Aplicação da Lei de Gauss em Elementos Diferenciais de Volume

P Aplicação da Lei de Gauss para um elemento diferencial de volume z x y 𝑣 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 P z x y 𝑣 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 No ponto P centro do cubo 𝐷𝑃 𝐷𝑥𝑝 𝑎𝑥 𝐷𝑦𝑝 𝑎𝑦 𝐷𝑧𝑝 𝑎𝑧 P z x y 𝑣 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 Aplicando a Lei de Gauss 𝐷𝑃 𝐷𝑥𝑝 𝑎𝑥 𝐷𝑦𝑝 𝑎𝑦 𝐷𝑧𝑝 𝑎𝑧 𝑄𝑖 ׯ𝑠 𝐷 𝑑Ԧ𝑠 𝑓𝑟𝑒𝑛𝑡𝑒 𝑎𝑡𝑟á𝑠 𝑑𝑖𝑟𝑒𝑖𝑡𝑎 𝑒𝑠𝑞𝑢𝑒𝑟𝑑𝑎 𝑡𝑜𝑝𝑜 𝑏𝑎𝑠𝑒 𝑄𝑖 𝑄 z x y 𝑣 𝑥 𝑦 𝑧 P 𝑥 𝑦 𝑧 𝐷𝑃 𝐷𝑥𝑝 𝑎𝑥 𝐷𝑦𝑝 𝑎𝑦 𝐷𝑧𝑝 𝑎𝑧 න 𝑓𝑟𝑒𝑛𝑡𝑒 𝐷𝑓𝑟𝑒𝑛𝑡𝑒 Ԧ𝑆 𝐷𝑓𝑟𝑒𝑛𝑡𝑒 𝑎𝑥 𝑦𝑧𝑎𝑥 𝐷𝑓𝑟𝑒𝑛𝑡𝑒 𝑦𝑧 𝐷𝑓𝑟𝑒𝑛𝑡𝑒 𝐷𝑥𝑝 𝑥 2 𝐷𝑥 𝑥 𝑄 z x y 𝑣 𝑥 𝑦 𝑧 P 𝑥 𝑦 𝑧 𝐷𝑃 𝐷𝑥𝑝 𝑎𝑥 𝐷𝑦𝑝 𝑎𝑦 𝐷𝑧𝑝 𝑎𝑧 න 𝑎𝑡𝑟á𝑠 𝐷𝑎𝑡𝑟á𝑠 Ԧ𝑆 𝐷𝑎𝑡𝑟á𝑠 𝑎𝑥 𝑦𝑧 𝑎𝑥 𝐷𝑎𝑡𝑟á𝑠 𝑦𝑧 𝐷𝑎𝑡𝑟á𝑠 𝐷𝑥𝑝 𝑥 2 𝐷𝑥 𝑥 z x y 𝑣 𝑥 𝑦 𝑧 P 𝑥 𝑦 𝑧 𝐷𝑃 𝐷𝑥𝑝 𝑎𝑥 𝐷𝑦𝑝 𝑎𝑦 𝐷𝑧𝑝 𝑎𝑧 න 𝑑𝑖𝑟𝑒𝑖𝑡𝑎 𝐷𝑑𝑖𝑟𝑒𝑖𝑡𝑎 Ԧ𝑆 𝐷𝑑𝑖𝑟𝑒𝑖𝑡𝑎 𝑎𝑦 𝑥𝑧 𝑎𝑦 𝐷𝑑𝑖𝑟𝑒𝑖𝑡𝑎 𝑥𝑧 𝐷𝑑𝑖𝑟𝑒𝑖𝑡𝑎 𝐷𝑦𝑝 𝑦 2 𝐷𝑦 𝑦 z x y 𝑣 𝑥 𝑦 𝑧 P 𝑥 𝑦 𝑧 𝐷𝑃 𝐷𝑥𝑝 𝑎𝑥 𝐷𝑦𝑝 𝑎𝑦 𝐷𝑧𝑝 𝑎𝑧 න 𝑒𝑠𝑞𝑢𝑒𝑟𝑑𝑎 𝐷𝑒𝑠𝑞𝑢𝑒𝑟𝑑𝑎 Ԧ𝑆 𝐷𝑒𝑠𝑞𝑢𝑒𝑟𝑑𝑎 𝑎𝑦 𝑥𝑧 𝑎𝑦 𝐷𝑒𝑠𝑞𝑢𝑒𝑟𝑑𝑎 𝑥𝑧 𝐷𝑒𝑠𝑞𝑢𝑒𝑟𝑑𝑎 𝐷𝑦𝑝 𝑦 2 𝐷𝑦 𝑦 z x y 𝑣 𝑥 𝑦 𝑧 P 𝑥 𝑦 𝑧 𝐷𝑃 𝐷𝑥𝑝 𝑎𝑥 𝐷𝑦𝑝 𝑎𝑦 𝐷𝑧𝑝 𝑎𝑧 න 𝑡𝑜𝑝𝑜 𝐷𝑡𝑜𝑝𝑜 Ԧ𝑆 𝐷𝑡𝑜𝑝𝑜 𝑎𝑧 𝑥𝑦𝑎𝑧 𝐷𝑡𝑜𝑝𝑜 𝑥𝑦 𝐷𝑡𝑜𝑝𝑜 𝐷𝑧𝑝 𝑧 2 𝐷𝑧 𝑧 z x y 𝑣 𝑥 𝑦 𝑧 P 𝑥 𝑦 𝑧 𝐷𝑃 𝐷𝑥𝑝 𝑎𝑥 𝐷𝑦𝑝 𝑎𝑦 𝐷𝑧𝑝 𝑎𝑧 න 𝑏𝑎𝑠𝑒 𝐷𝑏𝑎𝑠𝑒 Ԧ𝑆 𝐷𝑏𝑎𝑠𝑒 𝑎𝑧 𝑥𝑦𝑎𝑧 𝐷𝑏𝑎𝑠𝑒 𝑥𝑦 𝐷𝑏𝑎𝑠𝑒 𝐷𝑧𝑝 𝑧 2 𝐷𝑧 𝑧 𝑄𝑖 න 𝑓𝑟𝑒𝑛𝑡𝑒 න 𝑎𝑡𝑟á𝑠 න 𝑑𝑖𝑟𝑒𝑖𝑡𝑎 න 𝑒𝑠𝑞𝑢𝑒𝑟𝑑𝑎 න 𝑡𝑜𝑝𝑜 න 𝑏𝑎𝑠𝑒 𝑄 𝐷𝑥𝑝 𝑥 2 𝐷𝑥 𝑥 𝑦𝑧 𝐷𝑥𝑝 𝑥 2 𝐷𝑥 𝑥 𝑦𝑧 𝐷𝑦𝑝 𝑦 2 𝐷𝑦 𝑦 𝑥𝑧 𝐷𝑦𝑝 𝑦 2 𝐷𝑦 𝑦 𝑥𝑧 𝐷𝑧𝑝 𝑧 2 𝐷𝑧 𝑧 𝑥𝑦 𝐷𝑧𝑝 𝑧 2 𝐷𝑧 𝑦 𝑥𝑦 𝑄 𝐷𝑥 𝑥 𝑥𝑦𝑧 𝐷𝑦 𝑦 𝑥𝑦𝑧 𝐷𝑧 𝑧 𝑥𝑦𝑧 𝑄 𝐷𝑥 𝑥 𝐷𝑦 𝑦 𝐷𝑧 𝑧 𝑥𝑦𝑧 𝐷𝑥 𝑥 𝐷𝑦 𝑦 𝐷𝑧 𝑧 𝑄 𝑥𝑦𝑧 𝐷𝑥 𝑥 𝐷𝑦 𝑦 𝐷𝑧 𝑧 𝑄 𝑥𝑦𝑧 𝑄 𝑣 No limite lim 𝑣0 𝑄 𝑣 𝜌𝑣 𝐷𝑥 𝑥 𝐷𝑦 𝑦 𝐷𝑧 𝑧 𝜌𝑣 Operador vetorial Nabla 𝑥 𝑎𝑥 𝑦 𝑎𝑦 𝑧 𝑎𝑧 𝐷 𝑥 𝑎𝑥 𝑦 𝑎𝑦 𝑧 𝑎𝑧 𝐷𝑥 𝑎𝑥𝐷𝑦 𝑎𝑦𝐷𝑧 𝑎𝑧 𝐷 𝐷𝑥 𝑥 𝐷𝑦 𝑦 𝐷𝑧 𝑧 𝜌𝑣 𝐷 𝜌𝑣 O divergente pode ser expresso em coordenadas cilíndricas por 𝐷 1 𝜌 𝜌 𝜌𝐷𝜌 1 𝜌 𝐷𝜙 𝜙 𝐷𝑧 𝑧 𝜌𝑣 E em coordenadas esféricas 𝐷 1 𝑟2 𝑟 𝑟2𝐷𝑟 1 𝑟𝑠𝑒𝑛𝜃 𝜃 𝑠𝑒𝑛𝜃 𝐷𝜃 1 𝑟 𝑠𝑒𝑛𝜃 𝐷𝜙 𝜙 𝜌𝑣