Medan Vektor

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# Medan Vektor - PowerPoint PPT Presentation

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

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
Medan Vektor

Konsepfungsi yang sudahdipelajari :

• Fungsibernilairiildarisatupeubahriil
• Fungsibernilaivektordarisatupeubahriil
• Fungsibernilairiildaribeberapapeubahriil

Selanjutnyaakandipelajarikonsepfungsibernilaivektordaribeberapapeubahriil.

Fungsitersebutdinamakanmedanvektor.

Contoh:

Medan Vektor

Contoh:

Buatlahsketsasebuahmedanvektorberikutini :

1. 2.

Medan Skalar

Medan vektorinidisebutmedanvektorkonservatif, sedangkanfdisebutfungsipotensialnya.

Note :

merupakan operator dimana

Divergensi dan Curl dari Medan Vektor

Medan vektor :

berhubungandengan 2 medanpentinglainnya, yaitudivergensi (div) yang merupakanmedanskalar, dan curl yang merupakan medanvektor.

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
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

c. curl F h. curl(curl F)

e. div(div F) j. div(curl(grad f))

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