Curvilinear coordinate systems

Definition

A curvilinear coordinate system describes points using coordinates whose coordinate lines may be curved, rather than only straight Cartesian axes.

In vector calculus, coordinate systems are different ways of labelling the same point in space.

The main three are:

• Cartesian: $(x,y,z)$
• Cylindrical: $(r,\varphi ,z)$
• Spherical: $(r,\theta ,\varphi )$

They are not different physical spaces. They are different coordinate descriptions of the same 3D space.

Big idea

Cartesian coordinates use straight grid lines.

Cylindrical and spherical coordinates use curved grid lines:

• cylindrical is good when there is symmetry around an axis
• spherical is good when there is symmetry around a point

This is why they show up constantly in fluids, electromagnetism, gravity, heat, and any problem where the geometry screams “stop using rectangular boxes, you absolute spreadsheet-cel”.

Interactive sandbox

Explore this visually in the browser: Curvilinear coordinate sandbox.

Cartesian coordinates

Cartesian coordinates label a point by three perpendicular distances:

$$(x,y,z)$$

The basis vectors are constant:

$${\mathbf{e}}_x,{\mathbf{e}}_y,{\mathbf{e}}_z$$

This makes Cartesian coordinates the easiest system for basic derivatives:

$$\nabla =\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)$$

See Del operator ∇.

Cylindrical coordinates

Cylindrical coordinates are:

$$(r,\varphi ,z)$$

where:

• $r$ is distance from the $z$-axis
• $\varphi$ is the angle around the $z$-axis
• $z$ is vertical height

Conversion to Cartesian:

$$x=r\cos \varphi$$ $$y=r\sin \varphi$$ $$z=z$$

Conversion from Cartesian:

$$r=\sqrt{x^2+y^2}$$ $$\varphi =atan2(y,x)$$ $$z=z$$

Use cylindrical coordinates when the problem has rotational symmetry around an axis, like pipes, cylinders, vortices, solenoids, or fields around a wire.

Spherical coordinates

Spherical coordinates are:

$$(r,\theta ,\varphi )$$

where:

• $r$ is distance from the origin
• $\theta$ is the polar angle down from the positive $z$-axis
• $\varphi$ is the azimuthal angle around the $z$-axis

Conversion to Cartesian:

$$x=r\sin \theta \cos \varphi$$ $$y=r\sin \theta \sin \varphi$$ $$z=r\cos \theta$$

Conversion from Cartesian:

$$r=\sqrt{x^2+y^2+z^2}$$ $$\theta =\arccos \left(\frac{z}{r}\right)$$ $$\varphi =atan2(y,x)$$

Use spherical coordinates when the problem has symmetry around a central point, like gravity from a point mass, electric fields from a point charge, radiation, balls, shells, and planets.

Coordinate maps are not always linear

It is tempting to think of every coordinate conversion as a matrix transforming one 3D space into another.

That works for linear transformations like rotations, reflections, scales, and shears in Cartesian space:

$${\mathbf{x}}^{\prime }=A\mathbf{x}$$

But Cartesian to cylindrical or spherical coordinates is not globally linear:

$$x=r\cos \varphi ,y=r\sin \varphi$$

The $\sin$ and $\cos$ terms are the giveaway. This is a nonlinear coordinate map.

The matrix-like object that matters locally is the Jacobian:

$$J=\frac{\partial (x,y,z)}{\partial (q_1,q_2,q_3)}$$

The Jacobian tells you how tiny coordinate changes stretch, rotate, or squish nearby space. Basically: matrices are the global tool for linear maps; Jacobians are the local tool for nonlinear maps.

Scale factors

Curvilinear coordinates have scale factors because one unit of coordinate change does not always mean one unit of physical distance.

For cylindrical coordinates:

$$d{\mathbf{s}}^2=d{r}^2+r^{2}d{\varphi }^2+d{z}^2$$

So the scale factors are:

$$h_r=1,h_{\varphi }=r,h_z=1$$

For spherical coordinates:

$$d{\mathbf{s}}^2=d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\varphi }^2$$

So the scale factors are:

$$h_r=1,h_{\theta }=r,h_{\varphi }=r\sin \theta$$

These scale factors are why grad, div, curl, and volume integrals pick up extra terms in non-Cartesian coordinates.

Volume elements

Cartesian:

$$dV=dxdydz$$

Cylindrical:

$$dV=rdrd\varphi dz$$

Spherical:

$$dV=r^2\sin \theta drd\theta d\varphi$$

The extra $r$ and $r^2\sin \theta$ factors are not decoration. They account for the fact that curved coordinate grids spread out as you move away from the origin or axis.

Quick comparison

System	Coordinates	Best for	Volume element
Cartesian	$(x,y,z)$	boxes, planes, simple grids	$dxdydz$
Cylindrical	$(r,\varphi ,z)$	cylinders, pipes, axis symmetry	$rdrd\varphi dz$
Spherical	$(r,\theta ,\varphi )$	spheres, shells, radial symmetry	$r^2\sin \theta drd\theta d\varphi$

Common pitfalls

• Treating cylindrical or spherical conversion as a single global matrix transformation. Cope. It is nonlinear.
• Forgetting $atan2(y,x)$ and using plain $\arctan (y/x)$, which loses quadrant information.
• Mixing up spherical $\theta$ and $\varphi$. Convention matters. Here $\theta$ is polar angle from the $z$-axis and $\varphi$ is azimuth around the $z$-axis.
• Dropping the scale factors in vector calculus operators.
• Forgetting the volume element when doing integrals.

Links

• Vector Calculus index
• Del operator ∇
• Vector space
• Mapping
• Trigonometry