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Medan Vektor

Medan Vektor. Kalkulus Vektor. Vector calculus (or vector analysis ) is a branch of mathematics concerned with differentiation and integration of vector fields , primarily in 3 dimensional Euclidean space R 3

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Medan Vektor

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  1. Medan Vektor

  2. KalkulusVektor • Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space R3 • Vector calculus plays an important role in differential geometry and in the study of partial differential equations. • It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. • Vector calculus was developed by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis

  3. Medan Vektor Konsepfungsi yang sudahdipelajari : • Fungsibernilairiildarisatupeubahriil • Fungsibernilaivektordarisatupeubahriil • Fungsibernilairiildaribeberapapeubahriil Selanjutnyaakandipelajarikonsepfungsibernilaivektordaribeberapapeubahriil. Fungsitersebutdinamakanmedanvektor. Contoh:

  4. Medan Vektor Contoh: Buatlahsketsasebuahmedanvektorberikutini : 1. 2.

  5. Jawab :

  6. Medan Skalar Berlawanandenganmedanvektor, medanskalaradalahsuatufungsi F yang mengaitkansebuahbilanganpadasetiaptitikdalamruang.

  7. Gradien Medan Skalar Misalkanf(x,y,z) suatumedanskalardanfdapatdidifferensialkan, makagradienf ( ) adalahmedanvektor yang diberikanoleh : Medan vektorinidisebutmedanvektorkonservatif, sedangkanfdisebutfungsipotensialnya. Note : merupakan operator dimana Ketikaberoperasipadasebuahfungsif, operator tersebutmenghasilkangradien , dapatditulis jugasebagai grad f

  8. Divergensi dan Curl dari Medan Vektor Medan vektor : berhubungandengan 2 medanpentinglainnya, yaitudivergensi (div) yang merupakanmedanskalar, dan curl yang merupakan medanvektor. Definisi: Misalkanadalahmedanvektordanada, maka :

  9. Bentuk lain div Fdan curl F • 1. • 2.

  10. Makna div dan curl • JikaF melambangkanmedankecepatandarisuatufluida, maka div Fdititikpmengukurkecendrunganfluidatersebutuntukmenyebarmeninggalkanp (div F > 0) ataumengumpulmenuju p (div F < 0) • Curl F menyatakanarahsumbudimanafluidatersebutberotasi (melingkar) paling cepatdan |curl F| mengukurlajurotasiini. • Arahrotasiinimengikutiaturantangankanan

  11. Latihan • Gambarkan medan vektor untuk a. b. c. • Tentukandiv Fdan curl F dari a. F(x,y,z)=ex cos y i +ex sin y j +z k b.F(x,y,z)= x2e-zi + y3 ln x j + z cos y k 3. Misalkanfadalahsebuahmedanskalardan F adalahmedanvektor. Tentukanmana yang medanskalar, medanvektoratautidakberartiapa-apa a. div ff. curl(grad f) b. grad f g. grad(div F) c. curl F h. curl(curl F) d. div(grad f) i. grad(grad f) e. div(div F) j. div(curl(grad f))

  12. Latihan • Tunjukkan bahwa: a. div (curl F) = 0 b. div (fg) = f div (g) + gdiv (f) + 2 (f) . (g) c. div (fxg) = 0 d. curl (grad f) = 0 e. div (fF) = f (div F) + (grad f) .F f. curl (fF) = f(curl F) + (grad f) xF g. div (FxG) = G . curl F – F . curl G 5. Fungsi skalar div (grad f) =  . f (juga ditulis 2f) disebut Laplacian. Tunjukkan bahwa 2f = fxx + fyy + fzz

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