Formulas_Calculo de Varias Variaveis

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Formulário - Cálculo de várias variáveis – Matriz nova 
 
 
𝑆𝑒 𝑓(𝑥,𝑦) ≥ 0 
𝑉 = �𝑓(𝑥,𝑦)𝑑𝐴
𝑅
 𝑓𝑚é𝑑 = 1𝐴(𝑅)�𝑓(𝑥,𝑦)𝑑𝐴
𝑅
 
 𝑃𝑎𝑟𝑎 𝑅 = {(𝑥,𝑦)/𝑎 ≤ 𝑥 ≤ 𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑} 
�𝑓(𝑥,𝑦)𝑑𝐴
𝑅
= � � 𝑓(𝑥,𝑦)𝑑
𝑐
𝑑𝑦 𝑑𝑥𝑏
𝑎
= � � 𝑓(𝑥,𝑦)𝑏
𝑎
𝑑𝑥 𝑑𝑦𝑑
𝑐
 
 
�𝑔(𝑥)ℎ(𝑦) 𝑑𝐴 =
𝑅
� 𝑔(𝑥)𝑏
𝑎
𝑑𝑥 .� ℎ(𝑦)𝑑
𝑐
𝑑𝑦 𝑜𝑛𝑑𝑒 𝑅 = [𝑎, 𝑏] × [𝑐,𝑑] 
 
𝑃𝑎𝑟𝑎 𝐷 = {(𝑥,𝑦)/𝑎 ≤ 𝑥 ≤ 𝑏,𝑔1(𝑥) ≤ 𝑦 ≤ 𝑔2(𝑥)} 
�𝑓(𝑥, 𝑦)𝑑𝐴
𝐷
= � � 𝑓(𝑥,𝑦)𝑔2(𝑥)
𝑔1(𝑥) 𝑑𝑦 𝑑𝑥𝑏𝑎 
 
𝑃𝑎𝑟𝑎 𝐷 = {(𝑥,𝑦)/𝑐 ≤ 𝑦 ≤ 𝑑,ℎ1(𝑦) ≤ 𝑥 ≤ ℎ2(𝑦)} 
 
�𝑓(𝑥,𝑦)𝑑𝐴
𝐷
= � � 𝑓(𝑥,𝑦)ℎ2(𝑦)
ℎ1(𝑦) 𝑑𝑥 𝑑𝑦𝑑𝑐 
 
𝑟2 = 𝑥2 + 𝑦2 𝑥 = 𝑟𝑐𝑜𝑠𝜃 𝑦 = 𝑟𝑠𝑒𝑛𝜃 
�𝑓(𝑥,𝑦)𝑑𝐴
𝑅
= � � 𝑓(𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑒𝑛𝜃)𝑏
𝑎
𝑟𝑑𝑟 𝑑𝜃𝛽
𝛼
 
�𝑓(𝑥,𝑦)𝑑𝐴
𝐷
= � � 𝑓(𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑒𝑛𝜃)ℎ2(𝜃)
ℎ1(𝜃) 𝑟𝑑𝑟 𝑑𝜃𝛽𝛼 
 
𝑀𝑥 = �𝑦𝜌(𝑥,𝑦)𝑑𝐴
𝐷
 𝑀𝑦 = �𝑥𝜌(𝑥,𝑦)𝑑𝐴
𝐷
 
�̅� = 𝑀𝑦
𝑚
= 1
𝑚
�𝑥𝜌(𝑥,𝑦)𝑑𝐴
𝐷
 𝑦� = 𝑀𝑥
𝑚
= 1
𝑚
�𝑦𝜌(𝑥,𝑦)𝑑𝐴
𝐷
 𝑚 = �𝜌(𝑥,𝑦)𝑑𝐴
𝐷
 
 
�𝑓(𝑥,𝑦, 𝑧)𝑑𝑉 =
𝐵
� � � 𝑓(𝑥,𝑦, 𝑧)𝑑𝑥 𝑑𝑦 𝑑𝑧𝑏
𝑎
𝑑
𝑐
𝑠
𝑟
 
 
�𝑓(𝑥,𝑦, 𝑧)𝑑𝑉 =
𝐸
� � � 𝑓(𝑥,𝑦, 𝑧)𝑑𝑧 𝑑𝑦 𝑑𝑥𝑢2(𝑥,𝑦)
𝑢1(𝑥,𝑦)
𝑔2(𝑥)
𝑔1(𝑥)
𝑏
𝑎
 �𝑓(𝑥,𝑦, 𝑧)𝑑𝑉 =
𝐸
� � � 𝑓(𝑥,𝑦, 𝑧)𝑑𝑧 𝑑𝑥 𝑑𝑦𝑢2(𝑥,𝑦)
𝑢1(𝑥,𝑦)
ℎ2(𝑦)
ℎ1(𝑦)
𝑑
𝑐
 
 
 
𝑚 = �𝜌(𝑥, 𝑦, 𝑧)𝑑𝑉
𝐸
 𝑉(𝐸) = �𝑑𝑉
𝐸
 
 
Formulário - Cálculo de várias variáveis – Matriz nova 
 
 
𝑀𝑦𝑧 = �𝑥𝜌(𝑥,𝑦, 𝑧)𝑑𝑉
𝐸
 𝑀𝑥𝑧 = �𝑦𝜌(𝑥, 𝑦, 𝑧)𝑑𝑉
𝐸
 𝑀𝑥𝑦 = �𝑧𝜌(𝑥,𝑦, 𝑧)𝑑𝑉
𝐸
 
�̅� = 𝑀𝑦𝑧
𝑚
 𝑦� = 𝑀𝑥𝑧
𝑚
 𝑧̅ = 𝑀𝑥𝑦
𝑚
 
𝐼𝑥 = �(𝑦2 + 𝑧2)𝜌(𝑥,𝑦, 𝑧)𝑑𝑉
𝐸
 𝐼𝑦 = �(𝑥2 + 𝑧2)𝜌(𝑥,𝑦, 𝑧)𝑑𝑉
𝐸
 𝐼𝑧 = �(𝑥2 + 𝑦2)𝜌(𝑥,𝑦, 𝑧)𝑑𝑉
𝐸
 
 
�𝑓(𝑥,𝑦, 𝑧)𝑑𝑉 =
𝐸
� � � 𝑓(𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑒𝑛𝜃, 𝑧) 𝑟 𝑑𝑧 𝑑𝑟 𝑑𝜃𝑢2(𝑟𝑐𝑜𝑠𝜃,𝑟𝑠𝑒𝑛𝜃)
𝑢1(𝑟𝑐𝑜𝑠𝜃,𝑟𝑠𝑒𝑛𝜃)
ℎ2(𝜃)
ℎ1(𝜃)
𝛽
𝛼
 
 
𝑥 = 𝜌 𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜃 𝑦 = 𝜌 𝑠𝑒𝑛𝜙 𝑠𝑒𝑛𝜃 𝑧 = 𝜌 𝑐𝑜𝑠𝜙 
�𝑓(𝑥,𝑦, 𝑧)𝑑𝑉 = � � � 𝑓(𝜌 𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜃,𝑏
𝑎
𝛽
𝛼
𝑑
𝑐
𝐸
 𝜌 𝑠𝑒𝑛𝜙 𝑠𝑒𝑛𝜃,𝜌 𝑐𝑜𝑠𝜙) 𝜌2 𝑠𝑒𝑛𝜙 𝑑𝜌 𝑑𝜃 𝑑𝜙 
 
�⃗�(𝑥,𝑦) = 𝑃(𝑥,𝑦)𝚤 + 𝑄(𝑥,𝑦) 𝚥 ���⃗ �⃗�(𝑥,𝑦, 𝑧 ) = 𝑃(𝑥,𝑦, 𝑧)𝚤 + 𝑄(𝑥,𝑦, 𝑧) 𝚥 ���⃗ + 𝑟(𝑥,𝑦, 𝑧) 𝑘 ����⃗ 
 
 � 𝑓(𝑥,𝑦)
𝐶
𝑑𝑠 = � 𝑓(𝑥(𝑡),𝑦(𝑡))��𝑑𝑥
𝑑𝑡
�
2 + �𝑑𝑦
𝑑𝑡
�
2
𝑑𝑡
𝑏
𝑎
 
𝐿 = � ��𝑑𝑥
𝑑𝑡
�
2 + �𝑑𝑦
𝑑𝑡
�
2
𝑑𝑡
𝑏
𝑎
 𝑑𝑠 = ��𝑑𝑥
𝑑𝑡
�
2 + �𝑑𝑦
𝑑𝑡
�
2
𝑑𝑡 
� 𝑓(𝑥,𝑦)
𝐶
𝑑𝑥 = � 𝑓�𝑥(𝑡),𝑦(𝑡)� 𝑥´(𝑡) 𝑑𝑡𝑏
𝑎
 � 𝑓(𝑥, 𝑦)
𝐶
𝑑𝑦 = � 𝑓�𝑥(𝑡),𝑦(𝑡)� 𝑦´(𝑡) 𝑑𝑡𝑏
𝑎
 
 
� 𝛁𝑓
𝐶
 .𝑑𝒓 = 𝑓(𝒓(𝑏) − 𝑓(𝒓(𝑎)) 𝒓(𝑡) = (1 − 𝑡)𝒓𝟎 + 𝑡𝒓1 
 
𝑇 = � 𝑭 .𝒅𝒓
𝐶