16.1withScannedExamples

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

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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....
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## 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.

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

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