1 Lecture 4 Coordinate Systems: Rectangular, Cylindrical, Spherical

2 We will use all three of these coordinate systems to represent the position of a point P in 3- dimensional space. In all three cases, position is defined in terms of the intersection of 3 surfaces.

3 In each coordinate system, we shall define three coordinate variables and three corresponding “unit vectors”. Each unit vector has unit length, and points in the direction of increasing value of the coordinate variable to which it corresponds.

4 All three coordinate systems are “mutually orthogonal”. This means that their three unit vectors are mutually perpendicular. This makes it easy to calculate dot and cross vector products. (Eg: All dot products between pairs of unit vectors are zero!)

5 Rectangular (Cartesian) Coordinates Point P(x,y,z) is defined as intersection of 3 planes: plane of constant x, plane of constant y, and plane of constant z x = distance from y-z plane, y = distance from x-z plane, z = distance from x-y plane

6 Differential lengths, areas, volumes in Rectangular coordinate system Position vector drawn from origin to point (x,y,z) x ix + y iy + z iz Differential lengths dx, dy, dz Differential areas dxdy, dxdz, dydz Differential volume dxdydz Variable range x,y,z take on all values Note that the unit vectors are mutually orthogonal Unit vector direction remain constant at all positions in space. Unit vectors: ix, iy, iz (also called ax, ay, az)

7 Cylindrical Coordinate System Point P(r,Ф,z) defined as intersection of two planes and a cylinder: plane of constant z, plane of constant Ф, and cylinder of constant r (or ρ) r (or ρ) = distance from z axis, Ф = angle from positive x axis (positive wrt to fingers of right hand if thumb along z axis.) z = distance from x-y plane

8 Differential lengths, areas, volumes Unit vectors: i r, i Ф, i z (mutually orthogonal) Position vector: r i r + z i z Differential lengths: dr, rdФ, dz Differential areas: rdФdr, drdz, rdФdz Differential volume: rdФdrdz Coordinate range: r = (0 to infinity), Ф = (0 to 2π) z = (-infinity to +infinity) Useful for systems with cylindrical symmetry, but the i Ф and i r unit vectors do not point in constant directions, but rather, their direction is a function of position.

9 Spherical Coordinates Point P(r,Ф, θ) defined as intersection of a plane, sphere, and cone… constant Ф plane, constant r sphere, and constant θ cone. r = distance from the origin, Ф = angle from positive x axis (positive wrt to fingers of right hand if thumb along z axis.), θ = angle made with line drawn out to the point and the positive z axis

10 Differential lengths, areas, volumes Unit vectors: i r, i Ф, i θ (mutually orthogonal) Position vector: ri r Differential lengths: dr, rsin θ dФ, rdθ Differential areas: rsinθdrdФ, rdrd θ, r 2 sinθdθdФ Differential volume: r 2 sinθdФdθdr Coordinate range: r = (0 to infinity), Ф = (0 to 2π) θ = ( 0 to π ) Useful for systems with spherical symmetry, but none of the unit vectors have a constant direction. The direction of each unit vector is a function of position.

11 Conversion between coordinate systems Note that rc and rs distinguish between the r (or ρ) cylindrical coordinate and the r spherical coordinate.

12 All three coordinate systems will be useful in this course, but we shall choose to use rectangular coordinates unless a compelling cylindrical or spherical symmetry exists in the problem.