Coordinate Transformations

A coordinate transformation is a change of coordinate system: every point of a region keeps its position in space but gets a new address. The reason to do this is practical. Many shapes and functions are awkward to describe in standard Cartesian coordinates and become much simpler in a coordinate system that matches their symmetry. The circle $x^2 + y^2 = R^2$ — a quadratic equation in two variables — reduces to the single equation $r = R$ in polar coordinates. Spheres simplify the same way in spherical coordinates, cylinders in cylindrical coordinates. The relabelling itself isn’t the goal; the cleaner description it unlocks is.

In one dimension this idea is just ordinary substitution — replacing $x$ by some $g(x)$ inside an integral, for instance — and adds nothing on top of 1D calculus. Coordinate transformations only become genuinely useful in $\mathbb{R}^n$ for $n \ge 2$ , where the relabelling can mix several coordinates together at once. The local distortion this mixing introduces — how much the new coordinate grid is stretched or twisted near each point — is what the rest of this chapter is about.

Cartesian Coordinates

Before generalizing, it pays to name the coordinate system we have been silently using all along. The “default” understanding of coordinates — the perpendicular-grid picture so familiar that it disappears into the background — is itself a specific named construction.

The Cartesian coordinates of a point $\mathbf{x} \in \mathbb{R}^n$ are its signed perpendicular distances from $n$ mutually perpendicular axes, each carrying the same uniform unit of length. Equivalently, every point $\mathbf{x}$ has a unique representation

\mathbf{x} = \sum_{i=1}^{n} x_i\, \mathbf{e}_i

against the standard basis $\{\mathbf{e}_1, \ldots, \mathbf{e}_n\}$ , and the components $(x_1, \ldots, x_n)$ are its Cartesian coordinates. The system formed by these axes is the Cartesian coordinate system.

What singles out the Cartesian system is its rigidity: axes are straight lines, they are mutually perpendicular, the spacing along each is uniform, and the basis vectors are the same everywhere. That uniformity is exactly why descriptions of objects with rotational, radial, or spherical symmetry (a circle, a ball, a torus) come out clumsy in Cartesian form — the symmetry of the object doesn’t match the symmetry of the grid. Recognizing this as a choice of coordinate system, rather than a built-in feature of space itself, is what opens the door to the alternatives developed in this chapter.

Definition

The objects involved are two open subsets $B, D \subseteq \mathbb{R}^n$ and a function $\psi$ that converts addresses in $B$ into addresses in $D$ .

Let $B, D \subseteq \mathbb{R}^n$ be open and let $\psi : B \to D$ be a continuously differentiable bijection. Then $\psi$ is called a coordinate transformation if its inverse $\psi^{-1} : D \to B$ is also continuously differentiable.

The bijectivity is the load-bearing requirement: a coordinate transformation is, at heart, a bijective relabelling of points. Every point in $B$ gets exactly one new address in $D$ , and every address in $D$ corresponds to exactly one point in $B$ . Without it the new “coordinates” wouldn’t pin down points unambiguously — two distinct points could share the same address (failure of injectivity), or an address could refer to no point at all (failure of surjectivity onto the target region). The two-sided continuous-differentiability requirement adds that the relabelling distorts space smoothly in both directions: small steps in $B$ are mapped to small steps in $D$ and vice versa, so derivatives, gradients, and integrals carry over cleanly between the two coordinate systems.

Coordinate Transformation Matrix and the Jacobian

The first-order behavior of $\psi$ at a point is captured by exactly the same object as for any other differentiable multivariate function — its Jacobian matrix. In the context of a coordinate transformation, this matrix gets a context-specific name.

For a coordinate transformation $\psi : B \to D$ with $B, D \subseteq \mathbb{R}^n$ , the coordinate transformation matrix is the Jacobian matrix of $\psi$ :

J_\psi(\mathbf{x}) = \left( \frac{\partial \psi_i}{\partial x_j}(\mathbf{x}) \right)_{ij}, \qquad \mathbf{x} \in B

Because $\psi$ maps $\mathbb{R}^n$ to $\mathbb{R}^n$ , the matrix $J_\psi(\mathbf{x})$ is square ( $n \times n$ ), so its determinant is defined. That determinant gets its own name.

The Jacobian determinant (or simply the Jacobian) of a coordinate transformation $\psi : B \to D$ at $\mathbf{x} \in B$ is the determinant of its coordinate transformation matrix:

\det J_\psi(\mathbf{x})

Neither object is conceptually new. The Jacobian matrix is the same one defined for any differentiable multivariate function, and the determinant is the same determinant from linear algebra — only the role they play here is new. The matrix $J_\psi(\mathbf{x})$ is the local linear approximation of the relabelling $\psi$ at $\mathbf{x}$ , and the scalar $\det J_\psi(\mathbf{x})$ measures the local volume-scaling factor of that linear approximation: how much an infinitesimal volume element in $B$ is stretched or compressed when carried over to $D$ . This one number is what makes the change-of-variables formula in multivariate integration work, and it is the reason the Jacobian determinant is worth singling out in this setting.

The linear case: matrix multiplication. Multiplying by an invertible $n \times n$ matrix, which defines the transformation $\psi(\mathbf{x}) = A\mathbf{x}$ , is the simplest example of a coordinate transformation. The picture is concrete: $\psi$ takes the standard Cartesian grid and bends it into a new — in general skewed and stretched — grid. The new axes are nothing more than the images of the old axes under $\psi$ .

To see what those new axes look like, apply $\psi$ to the $j$ -th standard basis vector $\mathbf{e}_j$ . By a basic identity of matrix algebra, multiplying $A$ by $\mathbf{e}_j$ simply selects the $j$ -th column of $A$ :

\psi(\mathbf{e}_j) = A\,\mathbf{e}_j = (\text{the } j\text{-th column of } A)

So the columns of $A$ are the new axis directions — the basis vectors of the new coordinate system. Concretely, with $A = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}$ , the old $x$ -axis direction $\mathbf{e}_1 = (1,0)^\top$ is sent to $(2, 0)^\top$ — column 1 of $A$ — and the old $y$ -axis direction $\mathbf{e}_2 = (0,1)^\top$ is sent to $(1, 1)^\top$ — column 2 of $A$ . Those two columns are the new basis: the $x$ -axis stretched by a factor of 2, and the $y$ -axis tilted into a diagonal.

Because $\psi$ is itself linear, its local linear approximation is the global map: $J_\psi(\mathbf{x}) = A$ for every $\mathbf{x} \in \mathbb{R}^n$ , and the Jacobian determinant collapses to the single number $\det A$ — the volume-scaling factor of the transformation, uniform across the whole space rather than point-by-point. Nonlinear coordinate transformations are precisely those for which $J_\psi$ varies from point to point, which is why in the general case the Jacobian has to be re-evaluated at every $\mathbf{x}$ .

Common Coordinate Systems

Besides the usual Cartesian coordinates introduced above, three coordinate systems are useful enough to be worth knowing by heart:

the polar coordinates in 2D, parameterizing the plane by distance from the origin and an angle,
the cylindrical coordinates in 3D, extending the polar idea by an extra height axis,
the spherical coordinates in 3D, parameterizing space by distance from the origin and two angles.

Each is treated in its own section.

Polar Coordinates

The most familiar non-Cartesian system in 2D is the polar coordinate system. Instead of pinning a point down by its perpendicular distances along the $x$ - and $y$ -axes, polar coordinates use just two intuitive numbers: how far the point is from the origin, and which direction it lies in. The “how far” is the radius $r$ , and the “which direction” is the angle $\varphi$ measured counterclockwise from the positive $x$ -axis.

The conversion to Cartesian coordinates is right-triangle trigonometry: a point at distance $r$ along an angle $\varphi$ has horizontal component $r\cos\varphi$ and vertical component $r\sin\varphi$ . Packaging this as a coordinate transformation:

The polar coordinate transformation is the map

\psi : \mathbb{R}_{>0} \times [0, 2\pi) \to \mathbb{R}^2 \setminus \{\mathbf{0}\}, \quad \begin{pmatrix} r \\ \varphi \end{pmatrix} \mapsto \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} r \cos \varphi \\ r \sin \varphi \end{pmatrix}

with corresponding coordinate transformation matrix and Jacobian determinant

J_\psi(r, \varphi) = \begin{pmatrix} \cos \varphi & -r \sin \varphi \\ \sin \varphi & r \cos \varphi \end{pmatrix}, \qquad \det J_\psi(r, \varphi) = r

The pair $(r, \varphi)$ are called the polar coordinates of the point $(x, y)$ .

The domain of $\psi$ excludes $r = 0$ — the origin has no well-defined angle, so it can’t be assigned polar coordinates uniquely — and pins $\varphi$ to one full revolution $[0, 2\pi)$ to avoid wrapping the same point with infinitely many angles. Together these restrictions make $\psi$ a bijection onto $\mathbb{R}^2 \setminus \{\mathbf{0}\}$ , as required by the definition of a coordinate transformation.

Compute the Jacobian matrix and determinant

The coordinate transformation matrix is the matrix of partial derivatives of the components given $x = r\cos\varphi$ and $y = r\sin\varphi$ with respect to the inputs $r$ and $\varphi$ . Computing each entry directly inside the matrix:

J_\psi(r, \varphi) = \begin{pmatrix} \dfrac{\partial x}{\partial r} = \cos \varphi & \dfrac{\partial x}{\partial \varphi} = -r \sin \varphi \\ \dfrac{\partial y}{\partial r} = \sin \varphi & \dfrac{\partial y}{\partial \varphi} = r \cos \varphi \end{pmatrix} = \begin{pmatrix} \cos \varphi & -r \sin \varphi \\ \sin \varphi & r \cos \varphi \end{pmatrix}

The determinant of a $2 \times 2$ matrix expands as $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$ , giving:

\begin{aligned} \det J_\psi(r, \varphi) &= (\cos \varphi)(r \cos \varphi) - (-r \sin \varphi)(\sin \varphi) \\ &= r \cos^2 \varphi + r \sin^2 \varphi \\ &= r\,(\cos^2 \varphi + \sin^2 \varphi) \\ &= r \end{aligned}

using the Pythagorean identity $\cos^2 \varphi + \sin^2 \varphi = 1$ in the last step.

Geometric meaning of

\det J_\psi = r

The Jacobian determinant being $r$ has a clean geometric meaning. Recall that for any coordinate transformation, the Jacobian determinant measures its local area-scaling factor — the ratio between the area of a tiny patch in the input space and the area of its image in the output space. So there are two patches and two areas to compare: one in the input plane, one in the output plane, both connected by $\psi$ .

The polar transformation has two distinct 2D planes involved. The input plane is the $(r, \varphi)$ -plane — a regular Cartesian-looking plane with $r$ on one axis and $\varphi$ on the other. The output plane is the actual $(x, y)$ -plane that we live in.

Pick a small rectangle in the input plane anchored at $(r, \varphi)$ with sides $\mathrm{d}r$ and $\mathrm{d}\varphi$ . Its area, computed the usual way for a rectangle, is

\text{input area} = \mathrm{d}r \cdot \mathrm{d}\varphi

Now apply $\psi$ and see where this rectangle lands on the output plane. The image is a tiny “donut slice” anchored at radius $r$ , with two side lengths:

$\mathrm{d}r$ in the radial direction — just the change in radius;
$r\,\mathrm{d}\varphi$ in the angular direction — because an angle of $\mathrm{d}\varphi$ on a circle of radius $r$ traces an arc of length $r\,\mathrm{d}\varphi$ (arc length $=$ radius $\times$ angle).

So the donut slice has area

\text{output area} = \mathrm{d}r \cdot r\,\mathrm{d}\varphi = r\,\mathrm{d}r\,\mathrm{d}\varphi

The area-scaling factor of $\psi$ — i.e. the Jacobian determinant — is now just the ratio of the two:

\det J_\psi(r, \varphi) = \frac{\text{output area}}{\text{input area}} = \frac{r\,\mathrm{d}r\,\mathrm{d}\varphi}{\mathrm{d}r\,\mathrm{d}\varphi} = r

which matches the algebra exactly. The factor vanishes only at $r = 0$ , where the donut slice collapses to a point — the one place we had to exclude from the domain.

Cylindrical Coordinates

Cylindrical coordinates extend the polar idea to 3D in the simplest way imaginable: keep polar coordinates for the horizontal plane and bolt on an unchanged height axis. A point in space is described by where it lies in the $(x, y)$ -plane — the radius $r$ and angle $\varphi$ from polar coordinates — together with how high above (or below) that plane it sits, which is just the ordinary Cartesian $z$ . In that sense the system is more like 2.5D than full 3D: only two of the three coordinates do nontrivial work, and the third rides along untouched.

The conversion to Cartesian coordinates inherits the polar formulas for $x$ and $y$ , with $z$ passing through unchanged:

The cylindrical coordinate transformation is the map

\psi : \mathbb{R}_{>0} \times [0, 2\pi) \times \mathbb{R} \to \mathbb{R}^3 \setminus z\text{-axis}, \quad \begin{pmatrix} r \\ \varphi \\ z \end{pmatrix} \mapsto \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} r \cos \varphi \\ r \sin \varphi \\ z \end{pmatrix}

with corresponding coordinate transformation matrix and Jacobian determinant

J_\psi(r, \varphi, z) = \begin{pmatrix} \cos \varphi & -r \sin \varphi & 0 \\ \sin \varphi & r \cos \varphi & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad \det J_\psi(r, \varphi, z) = r

The triple $(r, \varphi, z)$ are called the cylindrical coordinates of the point $(x, y, z)$ .

The domain excludes the entire $z$ -axis (the $r = 0$ line) for the same reason polar coordinates exclude the origin: along this axis the angle $\varphi$ is undefined, so the assignment would not be a bijection. The angle is again pinned to one full revolution $[0, 2\pi)$ to avoid wrapping the same point with infinitely many addresses.

Compute the Jacobian matrix and determinant

Each entry of the $3 \times 3$ Jacobian is a partial derivative of one of the three component functions $x = r\cos\varphi$ , $y = r\sin\varphi$ , $z = z$ with respect to one of the three inputs $r$ , $\varphi$ , $z$ . Computing each entry directly inside the matrix:

J_\psi(r, \varphi, z) = \begin{pmatrix} \dfrac{\partial x}{\partial r} = \cos \varphi & \dfrac{\partial x}{\partial \varphi} = -r \sin \varphi & \dfrac{\partial x}{\partial z} = 0 \\ \dfrac{\partial y}{\partial r} = \sin \varphi & \dfrac{\partial y}{\partial \varphi} = r \cos \varphi & \dfrac{\partial y}{\partial z} = 0 \\ \dfrac{\partial z}{\partial r} = 0 & \dfrac{\partial z}{\partial \varphi} = 0 & \dfrac{\partial z}{\partial z} = 1 \end{pmatrix} = \begin{pmatrix} \cos \varphi & -r \sin \varphi & 0 \\ \sin \varphi & r \cos \varphi & 0 \\ 0 & 0 & 1 \end{pmatrix}

The third column and third row are mostly zero because $z$ is independent of $r$ and $\varphi$ , and $x, y$ are independent of $z$ . Expanding the determinant along the bottom row picks out only the $(3, 3)$ entry:

\begin{aligned} \det J_\psi(r, \varphi, z) &= 1 \cdot \det\begin{pmatrix} \cos \varphi & -r \sin \varphi \\ \sin \varphi & r \cos \varphi \end{pmatrix} \\ &= r \cos^2 \varphi + r \sin^2 \varphi \\ &= r\,(\cos^2 \varphi + \sin^2 \varphi) \\ &= r \end{aligned}

so the Jacobian determinant collapses to the polar result — as expected, since the $z$ -direction is left untouched and contributes a factor of $1$ .

Geometric meaning of

\det J_\psi = r

The Jacobian determinant being $r$ — the same as for polar coordinates — has the same geometric meaning. The transformation rescales infinitesimal volumes by a factor of $r$ : a tiny box $\mathrm{d}r \cdot \mathrm{d}\varphi \cdot \mathrm{d}z$ in the input maps to a tiny “wedge” of volume $r\,\mathrm{d}r\,\mathrm{d}\varphi\,\mathrm{d}z$ in the output, since the height direction is untouched and the $(r, \varphi)$ slice scales by $r$ exactly as in the polar case.

Spherical Coordinates

Spherical coordinates are the natural system for identifying a point inside a ball: instead of describing a point by perpendicular distances, describe it by how far out and which direction. The “how far out” is again a single radius $r$ — the distance from the origin, which is the same kind of radius as in polar/cylindrical coordinates. The “which direction” now needs two angles, because directions in 3D form a 2D sphere of possibilities (a single angle could only sweep out a circle). The two angles play complementary roles:

$\varphi \in [0, 2\pi)$ — the azimuthal angle, just as in polar and cylindrical coordinates: how far around the $z$ -axis we have rotated, measured in the $xy$ -plane from the positive $x$ -axis;
$\vartheta \in (0, \pi)$ — the polar angle, measured from the positive $z$ -axis down toward the point: $\vartheta = 0$ points to the north pole, $\vartheta = \pi/2$ lies on the equator, $\vartheta = \pi$ points to the south pole.

Converting to Cartesian coordinates is two nested right-triangle trigonometry steps. First, drop the radius $r$ along the polar angle $\vartheta$ : this gives a height $z = r \cos\vartheta$ and a horizontal projection of length $r \sin\vartheta$ . Then sweep that horizontal projection through the azimuth $\varphi$ as in polar coordinates, producing $x = r\sin\vartheta \cos\varphi$ and $y = r\sin\vartheta \sin\varphi$ .

The spherical coordinate transformation is the map

\psi : \mathbb{R}_{>0} \times [0, 2\pi) \times (0, \pi) \to \mathbb{R}^3 \setminus z\text{-axis}, \quad \begin{pmatrix} r \\ \varphi \\ \vartheta \end{pmatrix} \mapsto \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} r \cos \varphi \sin \vartheta \\ r \sin \varphi \sin \vartheta \\ r \cos \vartheta \end{pmatrix}

with corresponding coordinate transformation matrix and Jacobian determinant

J_\psi(r, \varphi, \vartheta) = \begin{pmatrix} \cos \varphi \sin \vartheta & -r \sin \varphi \sin \vartheta & r \cos \varphi \cos \vartheta \\ \sin \varphi \sin \vartheta & r \cos \varphi \sin \vartheta & r \sin \varphi \cos \vartheta \\ \cos \vartheta & 0 & -r \sin \vartheta \end{pmatrix}

\det J_\psi(r, \varphi, \vartheta) = -r^2 \sin \vartheta

The triple $(r, \varphi, \vartheta)$ are called the spherical coordinates of the point $(x, y, z)$ .

The domain again excludes the $z$ -axis. There two things go wrong simultaneously: $r = 0$ (the origin) has no well-defined direction, and $\vartheta \in \{0, \pi\}$ (the north and south poles, sitting on the $z$ -axis) has no well-defined azimuth $\varphi$ — every value of $\varphi$ produces the same point. Removing the $z$ -axis from the codomain rules out both failures at once and leaves a bijection. The azimuth is pinned to $[0, 2\pi)$ as before, and the polar angle to the open interval $(0, \pi)$ .

Compute the Jacobian matrix and determinant

Filling in the matrix from the same partial-derivative recipe as the polar and cylindrical cases — just with more chain-rule trig per entry:

J_\psi(r, \varphi, \vartheta) = \begin{pmatrix} \dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial \varphi} & \dfrac{\partial x}{\partial \vartheta} \\ \dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial \varphi} & \dfrac{\partial y}{\partial \vartheta} \\ \dfrac{\partial z}{\partial r} & \dfrac{\partial z}{\partial \varphi} & \dfrac{\partial z}{\partial \vartheta} \end{pmatrix} = \begin{pmatrix} \cos \varphi \sin \vartheta & -r \sin \varphi \sin \vartheta & r \cos \varphi \cos \vartheta \\ \sin \varphi \sin \vartheta & r \cos \varphi \sin \vartheta & r \sin \varphi \cos \vartheta \\ \cos \vartheta & 0 & -r \sin \vartheta \end{pmatrix}

For the determinant, expand along the bottom row — the middle entry is zero, so only two cofactors contribute:

\begin{aligned} \det J_\psi &= \cos\vartheta \cdot \det\begin{pmatrix} -r \sin \varphi \sin \vartheta & r \cos \varphi \cos \vartheta \\ r \cos \varphi \sin \vartheta & r \sin \varphi \cos \vartheta \end{pmatrix} \\ &\quad + (-r\sin\vartheta) \cdot \det\begin{pmatrix} \cos \varphi \sin \vartheta & -r \sin \varphi \sin \vartheta \\ \sin \varphi \sin \vartheta & r \cos \varphi \sin \vartheta \end{pmatrix} \end{aligned}

Each $2 \times 2$ determinant simplifies using $\sin^2 + \cos^2 = 1$ :

\begin{aligned} \det\begin{pmatrix} -r \sin \varphi \sin \vartheta & r \cos \varphi \cos \vartheta \\ r \cos \varphi \sin \vartheta & r \sin \varphi \cos \vartheta \end{pmatrix} &= -r^2 \sin^2 \varphi \sin \vartheta \cos \vartheta - r^2 \cos^2 \varphi \sin \vartheta \cos \vartheta \\ &= -r^2 \sin\vartheta \cos\vartheta\,(\sin^2 \varphi + \cos^2 \varphi) \\ &= -r^2 \sin\vartheta \cos\vartheta \end{aligned}

\begin{aligned} \det\begin{pmatrix} \cos \varphi \sin \vartheta & -r \sin \varphi \sin \vartheta \\ \sin \varphi \sin \vartheta & r \cos \varphi \sin \vartheta \end{pmatrix} &= r \cos^2 \varphi \sin^2 \vartheta + r \sin^2 \varphi \sin^2 \vartheta \\ &= r \sin^2 \vartheta\,(\cos^2 \varphi + \sin^2 \varphi) \\ &= r \sin^2 \vartheta \end{aligned}

Putting these back:

\begin{aligned} \det J_\psi &= \cos\vartheta \cdot (-r^2 \sin\vartheta \cos\vartheta) + (-r\sin\vartheta) \cdot r \sin^2 \vartheta \\ &= -r^2 \sin\vartheta \cos^2 \vartheta - r^2 \sin^3 \vartheta \\ &= -r^2 \sin\vartheta\,(\cos^2 \vartheta + \sin^2 \vartheta) \\ &= -r^2 \sin\vartheta \end{aligned}

Geometric meaning of

\det J_\psi = -r^2 \sin\vartheta

The Jacobian determinant scales infinitesimal volumes by $|{-r^2 \sin\vartheta}| = r^2 \sin\vartheta$ . Two factors compose to produce this: the $r^2$ comes from the radial direction stretching a fixed angular wedge into a patch whose area grows quadratically with $r$ (a small angular cone subtends a much larger surface area on a big sphere than on a small one); the $\sin\vartheta$ comes from circles of constant $\vartheta$ being smaller near the poles than at the equator — at the north pole ( $\vartheta = 0$ ) and south pole ( $\vartheta = \pi$ ) such circles collapse to a point, which is exactly where the determinant vanishes and the coordinates fail to be a bijection. The negative sign reflects an orientation flip in the chosen ordering $(r, \varphi, \vartheta)$ and is irrelevant to the absolute volume scaling.

Coordinate Transformations

Cartesian Coordinates

Definition

Coordinate Transformation Matrix and the Jacobian

Common Coordinate Systems

Polar Coordinates

Cylindrical Coordinates

Spherical Coordinates

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

V

W

#