P(x) \dx$. A resposta é na verdade simples. Estamos em busca de -uma função $I(x)$ tal que +$\int P(x) \dx$. A resposta é na verdade simples. Estamos em busca +de uma função $M(x)$ tal que % $$(M(x)y)' = M(x)y' + M(x)P(x)y.$$ % -Expandindo o lado esquerdo, obtemos $M'(x)y + M(x)y$. Agora iguale o +Expandindo o lado esquerdo, obtemos $M'(x)y + M(x)y'$. Agora iguale o lado esquerdo desejado com o lado direito pra obter a equação % - $$M'(x)y + M(x)y = M(x)y' + M(x)P(x)y = M(x)P(x)y + M(x)y'.$$ + $$M'(x)y + M(x)y' = M(x)P(x)y + M(x)y'.$$ % O que nos diz essa equação? Diz-nos que a função que estamos -procurando satisfaz $M'(x) = M(x) P(x)$. Essa é uma nova equação -diferencial que na verdade já conhecemos---ela é uma separável de -primeira ordem. Encontramos uma solução pra ela lendo-a em -palavras---que função $M(x)$ conhecemos que quando derivada resulta -nela própria vezes alguma coisa? Essa função $M(x)$ tem que ser -certamente uma exponencial porque só as exponenciais têm a propriedade -de que a velocidade de seu crescimento depende de si própria. Esse é -um fato importante aqui. Agora, pra solucionar a charada por completo, -declare $u'(t) = P(x)$ (pra alguma transformação que leve $t$ a $x$) -de forma que $u(t) = \int P(x) \dx$ e reescreva o problema como +procurando satisfaz $M'(x) = M(x) P(x)$. Que função é essa? Qual é a +função que quando derivada depende dela própria? É a função +exponencial de base $e$. Mas e o fator $P(x)$? O fator $P(x)$ só +aparece na derivada da exponencial quando ele é a derivada do +expoente. E $P(x)$ é a derivada de $\int P(x) \dx$. Logo, % - $$M'(x) = M(x) u'(t).$$ -% -Já sabemos que $M(x)$ é uma exponencial e a regra da cadeia implica -que a variável independente $x$ de $M(x)$ tem que ser $x = u(t)$. -Assim, obtemos -% - $$M'(x) = (M(u(t)))' = M'(u(t))u'(t)$$ -% -que implica -% - $$M(x) = \int M'(x) \dx = \exp\left(\int P(x) \dx\right) P(x) = \exp\left(P(x) \dx\right),$$ -% -como desejado. + $$M(x) = \exp\left(\int P(x) \dx\right).$$ -\begin{remark*} -Essa dedução ficou prolixa. Existe a resposta curta e de pouca emoção -que é o fato de que +Agora sabemos que se multiplicarmos ambos os lados de uma equação +diferencial linear de primeira ordem por $\exp\left(\int P(x) +\dx\right)$, obteremos no lado esquerdo da equação a derivada do +produto de $\exp\left(\int P(x) \dx\right)$ por $y$, ou seja, +obteremos a derivada de % - $$\exp\left(\int P(x) \dx\right)' = \exp(\Int P(x) \dx) P(x) + c.$$ + $$y \exp\left(\int P(x) \dx\right),$$ % -Mas todos nós sabemos que é bom exercício induzir a resposta a partir -de um problema anterior em vez de magicamente aparecer com a resposta -e mostrar que ela o soluciona. -\end{remark*} +restando apenas integrarmos o lado direito da equação com respeito a +$x$. \subsection{As lineares de segunda ordem} @@ -6935,11 +6933,17 @@ em que $f_2(x)$, $f_1(x)$, $f_0(x)$ são constantes, ou seja, só sabemos resolver as equações % $$ay'' + by' + cy = g(x).$$ -% -É possível resolver essa equação pra qualquer $g(x)$ diferenciável, -mas no momento só sabemos tratar o problema quando $g(x)$ toma as -formas de um polinômio, uma exponencial, uma função trigonométrica ou -uma combinação dessas. + +As mais fáceis de resolver são aquelas em que $g(x) = 0$ e essas são +chamadas de {\em homogêneas}. As outras em que $g(x) \neq 0$ são +chamadas de {\em não-homogêneas}. + +\ghostsection{As homogêneas}. \lipsum[1] + +É possível resolver as não-homogêneas pra qualquer $g(x)$ +diferenciável, mas no momento só sabemos tratar o problema quando +$g(x)$ toma as formas de um polinômio, uma exponencial, uma função +trigonométrica ou uma combinação dessas. Consideremos primeiro o caso de um polinômio como o da equação % @@ -6947,16 +6951,13 @@ Consideremos primeiro o caso de um polinômio como o da equação % Um fato curioso sobre essa equação é que a solução geral dela é obtida a partir de uma solução particular. E como uma solução particular é -obtida? Observe o que diz essa equação. Ela diz que $y(x)$ é uma +obtida? Observe o que diz essa equação. Ela diz que $y(x)$ é uma função tal que sua segunda derivada mais duas vezes sua primeira derivada mais três vezes ela própria é um polinômio quadrático. A gente pode especular que $y(x)$ é um polinômio também porque todas as -derivadas de um polinômio é um polinômio, incluindo suas somas e +derivadas de um polinômio são um polinômio, incluindo suas somas e constantes múltiplos. -Hm. Na verdade comecei pelas não-homogêneas. Vamos retroceder e -considerar as homogêneas primeiro, discutindo especialmente a palavra -``homogênea''. \newpage \tableofcontents \newpage \printbibliography