Product Identities for Curl and Divergence

Why Cross-Product Product Rules Matter

The divergence and curl of a vector field describe different features of its local variation. When two fields are combined through a cross product, differentiating that product brings in contributions from both fields. The resulting identities are useful because they express the divergence or curl of the combined field in terms of the original fields and their derivatives.

These formulas are not ordinary scalar product rules with a cross product substituted for multiplication. Their signs and directional-derivative terms depend on the order of the vectors. A careful derivation makes those details easier to remember and provides a reliable check on calculations.

Definition: Let \(\mathbf F=(F_1,F_2,F_3)\) and \(\mathbf G=(G_1,G_2,G_3)\) be vector fields on an open subset \(U\) of \(\mathbb R^3\). If their component functions have continuous first partial derivatives on \(U\), then the identities in this tutorial hold at every point of \(U\).

The Two Product Identities

The gradient operator is \(\nabla=(\partial_x,\partial_y,\partial_z)\). For a vector field \(\mathbf F=(F_1,F_2,F_3)\), its divergence and curl are

\nabla\cdot\mathbf F =\partial_x F_1+\partial_y F_2+\partial_z F_3, \qquad \nabla\times\mathbf F = \left( \partial_yF_3-\partial_zF_2,\, \partial_zF_1-\partial_xF_3,\, \partial_xF_2-\partial_yF_1 \right).

The cross product \(\mathbf F\times\mathbf G\) is another vector field. The two rules give its divergence and curl:

Theorem: If \(\mathbf F\) and \(\mathbf G\) have continuous first partial derivatives, then

\nabla\cdot(\mathbf F\times\mathbf G) = \mathbf G\cdot(\nabla\times\mathbf F) - \mathbf F\cdot(\nabla\times\mathbf G),

\nabla\times(\mathbf F\times\mathbf G) = \mathbf F(\nabla\cdot\mathbf G) -\mathbf G(\nabla\cdot\mathbf F) +(\mathbf G\cdot\nabla)\mathbf F -(\mathbf F\cdot\nabla)\mathbf G.

In the second identity, the directional derivatives act on each component of the vector field that follows them. Explicitly,

\big((\mathbf G\cdot\nabla)\mathbf F\big)_i = \sum_{j=1}^{3}G_j\,\partial_jF_i, \qquad \big((\mathbf F\cdot\nabla)\mathbf G\big)_i = \sum_{j=1}^{3}F_j\,\partial_jG_i.

For example, if \(\mathbf G=(G_1,G_2,G_3)\), then \(\mathbf G\cdot\nabla\) means \(G_1\partial_x+G_2\partial_y+G_3\partial_z\). This differential operator acts on the components of \(\mathbf F\); it does not mean the scalar dot product of two vector fields.

Why the Identities Hold

Both formulas follow from the ordinary product rule applied to components. Here is a compact derivation using index notation. Indices \(i,j,k,l,m\) range from 1 to 3, and a repeated index is summed over those three values. The symbol \(\varepsilon_{ijk}\) is \(1\) or \(-1\) according to whether \((i,j,k)\) is an even or odd permutation of \((1,2,3)\), and is \(0\) if any two indices match. It encodes the signs in the cross product and curl.

Differentiate the cross product for the divergence rule.
The \(i\)th component of \(\mathbf F\times\mathbf G\) is \(\varepsilon_{ijk}F_jG_k\). Applying the product rule to its divergence gives \(\partial_i(\varepsilon_{ijk}F_jG_k)=\varepsilon_{ijk}(\partial_iF_j)G_k+\varepsilon_{ijk}F_j(\partial_iG_k)\).

Recognize the two curl dot products.
Since \((\nabla\times\mathbf F)_k=\varepsilon_{kij}\partial_iF_j\) and \(\varepsilon_{ijk}=\varepsilon_{kij}\), the first term is \(\mathbf G\cdot(\nabla\times\mathbf F)\). Also \((\nabla\times\mathbf G)_j=\varepsilon_{jik}\partial_iG_k\) and \(\varepsilon_{ijk}=-\varepsilon_{jik}\), so the second term is \(-\mathbf F\cdot(\nabla\times\mathbf G)\).

Differentiate the cross product for the curl rule.
The \(i\)th component of its curl is \(\varepsilon_{ijk}\partial_j(\varepsilon_{klm}F_lG_m)\). Use \(\varepsilon_{ijk}\varepsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}\), where \(\delta_{ab}\) is 1 if \(a=b\) and 0 otherwise.

Apply the product rule and group terms.
The result is \(\partial_m(F_iG_m)-\partial_l(F_lG_i)\). Expanding gives \(G_m\partial_mF_i+F_i\partial_mG_m-G_i\partial_lF_l-F_l\partial_lG_i\), precisely the \(i\)th component of the stated curl identity.

This derivation also explains the hypotheses: differentiating each product requires first partial derivatives of the field components. The identities are pointwise formulas on a region where those derivatives exist; the stated continuous differentiability condition is a standard sufficient condition for using them throughout the region.

Worked Examples

Worked Example: Checking Both Rules with Nonzero Curls

Let \(\mathbf F=(y,0,0)\) and \(\mathbf G=(0,x,0)\). First find their cross product:

\mathbf F\times\mathbf G = \left(0,0,yx\right) = (0,0,xy).

Its divergence is \(\partial_x0+\partial_y0+\partial_z(xy)=0\), since \(xy\) does not depend on \(z\). Next,

\nabla\times\mathbf F=(0,0,-1), \qquad \nabla\times\mathbf G=(0,0,1).

Indeed, the third curl component of \(\mathbf F\) is \(\partial_xF_2-\partial_yF_1=0-1=-1\); that of \(\mathbf G\) is \(\partial_xG_2-\partial_yG_1=1-0=1\). Both dot products on the divergence rule's right-hand side vanish, because \(\mathbf F\) and \(\mathbf G\) have zero third components. Thus its right-hand side is \(0-0=0\), matching the directly computed divergence.

For the curl rule, the curl of \((0,0,xy)\) is

\nabla\times(\mathbf F\times\mathbf G) = (\partial_y(xy),-\partial_x(xy),0) = (x,-y,0).

Both divergences are zero: \(\nabla\cdot\mathbf F=\partial_x y=0\) and \(\nabla\cdot\mathbf G=\partial_y x=0\). The directional terms are

(\mathbf G\cdot\nabla)\mathbf F =x\partial_y(y,0,0)=(x,0,0), \qquad (\mathbf F\cdot\nabla)\mathbf G =y\partial_x(0,x,0)=(0,y,0).

The right-hand side is therefore \((x,0,0)-(0,y,0)=(x,-y,0)\), agreeing with the direct curl calculation.

Worked Example: A Nonzero Divergence of a Cross Product

Take \(\mathbf F=(0,x,0)\) and \(\mathbf G=(0,0,z)\). Their cross product is

\mathbf F\times\mathbf G=(xz,0,0), \qquad \nabla\cdot(\mathbf F\times\mathbf G)=\partial_x(xz)=z.

The curls are \(\nabla\times\mathbf F=(0,0,1)\) and \(\nabla\times\mathbf G=(0,0,0)\). Consequently,

\mathbf G\cdot(\nabla\times\mathbf F) - \mathbf F\cdot(\nabla\times\mathbf G) = (0,0,z)\cdot(0,0,1) - (0,x,0)\cdot(0,0,0) =z.

This matches the direct divergence. The example shows that the divergence of a cross product need not be zero, even when one of the fields has zero curl.

Worked Example: Applying the Curl Rule with Nonzero Divergences

Let \(\mathbf F=(x,y,0)\) and \(\mathbf G=(0,0,z)\). Their cross product and its curl are

\mathbf F\times\mathbf G=(yz,-xz,0), \qquad \nabla\times(\mathbf F\times\mathbf G) = (x,y,-2z).

For the direct curl, the components are \(\partial_y0-\partial_z(-xz)=x\), \(\partial_z(yz)-\partial_x0=y\), and \(\partial_x(-xz)-\partial_y(yz)=-z-z=-2z\).

Now \(\nabla\cdot\mathbf F=1+1+0=2\) and \(\nabla\cdot\mathbf G=1\). Since \(\mathbf F\) has no \(z\)-dependence, \((\mathbf G\cdot\nabla)\mathbf F=z\partial_z\mathbf F=(0,0,0)\). Since \(\mathbf G\) depends only on \(z\), \((\mathbf F\cdot\nabla)\mathbf G=(x\partial_x+y\partial_y)\mathbf G=(0,0,0)\). The identity's right-hand side is therefore

\mathbf F(1)-\mathbf G(2)+(0,0,0)-(0,0,0) =(x,y,0)-(0,0,2z) =(x,y,-2z).

This agrees with the direct calculation, including the negative third component.

Common Sign and Component Errors

A frequent error is to miscalculate one curl component and then trust the product identity to confirm it. For example, with \(\mathbf F=(y,0,0)\), it is incorrect to claim that \(\nabla\times\mathbf F=(0,0,0)\). The third component is not \(\partial_yF_1-\partial_xF_2\); the definition gives

(\nabla\times\mathbf F)_3 = \partial_xF_2-\partial_yF_1 = \partial_x0-\partial_y y = -1.

Thus \(\nabla\times\mathbf F=(0,0,-1)\). In the first worked example, its dot product with \(\mathbf G=(0,x,0)\) is still zero, but that does not make the curl itself zero. A dot product may hide an incorrect component when the corresponding vector component vanishes.

Another common mistake is to reverse the order of the two terms in the divergence identity, or to omit one directional derivative in the curl identity. A reliable procedure is to write the formula before substituting values, preserve the order of \(\mathbf F\) and \(\mathbf G\), and calculate each component of each directional derivative explicitly.

Takeaway: The divergence identity has two curl dot products with a minus sign between them. The curl identity has two divergence terms and two directional derivatives; the derivative along \(\mathbf G\) acts on \(\mathbf F\), while the derivative along \(\mathbf F\) acts on \(\mathbf G\), with a minus sign on the latter.

When These Rules Are Useful

The identities let you choose a calculation suited to the information available. If the cross product itself is cumbersome but the original curls are simple, the divergence rule may be shorter. If the original divergences and directional derivatives are easy to find, the curl rule can avoid expanding all three components of \(\mathbf F\times\mathbf G\) and differentiating them.

They are also useful for checking work. Compute one side directly and the other from the identity; agreement is a strong check on component arithmetic and signs. Such a check is meaningful only if the fields are differentiable where the calculation is performed and if the components are evaluated consistently at the same point.

Check Your Understanding

Use the definitions and identities above to answer the following questions. Show component calculations where appropriate.

For \(\mathbf F=(0,z,0)\) and \(\mathbf G=(x,0,0)\), find \(\nabla\cdot(\mathbf F\times\mathbf G)\) directly and using the divergence identity.
In \((\mathbf G\cdot\nabla)\mathbf F\), which field supplies the directional derivative and which field is differentiated?
State the differentiability condition used here and explain why the product identities need first partial derivatives.
For \(\mathbf F=(x,0,0)\) and \(\mathbf G=(0,y,0)\), calculate \(\nabla\times(\mathbf F\times\mathbf G)\) and the right-hand side of the curl identity.
Why does a zero dot product \(\mathbf G\cdot(\nabla\times\mathbf F)\) not imply that \(\nabla\times\mathbf F\) is the zero vector?