Teorema divergensi

Dalam matematika teorema divergensi, yang dikenal juga dengan sebutan teorema Gauss atau teorema Ostrogradsky memerikan hubungan antara aliran (fluks) medan vektor melalui permukaan dengan peri laku medan di dalam permukaan.

Tepatnya, teorema ini menyatakan bahwa fluks sebuah medan vektor melalui permukaan tertutup sama dengan integral volume dari divergensi pada daerah di dalam permukaan. Secara intuitif teorema ini menyatakan bahwa jumlah semua sumber dikurangi jumlah semua sumur memberikan aliran netto keluar dari daerah itu.

Teorema divergensi penting buat matematika rekayasa, terutama elektrostatika dan dinamika fluida. Dalam fisika dan rekayasa, teorema divergensi biasanya diterapkan dalam dimensi tiga. Namun teorema ini dapat digeneralisasi ke sembarang dimensi. Pada satu dimensi teorema ini ekivalen dengan teorema dasar Kalkulus. Pada ruang dua dimensi, ini setara dengan teorema Green.

Pernyataan matematika

Daerah V dibatasi oleh permukaan S=∂V dengan vektor normal n
Teorema divergensi dapat digunakan untuk menghitung fluks melalui permukaan tertutup yang sepenuhnya melingkupi sebuah volume, seperti semua permukaan di sebelah kiri. Teorema ini tidak dapat digunakan langsung untuk menghitung fluks melalui permukaan dengan batas, seperti di sebelah kanan (permukaan berwarna biru, sedangkan batas berwarna merah)

Misalkan V adalah himpunan bagian Rn (dalam kasus n = 3, V mewakili volume dalam ruang tiga dimensi) yang merupakan ruang kompak dan memiliki batas permukaan S. Bila F adalah medan vektor kontinu terdiferensialkan yang didefinisikan pada persekitaran V, maka kita mendapatkan:[1]

Catatan

  1. ^ M. R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis. Schaum’s Outlines (edisi ke-2nd). USA: McGraw Hill. ISBN 978-0-07-161545-7. 

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