Wolfram Alpha:

```Orthogonal Curvilinear Coordinates
----------------------------------

Curvilinear coordinates are a coordinate system for
Euclidean space in which the coordinate lines may
be curved.  If the intersections are all at right
angles, then the curvilinear coordinates are said
to form an orthogonal coordinate system.   If not,
they form a skew coordinate system.

Polar and spherical coordinates are both orthogonal
curvilinear coordinate systems.

Proof:

To prove orthogonality we require that tangent
vectors are orthogonal (i.e. their dot product
equals 0).  Therefore,

(∂f/∂u,∂g/∂u).(∂f/∂v,∂g/∂v) = 0

Where,

x = f(u,v) y = g(u,v) z = h(u,v)

Polar coordinates:

x = rcosθ

y = rsinθ

z = z

∂x/∂r = cosθ

∂x/∂θ = -rsinθ

∂y/∂r = sinθ

∂y/∂θ = rcosθ

∂z/∂r = 0

∂z/∂θ = 0

(∂x/∂r + ∂y/∂r).(∂x/∂θ + ∂y/∂θ)

er.eθ = (cosθi + sinθj).(-rsinθi + rcosθj)

= -rcosθsinθ + rsinθcosθ

= 0

Spherical coordinates:

x = rsinθcosφ

y = rsinθsinφ

z = rcosθ

∂x/∂r = sinθcosφ

∂x/∂θ = rcosθcosφ

∂x/∂φ = -rsinθsinφ

∂y/∂r = sinθsinφ

∂y/∂θ = rcosθsinφ

∂y/∂φ = rsinθcosφ

∂z/∂r = cosθ

∂z/∂θ = -rsinθ

∂z/∂φ = 0

eθ.eφ = (∂x/∂θ + ∂y/∂θ + ∂z/∂θ).(∂x/∂φ + ∂y/∂φ + ∂z/∂φ)

= (rcosθcosφi + rcosθsinφi - rsinθk)
.(-rsinθsinφi + rsinθcosφj + 0k)

= -r2cosθcosφsinθsinφ + r2cosθsinφsinθcosφ + 0

= 0```