# 10: Integral multivariável - Matemática

10: Integral multivariável - Matemática

Notas da aula - Resumo da semana 13 (PDF)

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## Bolas e esferas

• Leia o Capítulo 0 das notas online
• Jogue com os miniaplicativos SAGE na seção 0.3 das notas online ou diretamente no SageMathCell.
Você pode definir uma função de 2 variáveis ​​com:
def f (x, y):
return (x ^ 2) / 9 - (y ^ 2) / 4
Então você pode desenhar seu gráfico com:
plot3d (f, (- 5,5), (- 5,5), frame = True, axes = False, adaptive = True, color = rainbow (60, 'rgbtuple'))
Ou seu nível é definido com:
contour_plot (f, (-5,5), (-5,5), fill = False, contours = 20, labels = True, label_inline = True)
• Trabalhe nas questões em sala de aula sobre o interior / fechamento / limite de um conjunto.
• Trabalhe nas questões da seção 0.P.

## Recursos

• Escrito por um professor de matemática premiado com 30 anos de experiência de ensino
• Exposição realista
• Definições totalmente rigorosas, declarações de teoremas e provas ilustrativas
• Referências externas para provas mais técnicas
• As seções são divididas em Básico, Mais Profundidade e, quando apropriado, + Álgebra Linear
• Observações à margem e referências históricas
• Os hiperlinks são fornecidos para o conteúdo da Wikipedia sobre álgebra linear
• Soluções de vídeo para selecionar exercícios
• Formato PDF, compatível com todos os computadores, tablets e dispositivos móveis
• Baixo custo em formato eletrônico ou impresso

## 10: Integral multivariável - Matemática

10 melhores livros para cálculo multivariável pdf

### Melhores livros de texto de cálculo multivariável

MathSchoolinternational contém mais de 5000 livros em PDF gratuitos de matemática e livros em PDF gratuitos de física. Que cobrem quase todos os tópicos para alunos de Matemática, Física e Engenharia. Aqui está uma lista extensa de e-books de Cálculo. Esperamos que os alunos e professores gostem desses livros, notas e manuais de solução.

Como estudante de engenharia, economia, matemática ou física, você precisa fazer um curso de cálculo. mathschoolinternational fornece uma coleção abrangente dos seguintes melhores livros, melhores manuais de solução e notas resolvidas que serão úteis para você e aumentarão sua eficiência para qualquer curso de cálculo.
• Os 10 melhores cálculos de variável única
• Top 10 melhores cálculos mutivariáveis
• Top 5 melhores cálculos AP
• Os 7 melhores cálculos com geometria analítica
• Melhor cálculo transcendental inicial
• 25 melhores cálculos resolvidos

No cálculo multivariável (também conhecido como cálculo multivariado), estudamos funções de duas ou mais variáveis ​​independentes, por exemplo, f (x, y) = yx ou f (x, y, z) = xyz + yz enquanto o cálculo de variável única você estuda funções de uma única variável independente. Por exemplo, f (x) = 3x.

O cálculo tem muitas subdivisões, o cálculo de múltiplas variáveis ​​lida com funções de uma variável. Por exemplo, f (x) = 3x tem uma variável x, então o cálculo de variável única é incluído neste tipo de variável, enquanto o cálculo multivariável estuda funções de múltiplas variáveis ​​reais. Por exemplo f (x, y) = xyz ou f (x, y, z) = xy + yz.
Aqui estão mais de 10 melhores livros de múltiplas variáveis ​​que cobrem os seguintes tópicos importantes.
• Sequneces e séries infinitas
• Derivada parcial ou diferenciação parcial
• Diferenciação de funções variáveis ​​multivariáveis
• Integração de funções de variáveis ​​multivariáveis
• Integrais impróprios
• Funções vetoriais, curvas, superfícies e geometria em Sapace
• Cálculo e análise vetorial
• Equações diferenciais e tópicos adicionais em equações diferenciais
• O teorema do curl e de Stokes
• O teorema de Green é um teorema importante no cálculo multivariável, que é uma generalização do primeiro teorema fundamental do cálculo para duas dimensões.
livros didáticos de cálculo de múltiplas variáveis ​​são melhores para alunos de pós-graduação e de nível avançado.

Clique no título de cada livro para ver seu conteúdo, estudar online ou fazer download. Você pode baixá-los gratuitamente para seu celular, iPad, PC ou pen drive. Você também pode pesquisar um livro de matemática escrevendo o nome do livro ou o nome do autor na caixa de pesquisa da MathSchool fornecida no topo deste site. Observe também que esta lista será constantemente atualizada, portanto, mantenha contato com a página.

Bruchkov Yu.A., Glaeske H.-J., Prudnikov A.P., Vu K.T.,Transformadas Integrais Multidimensionais, Geest & amp Portig K.G., Leipzig e D. Reidel Publ., Amsterdam, (a aparecer).

Exton H.,Manual de integrais hipergeométricos, teoria, aplicativos, tabelas, programas de computador, Halsted Press (Ellis Horwood), John Wiley and Sons, Chichester-New York-Brisbane-Toronto, 1978.

Lauricella G.,Sulle funzioni ipergeometriche a più variabili, Rend. Circ. Esteira. Palermo7 (1893), 111–158.

## Termos

Observe que as informações a seguir estão sujeitas a alterações.

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• If a student misses a test without a valid reason, the mark for the test will remain a zero, regardless of exam performance. See Course Policy below.

### Course Policy

• All tests are mandatory, and can only be missed through pre-arrangement, or due to illness/family emergency. Notification by email is required within 7 days of the missed test in the case of illness/family emergency.
• Missed tests (with appropriate notification) will not be re-taken the 5% for the test will be added to the exam for the current term. Missed tests without a reason/notification will be given a grade of zero.
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### Final and Midterm Examination

Midterm exam: The date and time will be scheduled during the Fall term final examination period. You will write your midterm exam at the same exam centre location as the final proctored exam.

The final exam will be held during the Winter term final examination period. The exam schedule will be determined approximately eight weeks before the start of the exam period.

Students must write their exam on the day and time scheduled by the University. The start time may vary slightly depending on the off-campus exam centre. Do not schedule vacations, appointments, etc., during the exam period.

## Math Insight

Changing variables in triple integrals is nearly identical to changing variables in double integrals. You can view this story as a second chance to understand the basics underlying a change of variables. You can also read a more traditional introduction to changing variables in triple integrals.

Imagine that you are an engineer designing a new electrode tip that will be used to monitor brain activity in people with epilepsy. The end product will be an array of electrodes that will be implanted in the area of a patient's brain where doctors believe the epileptic seizures begin. Once the array of electrodes is implanted in the brain, doctors will be able to monitor the brain's activity before and during a seizure to pinpoint exactly where the seizure begins.

Currently, for severe cases of epilepsy that don't respond to other treatment, the recommended treatment is removing the small part of brain tissue that starts the seizure. Since doctors want to remove as little of the brain as possible, implanting electrode arrays is necessary to find exactly what part of the brain is causing the trouble.

You, however, are working with a team of scientists looking for methods to stop the seizure that don't involve removing parts of the brain. The idea is the following. The electrode arrays can detect patterns of brain activity that signal the seizure is about to begin. What if one can use the electrodes to disrupt that pattern of activity so it doesn't develop into a full blown epileptic seizure? If one passes small pulses of current through the electrodes with the right timing, the hope is that a seizure can be avoided.

With this goal and many other technical requirements in mind, you have developed a prototype electrode tip whose shape is given by the solid $dlv$ diagrammed below. You have optimized the electrical properties of the electrode tip both through developing the shape $dlv$ and through incorporating different metal alloys along different parts of the electrode.

A three dimensional domain of an electode trip. The solid $dlv$ represents a tip of an electrode.

You are quite excited about your new design. You show it to the chief scientist on the project. She likes your design, but has one concern. She wants you to calculate the total charge that may build up on the electrode to make sure it doesn't reach unsafe levels.

From the metal composition of your electrode tip and the information the chief scientist gives you about the current, you quickly calculate $g(x,y,z)$, the charge density at each point $(x,y,z)$ within the electrode tip. To find the total charge, all you need to do is calculate the triple integral egin exto = iiint_dlv g(x,y,z) , dV, end where $dlv$ is the solid representing the electrode tip.

You struggle with calculating the integral, but, especially given the complicated shape of the electrode, you find that the integral is too difficult to solve directly. Then, you remember something you learned in your multivariable calculus course: changing variables in triple integrals.

#### Step 1: Integrate over new region $dlv^*$

Instead of trying to directly integrate $g(x,y,z)$ over $dlv$, you realize you could solve your problem by finding a change of variables egin (x,y,z) = cvarf(cvarfv,cvarsv,cvartv) end that maps a simpler solid $dlv^*$ onto the complicated solid $dlv$. Then, rather than integrating over $dlv$, you could integrate over $dlv^*$.

After a long night's worth of calculations, you come up with a function $cvarf(cvarfv,cvarsv,cvartv)$ that looks promising. The effect of the function $cvarf$ is demonstrated below. The function $cvarf$ maps a simple solid $dlv^*$ (shown in the first panel, below) in $(cvarfv,cvarsv,cvartv)$-space to the electrode tip $dlv$ (shown in the second panel, below) in $(x,y,z)$-space. We often say that $cvarf(cvarfv,cvarsv,cvartv)$ parametrizes $dlv$ for $(cvarfv,cvarsv,cvartv)$ in $dlv^*$.

A change of variables for an electrode tip domain. A change of variables $cvarf$ maps a rectangular solid $dlv^*$ (first panel) onto the electrode tip $dlv$ (second panel). You can explore the mapping by moving with the mouse the red point in $dlv^*$, which represents the point $(cvarfv,cvarsv,cvartv)$, or the point blue point in $dlv$, which represents the point $(x,y,z)$. When you move one point, the other point moves to reflect the mapping so that $(x,y,z)= cvarf(cvarfv,cvarsv,cvartv)$.

#### Step 2: Compose function $g$ with change of variables function $cvarf$.

Since it is so late after you finished finding $cvarf(cvarfv,cvarsv,cvartv)$ and you are tired, you almost start to integrate $g$ directly over the solid $dlv^*$. Fortunately, you realize that such a procedure doesn't make sense because $g$ is a function of $(x,y,z)$ and hence is defined over the electrode tip $dlv$. Since it is not a function of $(cvarfv,cvarsv,cvartv)$, you can't integrate $g$ over the solid $dlv^*$ in $(cvarfv,cvarsv,cvartv)$-space. At this point, you are too tired to figure out what you were supposed to do, so you go to bed without finishing your calculation of what the total charge on the electrode would be. You sleep fitfully, dreaming of electrodes that were charged to dangerous levels attacking you.

The next morning, you wake up still groggy, but immediately realize the solution to your problem. Although $g(x,y,z)$ is defined only on your electrode $dlv$, you can simply compose $g(x,y,z)$ with $cvarf(cvarfv,cvarsv,cvartv)$ to obtain the function $g(cvarf(cvarfv,cvarsv,cvartv))$ that is defined in terms of $(cvarfv,cvarsv,cvartv)$. For a given point $(cvarfv,cvarsv,cvartv)$ (shown by the red point, above), the composition $g(cvarf(cvarfv,cvarsv,cvartv))$ gives the density at the point $(x,y,z)=cvarf(cvarfv,cvarsv,cvartv)$ (shown by the blue point, above) on the electrode $dlv$. By integrating $g(cvarf(cvarfv,cvarsv,cvartv))$ over the solid $dlv^*$ in $(cvarfv,cvarsv,cvartv)$-space, you will integrate $g(x,y,z)$ over the solid $dlv$ and compute the total charge. You feel pretty smug that you had this realization even before breakfast and coffee.

#### Step 3: Include a factor to account for change in volume

Unfortunately, you are still too groggy after your fitful sleep to remember everything you learned in multivariable calculus. You completely forget that $(x,y,z)=cvarf(cvarfv,cvarsv,cvartv)$ will change volume in $dlv^*$ compared to volume in $dlv$. Without compensating for this effect of the map $cvarf$, your calculations assume that volume in $dlv$ is the same as volume in $dlv^*$.

Glancing at the above diagram is enough to guess what the result of this error might be. The volume of $dlv^*$ is larger than the volume of $dlv$. Remember that total charge equals charge density times volume. If one uses the correct charge density $g$ of the electrode but multiplies by volume in $dlv^*$ rather than volume in $dlv$, the calculation for the total charge should be larger than the correct answer. If, for example, $dlv^*$ was uniformly twice as large as $dlv$, then this incorrect calculation for total charge would give an answer that was twice the actual value.

Not realizing this mistake, you proceed with your incorrect calculation and integrate $g(cvarf(cvarfv,cvarsv,cvartv))$ over $dlv^*$ without compensating for change in volume. Your result makes you so upset that you kick the table in anger, hurting your foot and spilling your coffee all over your calculations. But at that point, you don't care. You rip up your coffee-soaked notes and limp back to bed to cry. Your design is useless. The total charge on your electrode is much too large. Your hard work is more likely to fry people's brains than help them overcome epilepsy. Or so you think.

Once you stop crying, you discover that your throbbing foot has helped lift your grogginess and your mind is actually becoming clear. Finally, you remember the strict admonition of your calculus professor to never forget to compensate for change in volume when changing variables. You realize that maybe your electrode won't cook people's brains and you might be able to salvage your design.

You grab a fresh piece of paper, race back to your table, and find a dry spot on your table to work. You first sketch the following diagram to show how dividing $dlv^*$ into little boxes divides $dlv$ into small pieces. The diagram helps you visualize how changing variables changes volume. You see that, especially around the critical tip of the electrode, the volume in $dlv$ is much smaller than the volume in $dlv^*$. This gives you hope that less charge may build up on the electrode than you originally calculated.