Cartesian coordinates

Cartesian coordinates specify a point in 3D space by three real numbers $(x,y,z)$ along three mutually perpendicular axes. The right-handed convention has $\hat{\mathbf{i}}\times \hat{\mathbf{j}}=\hat{\mathbf{k}}$.

Cartesian is the right choice when the geometry has rectangular symmetry — boxes, planes parallel to axes, problems with no preferred axis of rotation or center.

Differential elements

For doing integrals, we need infinitesimal length, area, and volume elements.

Differential length vector (along an arbitrary curve):

$d\mathbf{l}=dx\hat{\mathbf{i}}+dy\hat{\mathbf{j}}+dz\hat{\mathbf{k}}.$

Differential area vectors, one per coordinate plane:

$d{\mathbf{s}}_x=dydz\hat{\mathbf{i}},d{\mathbf{s}}_y=dxdz\hat{\mathbf{j}},d{\mathbf{s}}_z=dxdy\hat{\mathbf{k}}.$

Each is a small patch in the corresponding coordinate plane, with normal pointing along the named axis.

Differential volume:

$dv=dxdydz.$

These differential elements feed directly into line integrals, surface integrals, and triple integrals.

When Cartesian is right

• Rectangular box, slab, or prism.
• Plane wave propagating along an axis.
• Problems with no natural center of rotation.
• Translation-invariant problems along any axis.

When to switch coordinates

• Spherical symmetry (around a point) → Spherical coordinates.
• Cylindrical symmetry (around an axis) → Cylindrical coordinates.

A volume integral that takes pages of work in Cartesian can collapse to two lines in spherical or cylindrical when the geometry matches. The skill is recognizing the symmetry before grinding.

Relation to other systems

Cylindrical coordinates relate to Cartesian by $x=\rho \cos \varphi$, $y=\rho \sin \varphi$, $z=z$.

Spherical coordinates relate to Cartesian by $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$ (using the engineering / physics convention with $\theta$ polar, $\varphi$ azimuthal).