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Curl and Divergence · Tutorial 917 of 999

Curl in Spherical Coordinates

Learn how spherical scale factors determine the three components of curl, and how to compute and check them in practice.

Advanced 10 min read

What You'll Learn

  • State the spherical-coordinate convention and its scale factors.
  • Derive the spherical curl components from circulation around small coordinate patches.
  • Compute curl for a radially symmetric vector field.
  • Evaluate all three curl components for angularly varying fields.
  • Identify and correct a common error involving the angular scale factor.

Why Curl Needs Scale Factors

Curl measures local rotation, but a spherical-coordinate change in an angle does not correspond to the same physical distance everywhere. A change in azimuth travels a longer distance far from the axis than near it, and both angular distances depend on the radius. Consequently, the spherical curl formula must account for the lengths of coordinate directions as well as for derivatives of vector components.

This tutorial uses spherical coordinates \((r,\theta,\phi)\), where \(r\) is distance from the origin, \(\theta\) is the polar angle measured down from the positive \(z\)-axis, and \(\phi\) is the azimuthal angle measured around the \(z\)-axis from the positive \(x\)-axis. The corresponding unit vectors are \(\mathbf e_r,\mathbf e_\theta,\mathbf e_\phi\), oriented so that \(\mathbf e_r\times\mathbf e_\theta=\mathbf e_\phi\).

Write a vector field as \(\mathbf A=A_r\mathbf e_r+A_\theta\mathbf e_\theta+A_\phi\mathbf e_\phi\). The components \(A_r,A_\theta,A_\phi\) may depend on all three coordinates. The coordinate scale factors are \(h_r=1\), \(h_\theta=r\), and \(h_\phi=r\sin\theta\): small coordinate changes have physical lengths \(dr\), \(r\,d\theta\), and \(r\sin\theta\,d\phi\), respectively.

Definition: At a point where the coordinates are regular and the field is differentiable, the component of \(\nabla\times\mathbf A\) along a unit normal is the limit of circulation around a small positively oriented loop, divided by the loop’s area. “Positively oriented” means that the loop direction follows the right-hand rule for that normal.

The Spherical Curl Formula

With the convention above, the curl components are

$$ \nabla\times\mathbf A = \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial\theta}(A_\phi\sin\theta) - \frac{\partial A_\theta}{\partial\phi} \right)\mathbf e_r + \frac{1}{r} \left( \frac{1}{\sin\theta}\frac{\partial A_r}{\partial\phi} - \frac{\partial}{\partial r}(rA_\phi) \right)\mathbf e_\theta + \frac{1}{r} \left( \frac{\partial}{\partial r}(rA_\theta) - \frac{\partial A_r}{\partial\theta} \right)\mathbf e_\phi. $$

Each derivative is a partial derivative: when differentiating with respect to one coordinate, the other two are held fixed. Notice that the formula differentiates products such as \(A_\phi\sin\theta\) and \(rA_\phi\). These factors encode the changing lengths of the coordinate directions; they must not be dropped.

Why the Formula Has These Factors and Signs

The formula follows by applying the circulation definition to small coordinate rectangles. For the radial component, hold \(r\) fixed and take a small rectangle in the \((\theta,\phi)\)-directions. Its side lengths are approximately \(r\,d\theta\) and \(r\sin\theta\,d\phi\), so its area is approximately \(r^2\sin\theta\,d\theta\,d\phi\). Orient its boundary positively for the normal \(\mathbf e_r\): first along increasing \(\theta\), then along increasing \(\phi\), and then back along the opposite sides.

The two \(\theta\)-directed sides contribute a net circulation, to leading order, of \(-\frac{\partial A_\theta}{\partial\phi}r\,d\theta\,d\phi\). The two \(\phi\)-directed sides contribute \(\frac{\partial}{\partial\theta}(A_\phi\sin\theta)r\,d\theta\,d\phi\). Dividing their sum by the patch area and taking the limit gives

$$ (\nabla\times\mathbf A)_r = \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial\theta}(A_\phi\sin\theta) - \frac{\partial A_\theta}{\partial\phi} \right). $$

For the other two components, use the same circulation-over-area definition on coordinate rectangles in the other coordinate planes. The side lengths are always the corresponding scale factors times the coordinate changes. The right-hand rule fixes the order of the two directions for each normal: \(\mathbf e_\theta\) is normal to the oriented \((\phi,r)\)-plane, and \(\mathbf e_\phi\) is normal to the oriented \((r,\theta)\)-plane. The resulting expressions are

$$ (\nabla\times\mathbf A)_\theta = \frac{1}{r\sin\theta} \left( \frac{\partial A_r}{\partial\phi} - \frac{\partial(r\sin\theta A_\phi)}{\partial r} \right), \qquad (\nabla\times\mathbf A)_\phi = \frac{1}{r} \left( \frac{\partial(rA_\theta)}{\partial r} - \frac{\partial A_r}{\partial\theta} \right). $$

In the \(\theta\)-component, \(\sin\theta\) is constant under the \(r\)-derivative, so the first expression becomes the \(\theta\)-component in the displayed curl formula. This circulation argument explains both the scale factors and the signs: the factors come from physical side lengths and areas, while the signs come from the orientation of the loops.

Worked Example: A Radially Symmetric Field

Worked Example: A Radially Symmetric Field

Let \(\mathbf A=r^2\mathbf e_r\), for \(r>0\). Its spherical components are \(A_r=r^2\), \(A_\theta=0\), and \(A_\phi=0\). Compute the curl component by component.

For the radial component, both terms vanish:

$$ (\nabla\times\mathbf A)_r = \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial\theta}(0\cdot\sin\theta) - \frac{\partial 0}{\partial\phi} \right) =0. $$

For the polar component, \(A_r\) has no \(\phi\)-dependence and \(A_\phi=0\):

$$ (\nabla\times\mathbf A)_\theta = \frac{1}{r} \left( \frac{1}{\sin\theta}\frac{\partial r^2}{\partial\phi} - \frac{\partial}{\partial r}(r\cdot 0) \right) = \frac{1}{r}(0-0) =0. $$

For the azimuthal component, \(A_\theta=0\) and \(A_r=r^2\) has no \(\theta\)-dependence:

$$ (\nabla\times\mathbf A)_\phi = \frac{1}{r} \left( \frac{\partial}{\partial r}(r\cdot 0) - \frac{\partial r^2}{\partial\theta} \right) = \frac{1}{r}(0-0) =0. $$

Thus \(\nabla\times\mathbf A=\mathbf 0\). Substitution into the component formulas confirms the result: every term is zero. More generally, wherever it is differentiable, a purely radial field \(f(r)\mathbf e_r\) has zero curl because its angular components vanish and its radial component has no angular dependence.

Worked Example: A Field with Angular Dependence

Worked Example: A Field with Angular Dependence

Consider \(\mathbf A=r\sin\theta\,\mathbf e_\phi\). Here \(A_r=0\), \(A_\theta=0\), and \(A_\phi=r\sin\theta\). The radial component is

$$ (\nabla\times\mathbf A)_r = \frac{1}{r\sin\theta} \frac{\partial}{\partial\theta} \bigl(r\sin^2\theta\bigr) = \frac{1}{r\sin\theta}(2r\sin\theta\cos\theta) = 2\cos\theta. $$

The polar component is

$$ (\nabla\times\mathbf A)_\theta = -\frac{1}{r}\frac{\partial}{\partial r}(r^2\sin\theta) = -\frac{1}{r}(2r\sin\theta) = -2\sin\theta. $$

The azimuthal component is

$$ (\nabla\times\mathbf A)_\phi = \frac{1}{r} \left( \frac{\partial}{\partial r}(r\cdot 0) - \frac{\partial 0}{\partial\theta} \right) =0. $$

Therefore

$$ \nabla\times\mathbf A = 2\cos\theta\,\mathbf e_r - 2\sin\theta\,\mathbf e_\theta. $$

To check, this field is \(\mathbf A=(-y,x,0)\) in Cartesian components, whose Cartesian curl is \((0,0,2)\). Since the positive \(z\)-direction is \(\cos\theta\,\mathbf e_r-\sin\theta\,\mathbf e_\theta\), the spherical answer is exactly \(2\mathbf e_z\), as required.

Worked Example: All Three Components from Angularly Varying Data

Worked Example: All Three Components from Angularly Varying Data

Let \(\mathbf A=r\sin\theta\cos\phi\,\mathbf e_r+r\cos\phi\,\mathbf e_\theta+r\sin\phi\,\mathbf e_\phi\). Thus \(A_r=r\sin\theta\cos\phi\), \(A_\theta=r\cos\phi\), and \(A_\phi=r\sin\phi\).

For the radial component, the two needed derivatives are

$$ \frac{\partial}{\partial\theta}(A_\phi\sin\theta) = \frac{\partial}{\partial\theta}(r\sin\theta\sin\phi) = r\cos\theta\sin\phi, \qquad \frac{\partial A_\theta}{\partial\phi} = -r\sin\phi. $$

Therefore

$$ (\nabla\times\mathbf A)_r = \frac{r\cos\theta\sin\phi+r\sin\phi}{r\sin\theta} = \frac{(1+\cos\theta)\sin\phi}{\sin\theta}. $$

For the polar component,

$$ \frac{1}{\sin\theta}\frac{\partial A_r}{\partial\phi} = \frac{1}{\sin\theta}(-r\sin\theta\sin\phi) = -r\sin\phi, \qquad \frac{\partial}{\partial r}(rA_\phi) = \frac{\partial}{\partial r}(r^2\sin\phi) = 2r\sin\phi. $$

Hence

$$ (\nabla\times\mathbf A)_\theta = \frac{-r\sin\phi-2r\sin\phi}{r} = -3\sin\phi. $$

For the azimuthal component,

$$ \frac{\partial}{\partial r}(rA_\theta) = \frac{\partial}{\partial r}(r^2\cos\phi) = 2r\cos\phi, \qquad \frac{\partial A_r}{\partial\theta} = r\cos\theta\cos\phi. $$

It follows that

$$ (\nabla\times\mathbf A)_\phi = \frac{2r\cos\phi-r\cos\theta\cos\phi}{r} = (2-\cos\theta)\cos\phi. $$

The result is

$$ \nabla\times\mathbf A = \frac{(1+\cos\theta)\sin\phi}{\sin\theta}\,\mathbf e_r - 3\sin\phi\,\mathbf e_\theta + (2-\cos\theta)\cos\phi\,\mathbf e_\phi. $$

As a direct check, at \(r=2\), \(\theta=\pi/2\), and \(\phi=0\), the component formulas give \(0\), \(0\), and \(2\), respectively. Substituting these angles into the answer gives the same components: \((1+0)\cdot0/1=0\), \(-3\cdot0=0\), and \((2-0)\cdot1=2\).

A Common Scale-Factor Error

A frequent mistake is to use a Cartesian-looking expression for the radial component and omit the angular scale factors. For the field in the preceding example, that incorrect shortcut would give

$$ \frac{1}{r} \left( \frac{\partial A_\phi}{\partial\theta} - \frac{\partial A_\theta}{\partial\phi} \right) = \frac{1}{r}(0-(-r\sin\phi)) = \sin\phi. $$

But the correct radial component is \(\frac{(1+\cos\theta)\sin\phi}{\sin\theta}\), not \(\sin\phi\). The shortcut omitted both the factor \(\sin\theta\) multiplying \(A_\phi\) inside the derivative and the denominator \(\sin\theta\). The correct procedure is to use the full spherical formula, including its scale factors, before differentiating.

Using the Formula Reliably

The order of calculation matters. First identify the three components relative to the spherical unit vectors. Then compute each component of curl from its own formula, keeping the products inside the derivatives intact. Finally, assemble the result using \(\mathbf e_r,\mathbf e_\theta,\mathbf e_\phi\); the three scalar components are not Cartesian components.

1
Resolve the field.
Write \(\mathbf A=A_r\mathbf e_r+A_\theta\mathbf e_\theta+A_\phi\mathbf e_\phi\).
2
Use the matching component formula.
Retain \(r\), \(\sin\theta\), and the products they multiply.
3
Differentiate with the other coordinates fixed.
Apply the product rule whenever the differentiated expression is a product.
4
Check the component values.
Substitute them into the formula or compare with an independent representation when available.

For radial fields, this calculation also explains why a changing magnitude alone does not create local rotation: \(f(r)\mathbf e_r\) has no angular components and does not vary with either angle. For fields with angular dependence, the angular derivatives and the changing lengths of the coordinate directions both contribute. Keeping these two sources of variation distinct is essential for a correct spherical-coordinate curl.

Check Your Understanding

For each question, identify the relevant components and scale factors before simplifying.

  1. Write the three spherical-coordinate curl component formulas, including every scale factor.
  2. Why does the radial curl component contain a derivative of \(A_\phi\sin\theta\), rather than just a derivative of \(A_\phi\)?
  3. Compute the curl of \(\mathbf A=f(r)\mathbf e_r\), assuming \(f\) is differentiable.
  4. For \(\mathbf A=r\sin\theta\,\mathbf e_\phi\), what are the radial and polar components of curl?
  5. In the azimuthal curl component, which product must be differentiated with respect to \(r\)?

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