1Derivatives and coordinates
IA Vector Calculus
1.3 Coordinate systems
Now we can apply the results above the changes of coordinates on Euclidean
space. Suppose
x
i
are the coordinates are Cartesian coordinates. Then we can
define an arbitrary new coordinate system
u
a
in which each coordinate
u
a
is a
function of x. For example, we can define the plane polar coordinates ρ, ϕ by
x
1
= ρ cos ϕ, x
2
= ρ sin ϕ.
However, note that
ρ
and
ϕ
are not components of a position vector, i.e. they
are not the “coefficients” of basis vectors like
r
=
x
1
e
1
+
x
2
e
2
are. But we can
associate related basis vectors that point to directions of increasing
ρ
and
ϕ
,
obtained by differentiating
r
with respect to the variables and then normalizing:
e
ρ
= cos ϕ e
1
+ sin ϕ e
2
, e
ϕ
= −sin ϕ e
1
+ cos ϕ e
2
.
e
1
e
2
ρ
e
ρ
e
ϕ
ϕ
These are not “usual” basis vectors in the sense that these basis vectors vary
with position and are undefined at the origin. However, they are still very useful
when dealing with systems with rotational symmetry.
In three dimensions, we have cylindrical polars and spherical polars.
Cylindrical polars Spherical polars
Conversion formulae
x
1
= ρ cos ϕ x
1
= r sin θ cos ϕ
x
2
= ρ sin ϕ x
2
= r sin θ sin ϕ
x
3
= z x
3
= r cos θ
Basis vectors
e
ρ
= (cos ϕ, sin ϕ, 0) e
r
= (sin θ cos ϕ, sin θ sin ϕ, cos θ)
e
ϕ
= (−sin ϕ, cos ϕ, 0) e
ϕ
= (−sin ϕ, cos ϕ, 0)
e
z
= (0, 0, 1) e
θ
= (cos θ cos ϕ, cos θ sin ϕ, −sin θ)