Orthogonal coordinates ($d{s}^2={\sum }_i{h}_i^{2}d{q}_i^2$)

Orthogonal coordinates have pairwise orthogonal coordinate tangent directions. Their metric is diagonal, with scale factors of orthogonal coordinates $h_i=|\partial \mathbf{x}/\partial q_i|$. Spherical coordinates and cylindrical coordinates are examples.

These plane orthogonal coordinates use two families of confocal parabolas. On the region $y\ge 0$, take $u,v\ge 0$ and defineThen $x=(u^2-v^2)/2$ and $y=uv$. The coordinate tangent vectors $(u,v)$ and $(-v,u)$ have zero dot product, and both have length $\sqrt{u^2+v^2}$. Thus the Jacobian determinant is $u^2+v^2$, the area element is $(u^2+v^2)dudv$, and these orthogonal coordinates degenerate only at the origin. Fixed $u=a>0$ gives $x=(a^2-y^2/a^2)/2$; fixed $v=b>0$ gives $x=(y^2/b^2-b^2)/2$. This normalization is a rescaling of the plane section of parabolic cylindrical coordinates.

Without a restricted parameter domain, simultaneous sign reversal $(u,v)\mapsto (-u,-v)$ preserves the image, so these planar parabolic coordinates are globally two-to-one. The inverse function theorem supplies local inverse branches away from the origin. The nonnegative quadrant is one useful branch chart for the upper half-plane.

For planar parabolic coordinates and $g(u,v)=f(x(u,v),y(u,v))$, the multivariable chain rule gives $g_u=u{f}_x+v{f}_y$ and $g_v=-v{f}_x+u{f}_y$. Differentiating once more and adding cancels the first partial derivatives and the mixed partial derivatives:Consequently, on every local coordinate patch away from the origin, $f$ is a harmonic function if and only if $g$ is a harmonic function. The Laplacian changes by a positive scale factor, so solutions of the Laplace equation are preserved. The restriction away from the origin is needed to divide by the Jacobian determinant.

These orthogonal coordinates extend two confocal families of parabolas parallel to an axis. In the displayed scaling their scale factors of orthogonal coordinates are $h_u=h_v=2\sqrt{u^2+v^2}$ and $h_w=1$, so the Laplacian is $\Delta \varphi =({\varphi }_{uu}+{\varphi }_{vv})/[4(u^2+v^2)]+{\varphi }_{ww}$ away from $u=v=0$. With $u\ge 0$, signed $v$ distinguishes the upper and lower half-planes; the two signs at $u=0$ describe the same negative $x$ half-axis. Fixed positive $u^2$ or $v^2$ describes a parabolic cylinder. This differs from the rotational three-dimensional parabolic coordinates, whose coordinate surfaces are elliptic paraboloids.

For a right-handed regular system of orthogonal coordinates, the gradient in orthogonal coordinates gives $\nabla u_i={\mathbf{e}}_i/h_i$. Thus $\varphi {\mathbf{e}}_1=(\varphi h_1)\nabla u_1$, and the curl product rule givesAlso $\nabla u_2\times \nabla u_3={\mathbf{e}}_1/(h_2{h}_3)$ and the divergence of this cross product is zero, by the divergence-of-a-cross-product identity and the vanishing curl of each gradient. Applying the divergence product rule to $\varphi {\mathbf{e}}_1=(\varphi h_2{h}_3)(\nabla u_2\times \nabla u_3)$ givesCyclic permutation and addition give the full component formulas for any vector field. This derivation accounts for the varying coordinate basis without differentiating its unit vectors explicitly. The factors $h_i>0$ are scale factors of orthogonal coordinates; the chart must avoid coordinate singularities.

For a regular chart of orthogonal coordinates $\mathbf{r}(q)$, define $h_i=|{\partial }_{q_i}\mathbf{r}|>0$ and ${\mathbf{e}}_i=h_i^{-1}{\partial }_{q_i}\mathbf{r}$. The tangent unit vectors form an orthonormal basis. Matching $d\Phi =\nabla \Phi \cdot d\mathbf{r}$ with $d\mathbf{r}={\sum }_i{h}_i{\mathbf{e}}_{i}d{q}_i$ proves the formula. Dividing by a scale factor converts a derivative per coordinate increment into a derivative per physical distance in that direction.

For a ring torus of major radius $A>0$, circular torus coordinates use distance $r$ and angle $\alpha$ in the circular cross-section, and azimuth $\beta$ around the central axis. The coordinate tangent vectors are the three partial derivatives of the displayed position. They are mutually orthogonal, with scale factors of orthogonal coordinates $1,r,A+r\sin \alpha$. For $0<r<A$ their ordered Jacobian determinant is $r(A+r\sin \alpha )>0$, so $dV=r(A+r\sin \alpha )drd\alpha d\beta$. At $r=0$ the cross-section angle degenerates. These are tubular coordinates for a circular torus, rather than the confocal coordinate convention also called toroidal coordinates.

For a length scale $c>0$, planar bipolar coordinates may be defined byThe common scale factor is $h=c/(\cosh \tau -\cos \sigma )$, so Laplace's equation becomes ${\Phi }_{\sigma \sigma }+{\Phi }_{\tau \tau }=0$ away from coordinate singularities. For $\tau >0$, constant-$\tau$ curves are circlesTheir radii decrease as $\tau$ increases; for $0<a<b$ the $\tau =b$ circle lies inside the $\tau =a$ circle. The region $a<\tau <b$ is therefore an eccentric circular annulus. This ordering is useful when translating coordinate boundary data into geometric language.

For a harmonic function between $\tau =a$ and $\tau =b$, $0<a<b$, in bipolar coordinates, suppose $\Phi (\sigma ,a)=0$ and $\Phi (\sigma ,b)=F(\sigma )$. Write the periodic boundary values as a Fourier seriesSeparation of variables givesEach radial factor solves $M^{\prime \prime }-n^{2}M=0$ and has endpoint values zero and one; the mean mode solves $M^{\prime \prime }=0$. Sufficiently regular boundary data give the required Fourier series solution. Uniqueness follows from the maximum principle for a harmonic function with zero Dirichlet boundary conditions on both circles. The formula follows the coordinate values; $\tau =b$ is the geometrically smaller circle.

For $a>0$, $u\ge 0$, $0\le v<2\pi$ and real $z$, these orthogonal coordinates have coordinate tangents $r_u=(a\sinh u\cos v,a\cosh u\sin v,0)$, $r_v=(-a\cosh u\sin v,a\sinh u\cos v,0)$ and $r_z=(0,0,1)$. Their scale factors of orthogonal coordinates are $h_u=h_v=a\sqrt{{\sinh }^{2}u+{\sin }^{2}v}$ and $h_z=1$; the ordered Jacobian determinant is positive away from the two focal lines. Positive constant $u$ gives confocal elliptical cylinders, while generic constant $v$ gives a hyperbolic-cylinder half-branch selected by the signs of $x$ and $y$. The limiting $u=0$ surface is the strip $y=0$, $|x|\le a$, with arbitrary $z$. Constant $z$ gives horizontal planes.

In elliptic cylindrical coordinates, write a vector field as $A=A_u{e}_u+A_v{e}_v+A_z{e}_z$ using the local orthonormal unit vectors. The scale factors of orthogonal coordinates are $(h,h,1)$, so the divergence and curl areApplying the divergence to the gradient of a scalar function gives $\Delta \varphi =({\varphi }_{uu}+{\varphi }_{vv})/h^2+{\varphi }_{zz}$. These formulas hold where $h>0$. They use physical components along unit vectors, rather than components along the unnormalized coordinate tangent vectors.

The displayed three-dimensional parabolic coordinates use $u,v\ge 0$ and an azimuthal angle $\varphi$. Their coordinate tangent vectors have pairwise zero dot products, with scale factors of orthogonal coordinates $h_u=h_v=\sqrt{u^2+v^2}$ and $h_{\varphi }=uv$. The Jacobian determinant is $uv(u^2+v^2)$. Constant $u$ or $v$ gives a family of confocal circular elliptic paraboloids; the coordinate system degenerates on the axis where $uv=0$. For example, $0\le u\le v\le 1$ fills the cap $(x^2+y^2-1)/2\le z\le 0$, whose rim is the unit circle and whose vertex is $(0,0,-1/2)$.

This collects the gradient, divergence, curl and Laplacian formulas in cylindrical and spherical coordinates. Coordinate unit vectors vary with position, so taking their derivatives is essential when translating vector calculus operators from Cartesian coordinates. Curl in spherical coordinates is one of these formulas.

Hyperspherical coordinates extend polar coordinates to $n$ dimensions with one radius and $n-1$ angles.

Spherical coordinates $(r,\theta ,\varphi )$ in ${\mathbb{R}}^3$ use

With polar angle $\theta$ measured from positive $z$ and azimuth $\varphi$, the Cartesian formulas $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$ have Jacobian determinant $r^2\sin \theta$. For $r\ge 0$ and $0\le \theta \le \pi$ this gives the unsigned volume element shown. The inverse determinant is $1/(r^2\sin \theta )$ off the origin and polar axis.

In the usual physics convention for spherical coordinates, the polar angle is the angle between the positive $z$ axis and the position vector. For $r=\sqrt{x^2+y^2+z^2}>0$, it satisfies $\cos \theta =z/r$ with $0\le \theta \le \pi$. The north and south axes have $\theta =0$ and $\theta =\pi$, respectively; the plane $z=0$ has $\theta =\pi /2$. The angle is undefined at the origin. In plane polar coordinates, the same phrase usually denotes the oriented in-plane angle instead, so the reference axis and convention should always be stated. The azimuth supplies the other angular coordinate in three dimensions.

For radial mass density $\rho (r)=A{r}^{-2}(1-e^{-r/r_0})$ within radius $R>0$, integration against the spherical volume element gives $M=4\pi ARf(r_0/R)$. The dimensionless quantity $q=r_0/R>0$ hasThus $f$ is positive, strictly decreasing and convex, with $f(0^{+})=1$ and $f(\infty )=0$. Its asymptotic forms are $f(q)=1-q+q{e}^{-1/q}$ near zero and $f(q)=1/(2q)-1/(6q^2)+O(q^{-3})$ at infinity. The density's central $1/r$ divergence is integrable against the factor $r^{2}dr$.

At fixed angular coordinates, differentiating the Cartesian coordinates with respect to radial distance gives this unit vector. It is the local direction of increasing radius in the spherical coordinate system.

If orthogonal coordinates $q_i$ satisfythen $h_i=|\partial \mathbf{x}/\partial q_i|$ are their scale factors.

These factors determine the diagonal metric and volume element of orthogonal coordinates.

For $\mathbf{A}=A_r{\mathbf{e}}_r+A_{\theta }{\mathbf{e}}_{\theta }+A_{\varphi }{\mathbf{e}}_{\varphi }$, the spherical-coordinate curl follows from the orthogonal-coordinate scale factors $1,r,r\sin \theta$.

Its coordinate expression is part of del in cylindrical and spherical coordinates.

The radius-$R$ ball in ${\mathbb{R}}^n$ has volume ${\pi }^{n/2}{R}^n/\Gamma (n/2+1)$.

Let $V_d$ be the volume of an n-ball of unit radius in ${\mathbb{R}}^d$. A radial shell has volume $d{V}_d{r}^{d-1}dr$, by differentiating $V_d{r}^d$. Integrating shells, or using hyperspherical coordinates, therefore reduces a radial integrand over a Euclidean ball to the displayed one-dimensional integral. The constant $d{V}_d$ is the surface area of an n-sphere of unit radius, with sphere dimension $d-1$.

The boundary of the radius-$R$ ball has area $2{\pi }^{n/2}{R}^{n-1}/\Gamma (n/2)$.