Coordinate Systems

Cartesian Coordinates

The standard Cartesian (rectangular) coordinate system uses three mutually perpendicular axes $x$, $y$, $z$ emanating from an origin $O$.

Position vector: $\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}$

Basis vectors $\hat{\mathbf{i}}$, $\hat{\mathbf{j}}$, $\hat{\mathbf{k}}$ are constant — they point in the same direction everywhere.

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Cylindrical Polar Coordinates ($r$, $\theta$, $z$)

Useful for systems with cylindrical symmetry (pipes, wires, cylindrical objects).

Coordinate Definitions

• $r \geq 0$: perpendicular distance from $z$-axis
• $\theta$: azimuthal angle from $x$-axis in the $xy$-plane
• $z$: height (same as Cartesian)

Relations to Cartesian

$x = r\cos\theta, \quad y = r\sin\theta, \quad z = z$

$r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\frac{y}{x}$

Basis Vectors

Unlike Cartesian, cylindrical basis vectors depend on position:

$\hat{\mathbf{e}}_r = \cos\theta\,\hat{\mathbf{i}} + \sin\theta\,\hat{\mathbf{j}}$

$\hat{\mathbf{e}}_\theta = -\sin\theta\,\hat{\mathbf{i}} + \cos\theta\,\hat{\mathbf{j}}$

$\hat{\mathbf{e}}_z = \hat{\mathbf{k}}$

Position vector: $\mathbf{r} = r\hat{\mathbf{e}}_r + z\hat{\mathbf{e}}_z$

Warning: $\hat{\mathbf{e}}_r$ and $\hat{\mathbf{e}}_\theta$ change direction as $\theta$ changes.

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Spherical Polar Coordinates ($r$, $\theta$, $\phi$)

Useful for systems with spherical symmetry (atoms, planets, central forces).

Coordinate Definitions

• $r \geq 0$: distance from origin
• $\theta \in [0, \pi]$: polar angle from $z$-axis (inclination)
• $\phi \in [0, 2\pi)$: azimuthal angle from $x$-axis

Note: Physics convention differs from mathematics — physicists use $\theta$ for polar angle and $\phi$ for azimuthal, while mathematicians often swap them.

Relations to Cartesian

$x = r\sin\theta\cos\phi, \quad y = r\sin\theta\sin\phi, \quad z = r\cos\theta$

$r = \sqrt{x^2 + y^2 + z^2}$

Basis Vectors

Again, position-dependent:

$\hat{\mathbf{e}}_r = \sin\theta\cos\phi\,\hat{\mathbf{i}} + \sin\theta\sin\phi\,\hat{\mathbf{j}} + \cos\theta\,\hat{\mathbf{k}}$

$\hat{\mathbf{e}}_\theta = \cos\theta\cos\phi\,\hat{\mathbf{i}} + \cos\theta\sin\phi\,\hat{\mathbf{j}} - \sin\theta\,\hat{\mathbf{k}}$

$\hat{\mathbf{e}}_\phi = -\sin\phi\,\hat{\mathbf{i}} + \cos\phi\,\hat{\mathbf{j}}$

Position vector: $\mathbf{r} = r\hat{\mathbf{e}}_r$

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Volume Elements

The volume element changes with coordinate system:

Cartesian: $dV = dx\,dy\,dz$

Cylindrical: $dV = r\,dr\,d\theta\,dz$

The extra factor $r$ accounts for the fact that rings at larger $r$ have larger circumference.

Spherical: $dV = r^2\sin\theta\,dr\,d\theta\,d\phi$

The factor $r^2\sin\theta$ comes from the Jacobian determinant. At the poles ($\theta = 0, \pi$), $\sin\theta = 0$ and the volume element vanishes — this reflects the coordinate singularity.

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Area Elements

For surface integrals:

Cylindrical (curved surface): $dS = r\,d\theta\,dz$

Spherical (surface of sphere): $dS = r^2\sin\theta\,d\theta\,d\phi$

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Why Different Coordinate Systems?

Choosing the right coordinates simplifies problems:

Symmetry	Best Coordinates	Example
Cylindrical	$(r, \theta, z)$	Electric field of a wire
Spherical	$(r, \theta, \phi)$	Gravitational field of Earth
Translational	Cartesian	Uniform field

Key insight: The physics remains the same; only the mathematical description changes. Choose coordinates that match the symmetry to reduce complexity.

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Warning: Basis Vectors Change

In curvilinear coordinates, derivatives of basis vectors are non-zero:

Cylindrical: $\frac{\partial \hat{\mathbf{e}}_r}{\partial \theta} = \hat{\mathbf{e}}_\theta, \quad \frac{\partial \hat{\mathbf{e}}_\theta}{\partial \theta} = -\hat{\mathbf{e}}_r$

Spherical: More complex relations involving all three basis vectors.

This means $\frac{\partial \mathbf{r}}{\partial \theta}$ is not simply $\hat{\mathbf{e}}_\theta$ — there are additional terms. Always compute vector products in Cartesian or use established formulas for operators like $\nabla$ in curvilinear coordinates.