1. School of EECS, SNU Photonic Systems Laboratory Generalized Coordinate Systems 2015.09.08 박현희 Photonic Systems Laboratory School of EE, Seoul National University

2. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Regular Cartesian Coordinate 3D Regular Cartesian coordinate system : describing each position with “perpendicular” “straight lines” (x-, y-, and z-)  Preconditions 1. “Perpendicular”  Orthogonality 2. “Straight lines”  Distance

3. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Other systems? Straight lines Orthogonal Straight lines Nonorthogonal Curved lines Orthogonal Curved lines Nonorthogonal Curvilinear

4. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor & transformations A= (a 1,a 2,a 3 ) under base vectors e i ’s A = a 1 e 1 + a 2 e 2 + a 3 e 3 Metric tensors: Scale factor  Defining the coordinate Consider the length vector ds e.g. ds = xdx + ydy + zdz = rdr + θr d θ + zdz In general form, For orthogonal coordinate system, Metric tensors

5. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor in Cartesian coordinate How about other coordinate?

6. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor in Spherical coordinate (1) Orthogonal & Curvilinear coordinate systems r: distance, θ, ϕ: Not a distance but an angle x = rsinθcosϕ y = rsinθsinϕ z = rcosθ

7. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor in Spherical coordinate (2) Cartesian Spherical x = rsinθcosϕ y = rsinθsinϕ z = rcosθ dx = drsinθcosϕ + rcosθdθcosϕ - rsinθsinϕdϕ dy = drsinθsinϕ + rcosθdθsinϕ + rsinθcosϕdϕ dz = drcosθ - rsinθdθ for z = f(x,y) Chain rule

8. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor in Spherical coordinate (3) To note, ds 2 = ds · ds is the scalar quantity  Coordinate-invariant (ds 2 ) xyz = (ds 2 ) r θ ϕ

9. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor in Spherical coordinate (4)

10. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor in Cylindrical coordinate (1) x = rcosθ y = rsinθ z = z Orthogonal & Curvilinear coordinate systems r, z: distance, θ: Not a distance but an angle

11. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor in Cylindrical coordinate (2) Example: Cartesian Cylindrical x = rcosθ y = rsinθ z = z dx = drcosθ - rsinθdθ dy = drsinθ + rcosθdθ dz = dz

12. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensor in Cylindrical coordinate (3)

13. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Metric tensors for 3 important coordinates

14. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Line, Surface & Volume in coordinates (1) Use of metric tensor  Defining the length-scale parameters e.g. Spherical coordinate (1) Length dl = e r h r dr + e θ h θ dθ + e ϕ h ϕ dϕ = e r dr + e θ rdθ + e ϕ rsinθdϕ (2) Surface dS r : Infinitesimal surface over r direction dS r = h θ dθ·h ϕ dϕ = r 2 sinθdθ·dϕ

15. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Line, Surface & Volume in coordinates (2) (2) Surface (cont) dS θ = h r dr·h ϕ dϕ = rsinθdr·dϕ dS ϕ = h r dr·h θ dθ = rdr·dθ (3) Volume dV = h r dr·h θ dθ·h ϕ dϕ = r 2 sinθdr·dθ·dϕ

16. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Line integral

17. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Surface integral

18. School of EECS, SNU Photonic Systems Laboratory Coordinate systems : Volume integral