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PHY481 - Lecture 3: Vector calculus
Griffiths: Chapter 1 (Pages 38-54), Also Appendix A of Griffiths
Vector calculus in three co-ordinate systems
We shall be using three orthogonal co-ordinate systems, cartesian, cylindrical and spherical polar that are defined
as follows
1. Cartesian co-ordinates
~r = (x, y, z) = xˆx+yˆy+zˆz(1)
2. Cylindrical co-ordinates
~r = (s, φ, z) = sˆs+zˆz=scos(φ)ˆx+ssin(φ) ˆy+ ˆz(2)
where,
x=scos(φ); y=ssin(φ); x2+y2=s2(3)
3. Spherical polar co-ordinates
~r = (r, θ, φ) = rˆr=rsin(θ)cos(φ) ˆx+rsin(θ)sin(φ)ˆy+rcos(θ)ˆz(4)
where,
x=rsin(θ)cos(φ); y=rsin(θ)sin(φ); z=rcos(θ); x2+y2+z2=r2(5)
Unit vector transformations
Relations between unit vectors in the three co-ordinate systems are very useful and can be derived using geometric
reasoning. First consider cylindrical co-ordinates. It is easy to see that,
ˆs=cos(φ)ˆx+sin(φ)ˆy;ˆ
φ=−sin(φ)ˆx+cos(φ) ˆy(6)
The first of these equations follows from Eq. (2) and the second is evident on taking projections of ˆ
φonto the x and
y directions. It is also easy to find the transformations from cylindrical to cartesian unit vectors;
ˆx=cos(φ)ˆr−sin(φ)ˆ
φ; ˆy=sin(φ)ˆr+cos(φ)ˆ
φ(7)
The transformations in the case of polar co-ordinates are,
ˆr=sin(θ)cos(φ)ˆx+sin(θ)sin(φ)ˆy+cos(θ)ˆz; (8)
which follows from Eq. (4),
ˆ
θ=cos(θ)cos(φ)ˆx+cos(θ)sin(φ) ˆy−sin(θ)ˆz; (9)
and
ˆ
φ=−sin(φ)ˆx+cos(φ) ˆy(10)
which is the same as the second of Eq. (6). The equations relating unit vectors in spherical polars to those in cartesian
co-ordinates are,
ˆx=sin(θ)cos(φ)ˆr+cos(θ)cos(φ)ˆ
θ−sin(φ)ˆ
φ; (11)
ˆy=sin(θ)sin(φ)ˆr+cos(θ)sin(φ)ˆ
θ+cos(φ)ˆ
φ; (12)
and
ˆz=cos(θ)ˆr−sin(θ)ˆ
θ(13)
