16.1withScannedExamples

16.1withScannedExamples - 16.1 VECTOR FIELDS KIAM HEONG KWA...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 16.1: VECTOR FIELDS KIAM HEONG KWA 1. Vector Fields Let D be a plane region in R 2 . A vector field on D is a function F that assigns to each point ( x,y ) ∈ D a two-dimensional vector F ( x,y ). In other words, F ( x,y ) is a vector that consists of two components P ( x,y ) and Q ( x,y ) for each ( x,y ) ∈ D . It is customary to write F ( x,y ) = P ( x,y ) i + Q ( x,y ) j = h P ( x,y ) ,Q ( x,y ) i for every ( x,y ) ∈ D . The scalar functions P and Q are called the first and the second components of F . Similarly, if E is a solid region in R 3 , a vector field on E is a function F that assigns to each point ( x,y,z ) ∈ E a three-dimensional vector F ( x,y,z ). The vector field F has three components P , Q , and R and we write F ( x,y,z ) = P ( x,y,z ) i + Q ( x,y,z ) j + R ( x,y,z ) k = h P ( x,y,z ) ,Q ( x,y,z ) ,R ( x,y,z ) i for each ( x,y,z ) ∈ E . Writing x for ( x,y ) ∈ D ⊂ R 2 or ( x,y,z ) ∈ E ⊂ R 3 , we also write F ( x ) for F ( x,y ) or for F ( x,y,x ). One way to visualize a vector field F is to draw the arrow repre- senting the vector F ( x ) at each point x in a fairly dense subset of the domain of F . As of Version 7.0, this can be done rather easily in Mathematica using VectorPlot and VectorPlot3D . Refer to the attached Mathematica notebook for details. Sometimes it will also be useful to consider the magnitude of the vector field F . In the two-dimensional case, the magnitude is | F ( x ) | = q [ P ( x )] 2 + [ Q ( x )] 2 , Date : October 27, 2010. 1 2 KIAM HEONG KWA while in the three-dimensional case, it is | F ( x ) | = q [ P ( x )] 2 + [ Q ( x )] 2 + [ R ( x )] 2 at each x in the domain of F . Example 1 (Problem 5 in the text) . Let F ( x,y ) = y i + x j p x 2 + y 2 . (1) Since | F ( x,y ) | = v u u t y p x 2 + y 2 ! 2 + x p x 2 + y 2 ! 2 = 1 , F has unit length everywhere in its domain. (2) By making the change of coordinate systems x = r cos θ , y = r sin θ , one has F ( r cos θ,r sin θ ) = sin θ i + cos θ j . This shows that the orientation of F at a point ( x,y ) = ( r cos θ,r sin θ ) depends only on the angular coordinate θ , i.e., for any two points ( x 1 ,y 1 ) and ( x 2 ,y 2 ) with the same angular coordinate θ , F ( x 1 ,y 1 ) and F ( x 2 ,y 2 ) point in the same direction....
View Full Document

This note was uploaded on 11/11/2011 for the course MATH 254.01 taught by Professor Kwa during the Fall '10 term at Ohio State.

Page1 / 10

16.1withScannedExamples - 16.1 VECTOR FIELDS KIAM HEONG KWA...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online