2.3.10 Differential operators in curvilinear coordinates

In Section 2.2 the gradient, divergence, curl and Laplacian were defined and computed in Cartesian coordinates, where the nabla operator is simply $\nabla = a_{x} \partial / \partial x + a_{y} \partial / \partial y + a_{z} \partial / \partial z$ . Most fields of electromagnetics, however, are much simpler in cylindrical or spherical coordinates: the field of a point charge is $\propto a_{r} / r^{2}$ , that of a line charge $\propto a_{ρ} / ρ$ . Converting such fields to Cartesian form only to differentiate them would be slow and would hide their simple structure. We therefore want the four operators directly in curvilinear coordinates.

It is tempting to write " $\nabla = a_{r} \partial / \partial r + a_{θ} \partial / \partial θ + a_{ϕ} \partial / \partial ϕ$ " in spherical coordinates, but this is wrong for two reasons.

Angles are not lengths. A derivative with respect to an angle is a change per radian, not a change per metre (Figure 2.3.28). The scale factors found in Section 2.3.6 must enter.
The unit vectors move. Even when the missing scale factors are put in, the unit vectors $a_{r}, a_{θ}, a_{ϕ}$ are themselves functions of position, see Eq. (2.3.38), so the divergence and the curl can no longer be obtained by simply "dotting" or "crossing" the operator with the components of the field, as was done in Cartesian coordinates.

The reliable way is to go back to the meaning of each operator, which does not refer to any coordinate system:

the gradient gives the change of a scalar along a displacement, $df = \nabla f \cdot d ℓ$ ,
the divergence is the net outward flux per unit volume of a vanishingly small volume, and
the curl is the circulation per unit area around a vanishingly small loop.

We now apply these definitions to the small curvilinear volume and surface elements of a general orthogonal system $(u_{1}, u_{2}, u_{3})$ with scale factors $(h_{1}, h_{2}, h_{3})$ . All results are valid for any orthogonal coordinate system. Cartesian, cylindrical and spherical coordinates are then obtained by inserting their scale factors from Table 2.3.4.

Why a derivative with respect to an angle must be divided by a scale factor. Let a scalar f depend only on the azimuth \phi (for example f=\phi). Increasing \phi by \Delta\phi changes f by the same amount at \mathrm P_1 and at \mathrm P_2, but the distance travelled, \rho\,\Delta\phi, is much longer at \mathrm P_2. The gradient measures change per metre, so its \phi component is (1/\rho)\,\partial f/\partial\phi=(1/h_\phi)\,\partial f/\partial\phi. — **Figure 2.3.28.** Why a derivative with respect to an angle must be divided by a scale factor. Let a scalar $f$ depend only on the azimuth $ϕ$ (for example $f = ϕ$ ). Increasing $ϕ$ by $Δ ϕ$ changes $f$ by the same amount at $P_{1}$ and at $P_{2}$ , but the distance travelled, $ρ Δ ϕ$ , is much longer at $P_{2}$ . The gradient measures change per metre, so its $ϕ$ component is $(1/ ρ) \partial f / \partial ϕ = (1/ h_{ϕ}) \partial f / \partial ϕ$ .

Common mistake

Writing $\nabla f = \frac{\partial f}{\partial r} a_{r} + \frac{\partial f}{\partial θ} a_{θ} + \frac{\partial f}{\partial ϕ} a_{ϕ}$ , or computing a spherical divergence as $\partial A_{r} / \partial r + \partial A_{θ} / \partial θ + \partial A_{ϕ} / \partial ϕ$ . A unit check already exposes the error: $\partial f / \partial θ$ has the units of $f$ , not of $f$ per metre.

Gradient

Recall the Cartesian form from Section 2.2,

\nabla f = \frac{\partial f}{\partial x} a_{x} + \frac{\partial f}{\partial y} a_{y} + \frac{\partial f}{\partial z} a_{z},

and its key property: for a small displacement $d ℓ = d x a_{x} + d y a_{y} + d z a_{z}$ ,

df = \frac{\partial f}{\partial x} d x + \frac{\partial f}{\partial y} d y + \frac{\partial f}{\partial z} d z = \nabla f \cdot d ℓ . (2.3.47)

The relation $df = \nabla f \cdot d ℓ$ involves only the scalar change $df$ and the geometric displacement $d ℓ$ , so it holds in every coordinate system and can serve as the definition of the gradient. We use it in three steps.

The change of

f

Let $f = f (u_{1}, u_{2}, u_{3})$ . By the chain rule,

df = \frac{\partial f}{\partial u _{1}} d u_{1} + \frac{\partial f}{\partial u _{2}} d u_{2} + \frac{\partial f}{\partial u _{3}} d u_{3} .

The dot product

Write the unknown gradient as $\nabla f = G_{1} a_{1} + G_{2} a_{2} + G_{3} a_{3}$ . With $d ℓ$ from Eq. (2.3.42) and the orthonormality of the unit vectors ( $a_{i} \cdot a_{j} = 0$ for $i \neq = j$ , $= 1$ for $i = j$ ),

\nabla f \cdot d ℓ = G_{1} h_{1} d u_{1} + G_{2} h_{2} d u_{2} + G_{3} h_{3} d u_{3} .

Compare

These two expressions must agree for arbitrary $d u_{1}, d u_{2}, d u_{3}$ , so the coefficients must agree: $G_{i} h_{i} = \partial f / \partial u_{i}$ , i.e. $G_{i} = (1/ h_{i}) \partial f / \partial u_{i}$ .

Gradient

\nabla f = \frac{1}{h _{1}} \frac{\partial f}{\partial u _{1}} a_{1} + \frac{1}{h _{2}} \frac{\partial f}{\partial u _{2}} a_{2} + \frac{1}{h _{3}} \frac{\partial f}{\partial u _{3}} a_{3} (general) (2.3.48)

\nabla f = \frac{\partial f}{\partial ρ} a_{ρ} + \frac{1}{ρ} \frac{\partial f}{\partial ϕ} a_{ϕ} + \frac{\partial f}{\partial z} a_{z} (cylindrical) (2.3.49)

\nabla f = \frac{\partial f}{\partial r} a_{r} + \frac{1}{r} \frac{\partial f}{\partial θ} a_{θ} + \frac{1}{r sin θ} \frac{\partial f}{\partial ϕ} a_{ϕ} (spherical) (2.3.50)

The meaning of the factors $1/ h_{i}$ is plain: $\partial f / \partial ϕ$ is the change of $f$ per radian, but a gradient component must be a change per metre. At distance $ρ$ from the axis one radian of $ϕ$ is $ρ$ metres of arc, so we divide by $ρ$ (Figure 2.3.28). Every component of the gradient therefore has the units of $f$ per metre (e.g. V/m when $f$ is a potential in volts).

In plain words

Each component of the gradient is "how fast $f$ changes per metre when you walk along that coordinate direction". Differentiate with respect to the coordinate, then divide by the number of metres that one unit of the coordinate represents ( $h_{i}$ ).

Symmetry pays

For $f = 1/ r$ only the radial derivative survives in Eq. (2.3.50): $\nabla (1/ r) = - a_{r} / r^{2} = - r / r^{3}$ . This is the result obtained with considerably more effort in Cartesian form in Section 2.2 (gradient of $1/ R$ with the source at the origin). Choosing the coordinate system that matches the symmetry of the problem is half of the solution. This lesson will be repeated many times in Sections 3.1 to 3.3.

Quick check

Compute the gradient of $f = r cos θ$ with Eq. (2.3.50). Why is the answer not a surprise?

$\nabla f = cos θ a_{r} + \frac{1}{r} (- r sin θ) a_{θ} = cos θ a_{r} - sin θ a_{θ}$ , which by Eq. (2.3.46) is exactly $a_{z}$ . No surprise: $r cos θ = z$ , and $\nabla z = a_{z}$ .

Divergence

The divergence of $A$ at a point is the net outward flux of $A$ per unit volume, in the limit of a vanishingly small volume $Δ V$ around the point:

\nabla \cdot A = Δ V \to 0 lim \frac{\oint _{S} A \cdot d S}{Δ V} . (2.3.51)

In Cartesian coordinates, a small box has opposite faces of equal area, and only the change of the field component across the box matters. In curvilinear coordinates there is a new effect: the opposite faces of a small box need not have the same area. Let us see this first in the most important case, and then in general.

The radial faces of a spherical box

Take the small spherical brick of Figure 2.3.23, between $r$ and $r + Δ r$ , $θ$ and $θ + Δ θ$ , $ϕ$ and $ϕ + Δ ϕ$ , and look only at its two faces on which $r$ is constant. On them only the radial component $A_{r}$ crosses the surface.

Outer face (at $r + Δ r$ , outward normal $+ a_{r}$ ): its area is $(r + Δ r)^{2} sin θ Δ θ Δ ϕ$ , so its outward flux is $[r^{2} A_{r}]_{r + Δ r} sin θ Δ θ Δ ϕ$ .
Inner face (at $r$ , outward normal $- a_{r}$ ): its area is $r^{2} sin θ Δ θ Δ ϕ$ , so its outward flux is $- [r^{2} A_{r}]_{r} sin θ Δ θ Δ ϕ$ .

The outer face is larger than the inner one, so what counts is not the change of $A_{r}$ alone but the change of the product $r^{2} A_{r}$ (field times area factor). To first order in $Δ r$ the net outward flux through the two faces is

([r^{2} A_{r}]_{r + Δ r} - [r^{2} A_{r}]_{r}) sin θ Δ θ Δ ϕ ≃ \frac{\partial ( r ^{2} A _{r} )}{\partial r} Δ r sin θ Δ θ Δ ϕ,

and dividing by the volume $Δ V = r^{2} sin θ Δ r Δ θ Δ ϕ$ gives the radial part of the spherical divergence,

\frac{1}{r ^{2}} \frac{\partial ( r ^{2} A _{r} )}{\partial r} .

The other two pairs of faces work in the same way. The faces $θ = const$ have area $r sin θ Δ r Δ ϕ$ , which depends on $θ$ through $sin θ$ , and this produces $\frac{1}{r s i n θ} \partial (sin θ A_{θ}) / \partial θ$ . The faces $ϕ = const$ have area $r Δ r Δ θ$ , which does not depend on $ϕ$ , so they produce simply $\frac{1}{r s i n θ} \partial A_{ϕ} / \partial ϕ$ .

Any orthogonal system

The same bookkeeping in a general orthogonal system gives the general formula below, with the details in the box. The result says: for each pair of faces, differentiate "component $\times$ area factor of that face", then divide by the volume factor $h_{1} h_{2} h_{3}$ .

Divergence

\nabla \cdot A = \frac{1}{h _{1} h _{2} h _{3}} [\frac{\partial ( h _{2} h _{3} A _{1} )}{\partial u _{1}} + \frac{\partial ( h _{1} h _{3} A _{2} )}{\partial u _{2}} + \frac{\partial ( h _{1} h _{2} A _{3} )}{\partial u _{3}}] (general) (2.3.52)

\nabla \cdot A = \frac{1}{ρ} \frac{\partial ( ρ A _{ρ} )}{\partial ρ} + \frac{1}{ρ} \frac{\partial A _{ϕ}}{\partial ϕ} + \frac{\partial A _{z}}{\partial z} (cylindrical) (2.3.53)

\nabla \cdot A = \frac{1}{r ^{2}} \frac{\partial ( r ^{2} A _{r} )}{\partial r} + \frac{1}{r sin θ} \frac{\partial ( sin θ A _{θ} )}{\partial θ} + \frac{1}{r sin θ} \frac{\partial A _{ϕ}}{\partial ϕ} (spherical) (2.3.54)

For Cartesian coordinates ( $h_{1} = h_{2} = h_{3} = 1$ ) Eq. (2.3.52) reduces to the familiar $\partial A_{x} / \partial x + \partial A_{y} / \partial y + \partial A_{z} / \partial z$ of Section 2.2. For cylindrical coordinates, $h_{1} h_{2} h_{3} = ρ$ and

\nabla \cdot A = \frac{1}{ρ} [\frac{\partial ( ρ A _{ρ} )}{\partial ρ} + \frac{\partial A _{ϕ}}{\partial ϕ} + \frac{\partial ( ρ A _{z} )}{\partial z}],

which is Eq. (2.3.53) because $ρ$ does not depend on $z$ . For spherical coordinates, $h_{1} h_{2} h_{3} = r^{2} sin θ$ and

\nabla \cdot A = \frac{1}{r ^{2} sin θ} [\frac{\partial ( r ^{2} sin θ A _{r} )}{\partial r} + \frac{\partial ( r sin θ A _{θ} )}{\partial θ} + \frac{\partial ( r A _{ϕ} )}{\partial ϕ}],

taking out of each derivative the factors that do not depend on the variable of differentiation gives Eq. (2.3.54), in agreement with the direct argument above.

(a) A small curvilinear box in an orthogonal coordinate system. The faces u_1=\text{const} have area h_2h_3\Delta u_2\Delta u_3, which in general is different on the two sides of the box (in spherical coordinates the outer face is larger than the inner one). The net outward flux through this pair of faces is therefore governed by the change of h_2h_3A_1, not of A_1 alone. — **Figure 2.3.29.** (a) A small curvilinear box in an orthogonal coordinate system. The faces $u_{1} = const$ have area $h_{2} h_{3} Δ u_{2} Δ u_{3}$ , which in general is different on the two sides of the box (in spherical coordinates the outer face is larger than the inner one). The net outward flux through this pair of faces is therefore governed by the change of $h_{2} h_{3} A_{1}$ , not of $A_{1}$ alone.

(b) A small loop on a surface u_1=\text{const}, traversed so that its normal is \vec a_1 (right-hand rule), which is used to compute the \vec a_1 component of the curl. — (b) A small loop on a surface $u_{1} = const$ , traversed so that its normal is $a_{1}$ (right-hand rule), which is used to compute the $a_{1}$ component of the curl.

Physical picture

The factor $1/ r^{2}$ in front of $\partial (r^{2} A_{r}) / \partial r$ has a direct physical meaning. Consider the purely radial field $A = a_{r} / r^{2}$ , whose magnitude falls off exactly as the area $4 π r^{2}$ of a sphere grows (Figure 2.3.30(a)). Its flux through any sphere centred at the origin is $(1/ r^{2}) (4 π r^{2}) = 4 π$ , the same for every radius: whatever enters a spherical shell through its inner face also flows out through its outer face. Accordingly Eq. (2.3.54) gives $\nabla \cdot (a_{r} / r^{2}) = r^{- 2} \partial (1) / \partial r = 0$ for every $r > 0$ , even though the field certainly decreases with $r$ . A naive " $\partial A_{r} / \partial r$ " would have given the wrong value $- 2/ r^{3}$ . The only place where the flux can originate is the origin itself, where the field is not defined. This is precisely the situation of a point charge, and it will be the heart of Gauss's law in Section 3.2. By contrast, for the position vector $r = r a_{r}$ we find $\nabla \cdot r = r^{- 2} \partial (r^{3}) / \partial r = 3$ , which agrees with the Cartesian value $\partial x / \partial x + \partial y / \partial y + \partial z / \partial z = 3$ .

$Divergence is about flux, not about whether the field grows or shrinks. Both fields are drawn in a plane through the origin (arrow length proportional to |\vec A|, capped near the origin in (a)). (a) \vec A=\vec a_r/r^2 weakens with distance, yet exactly as fast as the spheres grow: the same flux crosses the spheres r_1 and r_2, nothing is created in between, and the divergence is zero.$

Figure 2.3.30. Divergence is about flux, not about whether the field grows or shrinks. Both fields are drawn in a plane through the origin (arrow length proportional to

∣ A ∣

, capped near the origin in (a)). (a)

A = a_{r} / r^{2}

weakens with distance, yet exactly as fast as the spheres grow: the same flux crosses the spheres

r_{1}

and

r_{2}

, nothing is created in between, and the divergence is zero.

(b) \vec A=\vec a_r has constant magnitude, yet more flux leaves the larger sphere than enters through the smaller one: the spreading-out of the field lines is a source, and \nabla\cdot\vec a_r=r^{-2}\,\partial(r^2)/\partial r=2/r. — (b) $A = a_{r}$ has constant magnitude, yet more flux leaves the larger sphere than enters through the smaller one: the spreading-out of the field lines is a source, and $\nabla \cdot a_{r} = r^{- 2} \partial (r^{2}) / \partial r = 2/ r$ .

Quick check

Compute (i) $\nabla \cdot a_{r}$ in spherical coordinates and (ii) $\nabla \cdot a_{ρ}$ and $\nabla \cdot (a_{ρ} / ρ)$ in cylindrical coordinates.

(i) $r^{- 2} \partial (r^{2} \cdot 1) / \partial r = 2/ r$ . (ii) $ρ^{- 1} \partial (ρ \cdot 1) / \partial ρ = 1/ ρ$ and $ρ^{- 1} \partial (ρ \cdot ρ^{- 1}) / \partial ρ = 0$ for $ρ > 0$ . The field $a_{ρ} / ρ$ (the shape of the field of a line charge) plays for cylinders the role that $a_{r} / r^{2}$ plays for spheres.

The explorer below draws the field $a_{ρ} / ρ$ in a plane $z = const$ . Move the probe box around: wherever the box stays away from the axis, the flux entering it equals the flux leaving it and the divergence is zero, and only a box that encloses the axis $ρ = 0$ shows a net outward flux.

InteractiveLoading simulation…

Common mistake

Differentiating only the component: writing $\partial A_{r} / \partial r$ instead of $\frac{1}{r ^{2}} \partial (r^{2} A_{r}) / \partial r$ , or $\partial A_{ρ} / \partial ρ$ instead of $\frac{1}{ρ} \partial (ρ A_{ρ}) / \partial ρ$ . Multiply by the area factor first, then differentiate, then divide.

Curl

The component of the curl along a unit vector $a_{n}$ is the circulation of $A$ per unit area around a vanishingly small loop $C$ that bounds an area $Δ S$ with normal $a_{n}$ , the loop being traversed in the right-hand sense with respect to $a_{n}$ :

(\nabla \times A) \cdot a_{n} = Δ S \to 0 lim \frac{\oint _{C} A \cdot d ℓ}{Δ S} . (2.3.55)

As for the divergence, there is one new effect compared with Cartesian coordinates: opposite sides of a small curvilinear loop need not have the same length.

The z component of the curl in cylindrical coordinates

Take the small loop of Figure 2.3.31 in a plane $z = const$ , bounded by the circles $ρ$ and $ρ + Δ ρ$ and the half-planes $ϕ$ and $ϕ + Δ ϕ$ , traversed counter-clockwise seen from $+ z$ (so that its normal is $+ a_{z}$ ). Along each side only one component of $A$ contributes to $A \cdot d ℓ$ :

The two straight sides (length $Δ ρ$ ) contribute $A_{ρ} Δ ρ$ at $ϕ$ (travelled outwards) and $- A_{ρ} Δ ρ$ at $ϕ + Δ ϕ$ (travelled inwards). Together, to first order, they give $- \frac{\partial A _{ρ}}{\partial ϕ} Δ ϕ Δ ρ$ .
The two arcs contribute $[ρ A_{ϕ}]_{ρ + Δ ρ} Δ ϕ$ (outer arc, travelled forwards) and $- [ρ A_{ϕ}]_{ρ} Δ ϕ$ (inner arc, travelled backwards). The outer arc is longer, so it is the product $ρ A_{ϕ}$ that matters. Together the arcs give $\frac{\partial ( ρ A _{ϕ} )}{\partial ρ} Δ ρ Δ ϕ$ .

Dividing the circulation by the area $Δ S = ρ Δ ρ Δ ϕ$ gives

(\nabla \times A)_{z} = \frac{1}{ρ} [\frac{\partial ( ρ A _{ϕ} )}{\partial ρ} - \frac{\partial A _{ρ}}{\partial ϕ}] .

A small loop in a plane z=\text{const}, used for the z component of the cylindrical curl. It is traversed counter-clockwise (normal \vec a_z, out of the page). The two straight sides have the same length \Delta\rho, but the outer arc (\rho+\Delta\rho)\Delta\phi is longer than the inner arc \rho\,\Delta\phi. Hence the circulation involves the change of \rho A_\phi, not of A_\phi alone. — **Figure 2.3.31.** A small loop in a plane $z = const$ , used for the $z$ component of the cylindrical curl. It is traversed counter-clockwise (normal $a_{z}$ , out of the page). The two straight sides have the same length $Δ ρ$ , but the outer arc $(ρ + Δ ρ) Δ ϕ$ is longer than the inner arc $ρ Δ ϕ$ . Hence the circulation involves the change of $ρ A_{ϕ}$ , not of $A_{ϕ}$ alone.

Any orthogonal system

In general, the $a_{1}$ component of the curl comes from a small loop on a surface $u_{1} = const$ (Figure 2.3.29(b)), whose sides have lengths $h_{2} Δ u_{2}$ and $h_{3} Δ u_{3}$ . Exactly as above, the circulation involves the changes of the products $h_{2} A_{2}$ and $h_{3} A_{3}$ , and the area is $h_{2} h_{3} Δ u_{2} Δ u_{3}$ . The box gives the details.

Curl

\nabla \times A = \frac{1}{h _{1} h _{2} h _{3}} h_{1} a_{1} \frac{\partial}{\partial u _{1}} h_{1} A_{1} h_{2} a_{2} \frac{\partial}{\partial u _{2}} h_{2} A_{2} h_{3} a_{3} \frac{\partial}{\partial u _{3}} h_{3} A_{3} (2.3.57)

\nabla \times A = \frac{1}{ρ} a_{ρ} \frac{\partial}{\partial ρ} A_{ρ} ρ a_{ϕ} \frac{\partial}{\partial ϕ} ρ A_{ϕ} a_{z} \frac{\partial}{\partial z} A_{z} (cylindrical) (2.3.58)

\nabla \times A = \frac{1}{r ^{2} sin θ} a_{r} \frac{\partial}{\partial r} A_{r} r a_{θ} \frac{\partial}{\partial θ} r A_{θ} r sin θ a_{ϕ} \frac{\partial}{\partial ϕ} r sin θ A_{ϕ} (spherical) (2.3.59)

In the determinants the derivatives act only on the third row, and the coordinates must be listed in their right-handed order. Written out component by component, the cylindrical curl is

\nabla \times A = [\frac{1}{ρ} \frac{\partial A _{z}}{\partial ϕ} - \frac{\partial A _{ϕ}}{\partial z}] a_{ρ} + [\frac{\partial A _{ρ}}{\partial z} - \frac{\partial A _{z}}{\partial ρ}] a_{ϕ} + \frac{1}{ρ} [\frac{\partial ( ρ A _{ϕ} )}{\partial ρ} - \frac{\partial A _{ρ}}{\partial ϕ}] a_{z}, (2.3.60)

and the spherical curl is

\nabla \times A = \frac{1}{r sin θ} [\frac{\partial ( sin θ A _{ϕ} )}{\partial θ} - \frac{\partial A _{θ}}{\partial ϕ}] a_{r} + \frac{1}{r} [\frac{1}{sin θ} \frac{\partial A _{r}}{\partial ϕ} - \frac{\partial ( r A _{ϕ} )}{\partial r}] a_{θ} + \frac{1}{r} [\frac{\partial ( r A _{θ} )}{\partial r} - \frac{\partial A _{r}}{\partial θ}] a_{ϕ} . (2.3.61)

In plain words

The curl formulas look long, but each bracket is the same recipe as for the divergence, applied to a loop instead of a box: multiply each component by the length factor of its side ( $h_{i} A_{i}$ ), take the "crossed" derivatives, and divide by the area of the loop.

Physical picture

Two quick applications show what the curl measures (Figure 2.3.32). (i) A purely radial field that depends only on $r$ , $A = A_{r} (r) a_{r}$ , has $A_{θ} = A_{ϕ} = 0$ and $\partial A_{r} / \partial θ = \partial A_{r} / \partial ϕ = 0$ , so every bracket of Eq. (2.3.61) vanishes: $\nabla \times A = 0$ . Such a field has no circulation at all, and the field of a point charge will be of this kind.

(ii) The velocity field of a body rotating rigidly about the $z$ axis with angular velocity $ω$ is $v = ω ρ a_{ϕ}$ . By Eq. (2.3.60), $\nabla \times v = ρ^{- 1} \partial (ω ρ^{2}) / \partial ρ a_{z} = 2 ω a_{z}$ : a uniform curl, parallel to the axis of rotation, equal to twice the angular velocity. A small paddle wheel placed anywhere in this flow turns, as the picture of Section 2.2 suggests. On the other hand, the field $a_{ϕ} / ρ$ , which also "goes round", has $\nabla \times (a_{ϕ} / ρ) = ρ^{- 1} \partial (1) / \partial ρ a_{z} = 0$ for $ρ > 0$ : circulating field lines do not by themselves imply a non-zero curl.

$Two fields that both “go round” the z axis (drawn in a plane z=\text{const}, arrow length proportional to the field). (a) Rigid rotation \vec v=\omega\rho\,\vec a_\phi: the field grows with \rho, so a small paddle wheel feels a stronger push on its outer vanes and turns. The curl is 2\omega\,\vec a_z everywhere.$

Figure 2.3.32. Two fields that both “go round” the

z

axis (drawn in a plane

z = const

, arrow length proportional to the field). (a) Rigid rotation

v = ω ρ a_{ϕ}

: the field grows with

ρ

, so a small paddle wheel feels a stronger push on its outer vanes and turns. The curl is

2 ω a_{z}

everywhere.

(b) \vec A=\vec a_\phi/\rho: the field decreases as 1/\rho, exactly compensating the longer outer arcs of Figure 2.3.31. A paddle wheel off the axis is carried round but does not spin, and the curl is zero everywhere except on the axis itself. — (b) $A = a_{ϕ} / ρ$ : the field decreases as $1/ ρ$ , exactly compensating the longer outer arcs of Figure 2.3.31. A paddle wheel off the axis is carried round but does not spin, and the curl is zero everywhere except on the axis itself.

Quick check

The field $A = a_{ϕ}$ has constant magnitude $1$ and circles the $z$ axis. Is it curl-free?

No: $\nabla \times a_{ϕ} = ρ^{- 1} \partial (ρ \cdot 1) / \partial ρ a_{z} = a_{z} / ρ$ . Only the special decrease $A_{ϕ} \propto 1/ ρ$ makes a circulating field curl-free, see Figure 2.3.32(b).

Laplacian

The Laplacian of a scalar function is the divergence of its gradient, $Δ f = \nabla^{2} f = \nabla \cdot (\nabla f)$ . In Cartesian coordinates this is the sum of the three second derivatives. In a general orthogonal system we simply insert the components $G_{i} = (1/ h_{i}) \partial f / \partial u_{i}$ of the gradient, Eq. (2.3.48), as $A_{i}$ into the divergence, Eq. (2.3.52):

Laplacian

Δ f = \frac{1}{h _{1} h _{2} h _{3}} [\frac{\partial}{\partial u _{1}} (\frac{h _{2} h _{3}}{h _{1}} \frac{\partial f}{\partial u _{1}}) + \frac{\partial}{\partial u _{2}} (\frac{h _{1} h _{3}}{h _{2}} \frac{\partial f}{\partial u _{2}}) + \frac{\partial}{\partial u _{3}} (\frac{h _{1} h _{2}}{h _{3}} \frac{\partial f}{\partial u _{3}})] (2.3.62)

Δ f = \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial f}{\partial ρ}) + \frac{1}{ρ ^{2}} \frac{\partial ^{2} f}{\partial ϕ ^{2}} + \frac{\partial ^{2} f}{\partial z ^{2}} (cylindrical) (2.3.63)

Δ f = \frac{1}{r ^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial f}{\partial r}) + \frac{1}{r ^{2} sin θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial f}{\partial θ}) + \frac{1}{r ^{2} sin ^{2} θ} \frac{\partial ^{2} f}{\partial ϕ ^{2}} (spherical) (2.3.64)

For the spherical case, $h_{1} h_{2} h_{3} = r^{2} sin θ$ and the three ratios in Eq. (2.3.62) are $h_{θ} h_{ϕ} / h_{r} = r^{2} sin θ$ , $h_{r} h_{ϕ} / h_{θ} = sin θ$ and $h_{r} h_{θ} / h_{ϕ} = 1/ sin θ$ . Dividing each term by $r^{2} sin θ$ and moving constant factors out of the derivatives yields Eq. (2.3.64). The cylindrical form follows in the same way with $h_{1} h_{2} h_{3} = ρ$ and the ratios $ρ$ , $1/ ρ$ , $ρ$ .

Two simple but important checks:

Δ (\frac{1}{r}) = \frac{1}{r ^{2}} \frac{\partial}{\partial r} (r^{2} \cdot (- \frac{1}{r ^{2}})) = 0 (r > 0),

Δ (ln ρ) = \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \cdot \frac{1}{ρ}) = 0 (ρ > 0) .

Functions whose Laplacian vanishes are called harmonic, and these two will reappear as the potentials of a point charge and of a line charge (Section 3.3).

Quick check

Compute $Δ (r^{2})$ in spherical coordinates and check the result in Cartesian coordinates.

$\frac{1}{r ^{2}} \frac{\partial}{\partial r} (r^{2} \cdot 2 r) = \frac{6 r ^{2}}{r ^{2}} = 6$ . In Cartesian form $r^{2} = x^{2} + y^{2} + z^{2}$ and $Δ (r^{2}) = 2 + 2 + 2 = 6$ .

Remark

The formulas above are for the Laplacian of a scalar. The Laplacian of a vector field is defined through $\nabla^{2} A = \nabla (\nabla \cdot A) - \nabla \times (\nabla \times A)$ . Only in Cartesian coordinates does it reduce to the Laplacian of each component, $\nabla^{2} A = \nabla^{2} A_{x} a_{x} + \nabla^{2} A_{y} a_{y} + \nabla^{2} A_{z} a_{z}$ . In cylindrical and spherical coordinates $\nabla^{2} A_{r}$ is not the $r$ -component of $\nabla^{2} A$ , again because the unit vectors vary with position.

All four operators on one page

Every formula of this section is one and the same general expression with $h_{1}, h_{2}, h_{3}$ filled in. The following box is the page to keep beside you for the rest of the course.

The four operators in an orthogonal system (u₁, u₂, u₃)

Gradient (scalar $\to$ vector):

\nabla f = \frac{1}{h _{1}} \frac{\partial f}{\partial u _{1}} a_{1} + \frac{1}{h _{2}} \frac{\partial f}{\partial u _{2}} a_{2} + \frac{1}{h _{3}} \frac{\partial f}{\partial u _{3}} a_{3}

Divergence (vector $\to$ scalar):

\nabla \cdot A = \frac{1}{h _{1} h _{2} h _{3}} [\frac{\partial ( h _{2} h _{3} A _{1} )}{\partial u _{1}} + \frac{\partial ( h _{1} h _{3} A _{2} )}{\partial u _{2}} + \frac{\partial ( h _{1} h _{2} A _{3} )}{\partial u _{3}}]

Curl (vector $\to$ vector):

\nabla \times A = \frac{1}{h _{1} h _{2} h _{3}} h_{1} a_{1} \partial / \partial u_{1} h_{1} A_{1} h_{2} a_{2} \partial / \partial u_{2} h_{2} A_{2} h_{3} a_{3} \partial / \partial u_{3} h_{3} A_{3}

Laplacian (scalar $\to$ scalar), $Δ f = \nabla \cdot (\nabla f)$ , i.e. the first two formulas one after the other:

Δ f = \frac{1}{h _{1} h _{2} h _{3}} [\frac{\partial}{\partial u _{1}} (\frac{h _{2} h _{3}}{h _{1}} \frac{\partial f}{\partial u _{1}}) + \frac{\partial}{\partial u _{2}} (\frac{h _{1} h _{3}}{h _{2}} \frac{\partial f}{\partial u _{2}}) + \frac{\partial}{\partial u _{3}} (\frac{h _{1} h _{2}}{h _{3}} \frac{\partial f}{\partial u _{3}})]

scale factors	Cartesian $(x, y, z)$	cylindrical $(ρ, ϕ, z)$	spherical $(r, θ, ϕ)$
$h_{1}, h_{2}, h_{3}$	$1, 1, 1$	$1, ρ, 1$	$1, r, r sin θ$

Recipe · Applying an operator in curvilinear coordinates

Write the field in the system that matches its symmetry and read off $f$ or $A_{1}, A_{2}, A_{3}$ (in the right-handed order of the coordinates).
Take $h_{1}, h_{2}, h_{3}$ from the table and form the products that the formula asks for ( $h_{2} h_{3} A_{1}$ , $h_{i} A_{i}$ , …) before differentiating.
Differentiate with respect to one coordinate at a time, holding the other two fixed.
Divide by the indicated scale-factor products and simplify.
Check: units (each $\nabla$ adds "per metre"), and a special case you know (e.g. $\nabla \cdot r = 3$ , $\nabla \times r = 0$ , $\nabla r = a_{r}$ ).