Eng_Basico_P7-2_Gabarito-watermark
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Eng_Basico_P7-2_Gabarito-watermark


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1 
 
AVALIAÇÃO PRESENCIAL 
 
curso: Engenharia \u2013 Ciclo Básico bimestre: 12o bimestre data: / /2017 
P7-2 polo: mediador responsável: grupo (dia da semana, período, no): 
nome: RA: 
 
Utilize preferencialmente folhas sulfite, identificando em cada uma delas, frente e verso, com seu R.A. Evite 
escrever no canto superior direito das folhas de resposta. Boa prova! 
 
 
disciplina Cálculo II NOTA (0-10): 
 
Utilize o resumo/roteiro de estudos disponibilizado ao final da prova para consulta. 
 
 
Questão 1 (2,5 pontos) 
A temperatura em uma superfície é dada por T = \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66). Uma partícula se desloca sobre esta superfície 
pela curva \ud835\udefe\ud835\udefe(\ud835\udc61\ud835\udc61) = (2\ud835\udc61\ud835\udc61 \u2212 1, 4 + 2\ud835\udc61\ud835\udc612). Sabe-se que \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
(3,12) = \u22127 e \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
(3,12) = 2. Determine se no instante t = 
2 a temperatura na partícula está aumentando ou diminuindo e a que taxa? 
 
Questão 2 (2,5 pontos) 
Calcule a massa do sólido descrito a seguir: 
 
\ud835\udc37\ud835\udc37 = {(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67) \u2208 \u211d3|\ud835\udc65\ud835\udc652 + \ud835\udc66\ud835\udc662 \u2264 1, 0 \u2264 \ud835\udc67\ud835\udc67 \u2264 12 \u2212 \ud835\udc65\ud835\udc652 \u2212 \ud835\udc66\ud835\udc662} com densidade \ud835\udeff\ud835\udeff(\ud835\udc65\ud835\udc65, \ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67) = \ud835\udc65\ud835\udc652 + \ud835\udc66\ud835\udc662 
 
Questão 3 (2,5 pontos) 
Considere o campo vetorial no plano dado por \ufffd\u20d7\ufffd\ud835\udc39(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66) = (4\ud835\udc65\ud835\udc653\ud835\udc66\ud835\udc663 \u2212 2\ud835\udc66\ud835\udc66)\ud835\udea4\ud835\udea4 + 3\ud835\udc65\ud835\udc654\ud835\udc66\ud835\udc662\ud835\udea5\ud835\udea5 . 
 
a) Calcule o rotacional de \ufffd\u20d7\ufffd\ud835\udc39. 
b) \ufffd\u20d7\ufffd\ud835\udc39 é um campo conservativo? 
c) Calcule o trabalho realizado pelo campo \ufffd\u20d7\ufffd\ud835\udc39 sobre a trajetória formada pela circunferência de centro 
(3, 5) e raio 9, percorrida no sentido anti-horário, e pela circunferência de centro (3, 5) e raio 4 
percorrida no sentido horário. 
 
Questão 4 (2,5 pontos) 
Considere o campo \ufffd\u20d7\ufffd\ud835\udc39(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67) = 4\ud835\udc67\ud835\udc67\ud835\udea4\ud835\udea4 + 3\ud835\udc65\ud835\udc652\ud835\udea5\ud835\udea5 + 5
3
\ud835\udc67\ud835\udc673\ud835\udc58\ud835\udc58\ufffd\u20d7 e a superfície da esfera \ud835\udc46\ud835\udc46 \u2236 \ud835\udc65\ud835\udc652 + \ud835\udc66\ud835\udc662 + \ud835\udc67\ud835\udc672 = 4, orientada 
para fora. 
 
a) Calcule o divergente de \ufffd\u20d7\ufffd\ud835\udc39 , div\ufffd\u20d7\ufffd\ud835\udc39. 
b) Calcule o fluxo de \ufffd\u20d7\ufffd\ud835\udc39 através de \ud835\udc46\ud835\udc46, \u222c \ufffd\u20d7\ufffd\ud835\udc39 \u2219 \ud835\udc5b\ud835\udc5b\ufffd\u20d7 \ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc46\ud835\udc46 . 
 
2 
 
_____________________ 
ROTEIRO DE ESTUDOS 
 
Funções contínuas. Uma função \ud835\udc53\ud835\udc53 é contínua em (\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f) se ela está definida em (\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f), tem limite quando 
(\ud835\udc65\ud835\udc65, \ud835\udc66\ud835\udc66) \u2192 (\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f) e vale 
lim
(\ud835\udf15\ud835\udf15,\ud835\udf15\ud835\udf15)\u2192(\ud835\udc4e\ud835\udc4e,\ud835\udc4f\ud835\udc4f)
\ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66) = \ud835\udc53\ud835\udc53(\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f). 
 
Regra da Cadeia para funções de várias variáveis: 
 
1º Caso: Suponha \ud835\udc67\ud835\udc67 = \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65, \ud835\udc66\ud835\udc66) seja uma função \ud835\udc36\ud835\udc361 , 
 \ud835\udc65\ud835\udc65 = \ud835\udc54\ud835\udc54(\ud835\udc61\ud835\udc61) e \ud835\udc66\ud835\udc66 = \u210e(\ud835\udc61\ud835\udc61) são funções deriváveis de \ud835\udc61\ud835\udc61. 
Então \ud835\udc67\ud835\udc67 é uma função derivável em \ud835\udc61\ud835\udc61 e 
\ud835\udc85\ud835\udc85\ud835\udc85\ud835\udc85
\ud835\udc85\ud835\udc85\ud835\udc85\ud835\udc85
=
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
.
\ud835\udc85\ud835\udc85\ud835\udf4f\ud835\udf4f
\ud835\udc85\ud835\udc85\ud835\udc85\ud835\udc85
+
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
.
\ud835\udc85\ud835\udc85\ud835\udf4f\ud835\udf4f
\ud835\udc85\ud835\udc85\ud835\udc85\ud835\udc85
 
 
2º Caso: Suponha \ud835\udc67\ud835\udc67 = \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66) seja uma função \ud835\udc36\ud835\udc361, 
 \ud835\udc65\ud835\udc65 = \ud835\udc54\ud835\udc54(\ud835\udc60\ud835\udc60, \ud835\udc61\ud835\udc61) e \ud835\udc66\ud835\udc66 = \u210e(\ud835\udc60\ud835\udc60, \ud835\udc61\ud835\udc61) são funções de classe \ud835\udc36\ud835\udc361 
então \ud835\udc67\ud835\udc67 é uma função derivável de \ud835\udc60\ud835\udc60 e \ud835\udc61\ud835\udc61 e temos: 
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
(\ud835\udf4f\ud835\udf4f, \ud835\udc85\ud835\udc85) =
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
(\ud835\udf4f\ud835\udf4f,\ud835\udf4f\ud835\udf4f).
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
(\ud835\udf4f\ud835\udf4f, \ud835\udc85\ud835\udc85) +
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
(\ud835\udf4f\ud835\udf4f,\ud835\udf4f\ud835\udf4f).
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
(\ud835\udf4f\ud835\udf4f, \ud835\udc85\ud835\udc85) 
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
(\ud835\udf4f\ud835\udf4f, \ud835\udc85\ud835\udc85) =
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
(\ud835\udf4f\ud835\udf4f,\ud835\udf4f\ud835\udf4f).
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
(\ud835\udf4f\ud835\udf4f, \ud835\udc85\ud835\udc85) +
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
(\ud835\udf4f\ud835\udf4f,\ud835\udf4f\ud835\udf4f).
\ud835\udf4f\ud835\udf4f\ud835\udf4f\ud835\udf4f
\ud835\udf4f\ud835\udf4f\ud835\udc85\ud835\udc85
(\ud835\udf4f\ud835\udf4f, \ud835\udc85\ud835\udc85) 
 
Polinômio de Taylor. Ordem 2: 
Q(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66) = \ud835\udc53\ud835\udc53(\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f) + \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
(\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f)(\ud835\udc65\ud835\udc65 \u2212 \ud835\udc4e\ud835\udc4e) + \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
(\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f)(\ud835\udc66\ud835\udc66 \u2212 \ud835\udc4f\ud835\udc4f) + 
1
2
\ufffd
\ud835\udf15\ud835\udf152\ud835\udc53\ud835\udc53
\ud835\udf15\ud835\udf15\ud835\udc65\ud835\udc652
(\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f)(\ud835\udc65\ud835\udc65 \u2212 \ud835\udc4e\ud835\udc4e)2 + 2
\ud835\udf15\ud835\udf152\ud835\udc53\ud835\udc53
\ud835\udf15\ud835\udf15\ud835\udc65\ud835\udc65\ud835\udf15\ud835\udf15\ud835\udc66\ud835\udc66
(\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f)(\ud835\udc65\ud835\udc65 \u2212 \ud835\udc4e\ud835\udc4e)(\ud835\udc66\ud835\udc66 \u2212 \ud835\udc4f\ud835\udc4f) +
\ud835\udf15\ud835\udf152\ud835\udc53\ud835\udc53
\ud835\udf15\ud835\udf15\ud835\udc66\ud835\udc662
(\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f)(\ud835\udc66\ud835\udc66 \u2212 \ud835\udc4f\ud835\udc4f)2\ufffd 
Pontos de máximo/mínimo: (\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f) ponto crítico de \ud835\udc67\ud835\udc67 = \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66). 
 D = \ufffd
\ud835\udf15\ud835\udf152\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf152
\ud835\udf15\ud835\udf152\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf152\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf152\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf152
\ufffd=\ud835\udf15\ud835\udf15
2\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf152
\u2219 \ud835\udf15\ud835\udf15
2\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf152
\u2212 \ud835\udf15\ud835\udf15
2\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\u2219 \ud835\udf15\ud835\udf15
2\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
 
D > 0 e \ud835\udf15\ud835\udf15
2\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf152
> 0 então (\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f) é ponto de mínimo (local) 
D > 0 e \ud835\udf15\ud835\udf15
2\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf152
< 0 então (\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f) é ponto de máximo (local) 
D < 0 então (\ud835\udc4e\ud835\udc4e, \ud835\udc4f\ud835\udc4f) não é ponto de máximo nem de mínimo 
 
Teorema de Fubini em 2 variáveis (caso geral): 
\ud835\udc37\ud835\udc37 = {(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66)|\ud835\udc4e\ud835\udc4e \u2264 \ud835\udc65\ud835\udc65 \u2264 \ud835\udc4f\ud835\udc4f \ud835\udc52\ud835\udc52 \ud835\udc54\ud835\udc54(\ud835\udc65\ud835\udc65) \u2264 \ud835\udc66\ud835\udc66 \u2264 \u210e(\ud835\udc65\ud835\udc65)} 
\ufffd\ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65, \ud835\udc66\ud835\udc66)\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66
\ud835\udc37\ud835\udc37
= \ufffd \ufffd \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66)\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65
\u210e(\ud835\udf15\ud835\udf15)
\ud835\udc54\ud835\udc54(\ud835\udf15\ud835\udf15)
\ud835\udc4f\ud835\udc4f
\ud835\udc4e\ud835\udc4e
 
 
Teorema de Fubini em 3 variáveis. Domínio paralelepípedo reto retângulo: 
\ufffd \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65, \ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67)\ud835\udc51\ud835\udc51\ud835\udc67\ud835\udc67\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65
\ud835\udc37\ud835\udc37
= \ufffd \ufffd \ufffd \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67)\ud835\udc51\ud835\udc51\ud835\udc67\ud835\udc67\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65
\ud835\udf15\ud835\udf15
\ud835\udc52\ud835\udc52
\ud835\udc51\ud835\udc51
\ud835\udc50\ud835\udc50
\ud835\udc4f\ud835\udc4f
\ud835\udc4e\ud835\udc4e
 
Em 3 variáveis, caso geral: 
\ufffd\ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65, \ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67)\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51 = \ufffd \ufffd \ufffd \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65, \ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67)\ud835\udc51\ud835\udc51\ud835\udc67\ud835\udc67\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65
\ud835\udc63\ud835\udc63(\ud835\udf15\ud835\udf15,\ud835\udf15\ud835\udf15)
\ud835\udc62\ud835\udc62(\ud835\udf15\ud835\udf15,\ud835\udf15\ud835\udf15)
\u210e(\ud835\udf15\ud835\udf15)
\ud835\udc54\ud835\udc54(\ud835\udf15\ud835\udf15)
\ud835\udc4f\ud835\udc4f
\ud835\udc4e\ud835\udc4e\ud835\udc37\ud835\udc37
 
 
 
 
3 
 
Significado físico da integral tripla: massa de um sólido calculada a partir da distribuição de massa 
(densidade). 
Volume de um sólido espacial D = \u222d 1\ud835\udc51\ud835\udc51\ud835\udc67\ud835\udc67\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65\ud835\udc37\ud835\udc37 
 
Coordenadas polares no plano: 
\ud835\udc65\ud835\udc65 = \ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc60\ud835\udc60\ud835\udc5f\ud835\udc5f 0 \u2264 \ud835\udc5f\ud835\udc5f \u2264 2\ud835\udf0b\ud835\udf0b 
\ud835\udc66\ud835\udc66 = \ud835\udc5f\ud835\udc5f\ud835\udc60\ud835\udc60\ud835\udc52\ud835\udc52\ud835\udc5b\ud835\udc5b\ud835\udc5f\ud835\udc5f 0 \u2264 \ud835\udc5f\ud835\udc5f 
Jacobiano = \ud835\udc5f\ud835\udc5f \ud835\udc65\ud835\udc652 + \ud835\udc66\ud835\udc662 = \ud835\udc5f\ud835\udc5f2 
\u222c \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66)\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66 =\ud835\udc37\ud835\udc37\ud835\udc65\ud835\udc65\ud835\udc65\ud835\udc65 \u222c \ud835\udc53\ud835\udc53(\ud835\udc5f\ud835\udc5f,\ud835\udc5f\ud835\udc5f) \u2219 \ud835\udc5f\ud835\udc5f \u2219 \ud835\udc51\ud835\udc51\ud835\udc5f\ud835\udc5f\ud835\udc51\ud835\udc51\ud835\udc5f\ud835\udc5f\ud835\udc37\ud835\udc37\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f . 
 
Coordenadas Cilíndricas no espaço tridimensional 
\ud835\udc65\ud835\udc65 = \ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc60\ud835\udc60\ud835\udc5f\ud835\udc5f 0 \u2264 \ud835\udc5f\ud835\udc5f \u2264 2\ud835\udf0b\ud835\udf0b 
\ud835\udc66\ud835\udc66 = \ud835\udc5f\ud835\udc5f\ud835\udc60\ud835\udc60\ud835\udc52\ud835\udc52\ud835\udc5b\ud835\udc5b\ud835\udc5f\ud835\udc5f 0 \u2264 \ud835\udc5f\ud835\udc5f 
\ud835\udc67\ud835\udc67 = \ud835\udc67\ud835\udc67 \ud835\udc67\ud835\udc67 \u2208 \u211d Jacobiano = \ud835\udc5f\ud835\udc5f 
 \ud835\udc65\ud835\udc652 + \ud835\udc66\ud835\udc662 = \ud835\udc5f\ud835\udc5f2 
 
Coordenadas Esféricas no espaço tridimensional 
\ud835\udc65\ud835\udc65 = \ud835\udf0c\ud835\udf0c\ud835\udc60\ud835\udc60\ud835\udc52\ud835\udc52\ud835\udc5b\ud835\udc5b\ud835\udc60\ud835\udc60\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc60\ud835\udc60\ud835\udc5f\ud835\udc5f 0 \u2264 \ud835\udc5f\ud835\udc5f \u2264 2\ud835\udf0b\ud835\udf0b 
\ud835\udc66\ud835\udc66 = \ud835\udf0c\ud835\udf0c\ud835\udc60\ud835\udc60\ud835\udc52\ud835\udc52\ud835\udc5b\ud835\udc5b\ud835\udc60\ud835\udc60\ud835\udc60\ud835\udc60\ud835\udc52\ud835\udc52\ud835\udc5b\ud835\udc5b\ud835\udc5f\ud835\udc5f 0 \u2264 \ud835\udf0c\ud835\udf0c 
\ud835\udc67\ud835\udc67 = \ud835\udf0c\ud835\udf0c\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc5f\ud835\udc60\ud835\udc60\ud835\udc60\ud835\udc60 0 \u2264 \ud835\udc60\ud835\udc60 \u2264 \ud835\udf0b\ud835\udf0b Jacobiano = \ud835\udf0c\ud835\udf0c2\ud835\udc60\ud835\udc60\ud835\udc52\ud835\udc52\ud835\udc5b\ud835\udc5b\ud835\udc60\ud835\udc60 
 \ud835\udc65\ud835\udc652 + \ud835\udc66\ud835\udc662 + \ud835\udc67\ud835\udc672 = \ud835\udf0c\ud835\udf0c2 
 
Integral de linha de campo escalar: 
\u222b \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67)\ud835\udc51\ud835\udc51\ud835\udc60\ud835\udc60 =\ud835\udefe\ud835\udefe \u222b \ud835\udc53\ud835\udc53(\ud835\udefe\ud835\udefe(\ud835\udc61\ud835\udc61)) \u2219 \ufffd\ud835\udefe\ud835\udefe
\u2032\ufffd\ufffd\ufffd\u20d7 (\ud835\udc61\ud835\udc61)\ufffd| \ud835\udc51\ud835\udc51\ud835\udc61\ud835\udc61\ud835\udc4f\ud835\udc4f\ud835\udc4e\ud835\udc4e significado: massa de um fio a partir da densidade linear. 
Na geometria o comprimento de uma curva é \u222b 1\ud835\udc51\ud835\udc51\ud835\udc60\ud835\udc60 =\ud835\udefe\ud835\udefe \u222b 1 \u2219 \ufffd\ud835\udefe\ud835\udefe\u2032\ufffd\ufffd\ufffd\u20d7 (\ud835\udc61\ud835\udc61)\ufffd| \ud835\udc51\ud835\udc51\ud835\udc61\ud835\udc61
\ud835\udc4f\ud835\udc4f
\ud835\udc4e\ud835\udc4e 
 
Integral de linha de campo vetorial: 
\ud835\udf0f\ud835\udf0f = \u222b \ufffd\u20d7\ufffd\ud835\udc39\ud835\udc51\ud835\udc51\ud835\udc5f\ud835\udc5f =\ud835\udefe\ud835\udefe \u222b \ufffd\u20d7\ufffd\ud835\udc39(\ud835\udefe\ud835\udefe(\ud835\udc60\ud835\udc60)) \u2219 \ud835\udefe\ud835\udefe
\u2032\ufffd\ufffd\ufffd\u20d7 (\ud835\udc60\ud835\udc60)\ud835\udc51\ud835\udc51\ud835\udc60\ud835\udc60\ud835\udc4f\ud835\udc4f\ud835\udc4e\ud835\udc4e significado: trabalho realizado pela campo de forças \ufffd\u20d7\ufffd\ud835\udc39, sobre a trajetória 
\ud835\udefe\ud835\udefe. 
Outra notação: \u222b \ufffd\u20d7\ufffd\ud835\udc39\ud835\udc51\ud835\udc51\ud835\udc5f\ud835\udc5f =\ud835\udefe\ud835\udefe \u222b \ud835\udc43\ud835\udc43\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65 + \ud835\udc44\ud835\udc44\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66 + \ud835\udc45\ud835\udc45\ud835\udc51\ud835\udc51\ud835\udc67\ud835\udc67\ud835\udefe\ud835\udefe 
 
Gradiente; Divergente; Rotacional. 
 
Gradiente: \ud835\udefb\ud835\udefb\ufffd\u20d7 \ud835\udc53\ud835\udc53(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67) = (\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
, \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
, \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
) 
Divergente: \ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51\ufffd\u20d7\ufffd\ud835\udc39 = \ud835\udefb\ud835\udefb \u2219 \ufffd\u20d7\ufffd\ud835\udc39 = \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
+ \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
+ \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
 
Rotacional no plano: \ud835\udc45\ud835\udc45\ud835\udc5f\ud835\udc5f\ud835\udc61\ud835\udc61\ufffd\u20d7\ufffd\ud835\udc39 = \ufffd\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\u2212 \ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ufffd \ud835\udc58\ud835\udc58\ufffd\u20d7 
Rotacional no Espaço: \ud835\udc45\ud835\udc45\ud835\udc5f\ud835\udc5f\ud835\udc61\ud835\udc61\ufffd\u20d7\ufffd\ud835\udc39 = \ufffd
\ud835\udea4\ud835\udea4 \ud835\udea5\ud835\udea5 \ud835\udc58\ud835\udc58\ufffd\u20d7
\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15
\ud835\udf15\ud835\udf15\ud835\udf15\ud835\udf15
\ud835\udc43\ud835\udc43 \ud835\udc44\ud835\udc44 \ud835\udc45\ud835\udc45
\ufffd 
Campos conservativos 
(I) \ud835\udefb\ud835\udefb\ufffd\u20d7 \ud835\udc60\ud835\udc60 = \ufffd\u20d7\ufffd\ud835\udc39 \u21d4 (II) \u222b \ufffd\u20d7\ufffd\ud835\udc39\ud835\udc51\ud835\udc51\ud835\udc5f\ud835\udc5f\ufffd\ufffd\ufffd\ufffd\u20d7\ud835\udefe\ud835\udefe = \ud835\udc60\ud835\udc60\ufffd\ud835\udefe\ud835\udefe(\ud835\udc4f\ud835\udc4f)\ufffd \u2212 \ud835\udc60\ud835\udc60(\ud835\udefe\ud835\udefe(\ud835\udc4e\ud835\udc4e)) \u21d4 (III) \u222e \ufffd\u20d7\ufffd\ud835\udc39\ud835\udc51\ud835\udc51\ud835\udc5f\ud835\udc5f = 0 
 Conservativo \u21d2 \ud835\udc45\ud835\udc45\ud835\udc5f\ud835\udc5f\ud835\udc61\ud835\udc61(\ud835\udc39\ud835\udc39)\ufffd\ufffd\ufffd\ufffd\u20d7 = 0\ufffd\u20d7 
\ud835\udc45\ud835\udc45\ud835\udc5f\ud835\udc5f\ud835\udc61\ud835\udc61(\ud835\udc39\ud835\udc39)\ufffd\ufffd\ufffd\ufffd\u20d7 = 0\ufffd\u20d7 e domínio simplesmente conexo \u21d2 conservativo 
 
 
4 
 
Integral de superfície de campo escalar 
\u222c \ud835\udc60\ud835\udc60(\ud835\udc65\ud835\udc65,\ud835\udc66\ud835\udc66, \ud835\udc67\ud835\udc67)\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51 = \u222c \ud835\udc60\ud835\udc60(\ud835\udc62\ud835\udc62, \ud835\udc51\ud835\udc51) \u2219 \ufffd\ud835\udc4b\ud835\udc4b\ud835\udc62\ud835\udc62\ufffd\ufffd\ufffd\ufffd\u20d7 \u2227 \ud835\udc4b\ud835\udc4b\ud835\udc63\ud835\udc63\ufffd\ufffd\ufffd\ufffd\u20d7 \ufffd\ud835\udc51\ud835\udc51\ud835\udc62\ud835\udc62\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc37\ud835\udc37\ud835\udc46\ud835\udc46 significado: massa de uma 
região a partir da densidade superficial. 
Na geometria a área de uma superfície é: 
\ud835\udc51\ud835\udc51 = \u222c 1\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51 = \u222c \ufffd\ud835\udc4b\ud835\udc4b\ud835\udc62\ud835\udc62\ufffd\ufffd\ufffd\ufffd\u20d7 \u2227 \ud835\udc4b\ud835\udc4b\ud835\udc63\ud835\udc63\ufffd\ufffd\ufffd\ufffd\u20d7 \ufffd \u2219 \ud835\udc51\ud835\udc51\ud835\udc62\ud835\udc62\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc37\ud835\udc37\ud835\udc46\ud835\udc46 
 
Integral de superfície de campo vetorial \u222c \ufffd\ufffd\u20d7\ufffd\ud835\udc39\ufffd\ud835\udc5b\ud835\udc5b\ufffd\u20d7 \ufffd\ud835\udc46\ud835\udc46 \ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51 = ±\u222c \ufffd\ufffd\u20d7\ufffd\ud835\udc39\ufffd\ud835\udc4b\ud835\udc4b\ud835\udc62\ud835\udc62\ufffd\ufffd\ufffd\ufffd\u20d7 \u2227 \ud835\udc4b\ud835\udc4b\ud835\udc63\ud835\udc63\ufffd\ufffd\ufffd\ufffd\u20d7 \ufffd\ud835\udc51\ud835\udc51\ud835\udc62\ud835\udc62\ud835\udc37\ud835\udc37 \ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51 significado: fluxo de um 
campo de vetores através de uma superfície. 
Outra notação: \u222c \ufffd\ufffd\u20d7\ufffd\ud835\udc39\ufffd\ud835\udc5b\ud835\udc5b\ufffd\u20d7 \ufffd\ud835\udc46\ud835\udc46 \ud835\udc51\ud835\udc51\ud835\udc51\ud835\udc51 = \u222c \ud835\udc43\ud835\udc43\ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66 \u2227 \ud835\udc51\ud835\udc51\ud835\udc67\ud835\udc67 + \ud835\udc44\ud835\udc44\ud835\udc51\ud835\udc51\ud835\udc67\ud835\udc67 \u2227 \ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65 + \ud835\udc45\ud835\udc45\ud835\udc51\ud835\udc51\ud835\udc65\ud835\udc65 \u2227 \ud835\udc51\ud835\udc51\ud835\udc66\ud835\udc66 \ud835\udc46\ud835\udc46 
 
Superfície de um tronco de Cilindro no espaço tridimensional 
 \ud835\udc65\ud835\udc652 + \ud835\udc66\ud835\udc662 = \ud835\udc4e\ud835\udc4e2 (o raio está