Lecture 4 Coordinate Systems: Transformations of Coordinates and Vectors. Sections: 1.8, 1.9 Homework: See homework file

Transcription

1 Lecture 4 Coordinte Systems: Trnsformtions of Coordintes nd Vectors Sections: 1.8, 1.9 Homework: See homework file

2 Trnsformtion of Coordintes Rectngulr Cylindricl x y = = = ρcos ρsin x = y = 2 2 ρ = x + y y = rctn x = LECTURE 4 slide 2

3 Trnsformtion of Coordintes 2 Rectngulr Sphericl x = r sinθ cos y = r sinθsin = r cosθ r = x + y + θ = rccos x + y + y = rctn x x r cos θ r sinθcos θ r P r sinθsin r sin θ LECTURE 4 slide 3 y

4 Trnsformtion of Coordintes 3 Cylindricl Sphericl ρ = r sinθ = = rcosθ r cos θ θ r P 2 2 r = ρ + = θ = rccos ρ x r sin θ ρ y LECTURE 4 slide 4

5 Trnsformtion of Coordintes: Exmple A point lies on sphere of rdius r = 1 m. The sphere is centered on the origin. The point s ngulr position is given by θ = 45 nd ϕ = 45. Find its rectngulr nd cylindricl coordintes. LECTURE 4 slide 5

6 Trivi Wht re the ltitude nd the longitude of Toronto? Toronto LECTURE 4 slide 6

7 TRUE OR FALSE? Q1: The point (0,0,0) in rectngulr CS when trnsformed into sphericl coordintes results in r = 1, θ = 0, ϕ = 0. Q2: The point (r,0,0), r 0, in sphericl coordintes lies on the x xis. LECTURE 4 slide 7

8 RCS Unit Vectors s Functions of Position the unit vectors of the rectngulr coordinte system point in the sme direction t ny position, i.e., they re constnt in spce 0 x x x P y 3 P y 1 y x P y 2 LECTURE 4 slide 8

9 CCS Unit Vectors s Functions of Position the unit vectors ρ nd ϕ of the CCS do NOT point in the sme direction t different points of spce: they depend on ϕ sin 1 P 1 ρ ρ P 2 x LECTURE 4 slide 9 y

10 How to Red Dot-product Tble sin 1 x = (cos ) ρ (sin ) y = (sin ) ρ + (cos ) = cos x sin ρ y y ρ = (cos ) x + (sin ) y = ( sin ) x + (cos ) = y x ρ LECTURE 4 slide 10

11 SCS Unit Vectors s Functions of Position the unit vectors of the SCS do NOT point in the sme direction t different points of spce: they depend on θ nd ϕ θ θ P 1 r θ x P 2 r θ LECTURE 4 y slide 11

12 Unit Vectors in SCS nd CCS: Trnsformtion dot-product tble between unit vectors of CCS nd SCS r θ ρ sinθ 0 cosθ cosθ 0 sinθ derivtion is esy using the RCS SCS nd RCS SCS dot-product tbles = sinθcos + cosθcos sin = cos sin x r θ ρ = cosθ sinθ = r θ = sinθ + cosθ ρ r θ ρ = sinθ + cosθθ = sinθρ + cosθ = θ = cosθρ sin = cosθ sinθ = r r r θ LECTURE 4 slide 12

13 Trnsformtion of Unit Vectors in RCS, SCS, nd CCS Express the unit vectors θ nd in terms of x, y, nd t the point P(1, 90, 270 ). LECTURE 4 slide 13

14 Vectors in the RCS vector components re projections onto the unit vectors of the respective CS t the given position vector components in generl depend on position A( xy,, ) = A( xy,, ) + A( xy,, ) + A( xy,, ) x x y y x 0 A y A A + A y A x x x x y y y A LECTURE 4 slide 14

15 Vectors in the CCS A( ρ,, ) = A ( ρ,, ) + A ( ρ,, ) + A ( ρ,, ) ρ ρ do not forget tht the orienttions of ρ nd ϕ depend on the ngulr position ϕ of the observtion point A ρ A x A ρ A y LECTURE 4 slide 15

16 Vectors in the SCS A(, r θ, ) = A (, r θ, ) + A (, r θ, ) + A (, r θ, ) r r θ θ do not forget tht the orienttions of r, θ, nd ϕ depend on the ngulr position (θ, ϕ) of the observtion point θ A θ A r r θ A A y x LECTURE 4 slide 16

17 RCS CCS Ax = cos A Ay = Aρsin + A A = A sin cos Aρ = Axcos+ Aysin A = Axsin + Aycos A = A Vector Trnsformtions 1 A = Aρρ + A + A / x, y, A= A + A + A / ρ,, x x y y Notice: vector components depend on the ngulr coordinte ϕ! dot product tble sin LECTURE 4 slide 17 1

18 RCS SCS A = A + Aθθ + A /,, Ax = Arsin θcos+ Aθ cosθcos Asin Ay = Arsinθsin+ Aθ cosθsin+ A cos A = A cosθ A sinθ A = Axx + Ayy + A / r, θ, Ar = Axsinθcos+ Aysinθsin+ Acosθ Aθ = Axcosθcos+ Aycosθsin Asin θ A = A sin+ A cos r x Vector Trnsformtions 2 θ y r r x y Notice: vector components depend on the ngulr coordintes! θ LECTURE 4 slide 18

19 Vector Trnsformtions 3 CCS SCS Aρ = Ar sinθ + A A = A A = A cosθ A r θ θ cosθ sinθ Ar = Aρ sinθ + Acosθ Aθ = Aρ cosθ A sinθ A = A r θ ρ sinθ 0 cosθ cosθ 0 sinθ LECTURE 4 slide 19

20 CAN YOU SOLVE IN YOUR MIND? A vector A = 5 ϕ exists t the point P(ρ = 1, ϕ = 90, = 0). () Give the coordintes of P in RCS. x = y = = (b) Wht re the components of A in RCS? A x = A y = A = LECTURE 4 slide 20

21 Dot nd Cross Products in CCS nd SCS dot nd cross products of vectors expressed in CCS or SCS components re tken in exctly the sme wy s in RCS the result of dot nd cross products does not depend on the choice of coordinte system in CCS nd SCS, dot nd cross products re tken only with vectors tht re defined t the sme point, becuse the bse unit vectors chnge with position dot product pdot ( R ) = AR ( ) BR ( ) = Ax( R) Bx( R) + Ay( R) By( R) + A( R) B( R) = Aρ( R) Bρ( R) + A( R) B( R) + A( R) B( R) = A ( R) B ( R) + A ( R) B ( R) + A ( R) B ( R) r r θ θ LECTURE 4 slide 21

22 Dot Product: Exmple The point P hs coordintes x = 0, y = 1 m, = 0. The vectors A nd B t this point re given by A( P) = 1x + 0y + 0 B( P) = 1x 1y + 0 Find the dot product of A nd B t P in RCS, CCS, nd SCS. LECTURE 4 slide 22

23 Dot nd Cross Products in CCS nd SCS 2 cross product ( R) ( R) ( R) p ( R) = AR ( ) BR ( ) = A1( R) A2( R) A3( R) B ( R) B ( R) B ( R) x y ρρ r θθ LECTURE 4 slide 23

24 Cross Products: The Esy Wy 1) use the right-hnd rule to obtin your first term p = A B = + + ( AB AB ) ( AB AB ) ( AB AB ) ( AB AB ) ( ) ( ) AB AB AB AB ( AB ) ( ) ( ) r ABr AB AB r ABr AB r x y y x y y x x x y = + + ρ ρ ρ ρ ρ = + + θ θ θ θ θ 2) the remining terms re obtined vi the circulr substitution LECTURE 4 slide

25 Summry your cn esily trnsform the coordintes of points nd the vector components from coordinte system to nother formul sheet is provided t exms the rules of vector multipliction (dot nd cross products) re the sme in ll orthogonl coordinte systems BUT the vectors must be defined t the sme point LECTURE 4 slide 25