medan vektor n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Medan Vektor PowerPoint Presentation
Download Presentation
Medan Vektor

Loading in 2 Seconds...

play fullscreen
1 / 17

Medan Vektor - PowerPoint PPT Presentation


  • 479 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Medan Vektor' - ilya


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
kalkulus 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 vektor1
Medan Vektor

Konsepfungsi yang sudahdipelajari :

  • Fungsibernilairiildarisatupeubahriil
  • Fungsibernilaivektordarisatupeubahriil
  • Fungsibernilairiildaribeberapapeubahriil

Selanjutnyaakandipelajarikonsepfungsibernilaivektordaribeberapapeubahriil.

Fungsitersebutdinamakanmedanvektor.

Contoh:

medan vektor2
Medan Vektor

Contoh:

Buatlahsketsasebuahmedanvektorberikutini :

1. 2.

medan skalar
Medan Skalar

Berlawanandenganmedanvektor, medanskalaradalahsuatufungsi F yang mengaitkansebuahbilanganpadasetiaptitikdalamruang.

gradien medan skalar
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

divergensi dan curl dari medan vektor
Divergensi dan Curl dari Medan Vektor

Medan vektor :

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

Definisi: Misalkanadalahmedanvektordanada, maka :

makna div dan curl
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
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))

latihan1
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