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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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