Two Different Questions About a Vector Field
A vector field can describe motion, flow, or another quantity that varies from point to point. Two local questions are especially useful: does the field tend to expand or contract a small region near a point, and does it have a local tendency to rotate? Divergence addresses the first question; curl addresses the second.
These questions are related to different combinations of the field’s component derivatives. A field may have a positive divergence and nonzero curl at the same point, or either quantity may vanish while the other does not. The classification must therefore be based on both calculations, not on the appearance of the field or on one component alone.
Component Formulas and Local Classification
In Cartesian coordinates, write a differentiable vector field as \(\mathbf{F}(x,y,z)=\langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle\). The formulas below apply wherever the indicated first partial derivatives exist. The curl convention uses the standard right-handed orientation.
A positive divergence at a point indicates local outward or source-like behavior there; a negative divergence indicates local inward or sink-like behavior. A zero divergence means there is no first-order local volume expansion or contraction at that point. It does not mean the vector field itself is zero.
A nonzero curl indicates local rotational tendency. Its direction gives the axis and orientation of that tendency according to the right-hand rule, and its magnitude measures its strength. If the curl is zero at a point, the local rotational measure vanishes there; this does not by itself rule out curved paths or rotation elsewhere.
Why the Two Quantities Measure Different Behavior
The distinction can be justified from the first-order behavior of a differentiable field. Near a point \(\mathbf{x}_0\), its change is approximated by its Jacobian matrix \(\mathbf{J}\), whose entries are the partial derivatives of the components. For a small displacement \(\mathbf{h}\), the linear approximation is \(\mathbf{F}(\mathbf{x}_0+\mathbf{h})\approx\mathbf{F}(\mathbf{x}_0)+\mathbf{J}\mathbf{h}\).
The trace of \(\mathbf{J}\), the sum of its diagonal entries, is \(P_x+Q_y+R_z\). To see why this is the local expansion measure, consider the small displacement map \(\mathbf{x}\mapsto\mathbf{x}+t\mathbf{F}(\mathbf{x})\). Its derivative matrix at \(\mathbf{x}_0\) is \(\mathbf{I}+t\mathbf{J}\). The determinant expansion at \(t=0\) is \(\det(\mathbf{I}+t\mathbf{J})=1+t\,\operatorname{tr}(\mathbf{J})+O(t^2)\). Thus divergence is the first-order signed change in volume under this displacement.
In contrast, the antisymmetric part of \(\mathbf{J}\), namely \((\mathbf{J}-\mathbf{J}^{\top})/2\), records the rotational part of the linear change. Directly grouping the off-diagonal entries gives
This equality follows by substituting the three curl components from the definition. It shows why divergence and curl answer different questions: divergence is the trace, while curl is determined by the antisymmetric cross-component differences. Neither quantity alone describes all the local behavior.
Worked Examples
Worked Example: Positive Divergence and Nonzero Curl
Let \(\mathbf{F}(x,y,z)=\langle x^2-y,\;2y+z,\;3z-x\rangle\). Classify the field at \((1,0,0)\).
Set \(P=x^2-y\), \(Q=2y+z\), and \(R=3z-x\). For divergence, the component derivatives are \(P_x=2x\), \(Q_y=2\), and \(R_z=3\). Therefore, throughout the domain,
At the specified point, \(\nabla\cdot\mathbf{F}(1,0,0)=2(1)+5=7\), which is positive. The field is locally source-like there.
For curl, compute each component: \(R_y-Q_z=0-1=-1\), \(P_z-R_x=0-(-1)=1\), and \(Q_x-P_y=0-(-1)=1\). Hence
This curl is constant, so at \((1,0,0)\) it is also \(\langle-1,1,1\rangle\), which is nonzero. The classification is therefore source-like and rotational. As a check, the derivatives used above come directly from \(P=x^2-y\), \(Q=2y+z\), and \(R=3z-x\); substituting the point gives divergence \(7\), while the three curl components remain \(-1,1,1\).
Worked Example: Zero Curl with Divergence That Changes Sign
Let \(\mathbf{F}(x,y,z)=\langle x^2,y^2,z^2\rangle\). First compute the divergence throughout space. The diagonal component derivatives are \(P_x=2x\), \(Q_y=2y\), and \(R_z=2z\), so
The curl components are \(R_y-Q_z=0-0=0\), \(P_z-R_x=0-0=0\), and \(Q_x-P_y=0-0=0\). Thus \(\nabla\times\mathbf{F}=\mathbf{0}\) everywhere.
At \((1,1,1)\), the divergence is \(2(1+1+1)=6\), so the field is source-like there, with no local rotational measure. At \((-1,-1,-1)\), it is \(2(-1-1-1)=-6\), so it is sink-like there, again with zero curl. On the plane \(x+y+z=0\), divergence is zero; the curl remains zero there as well. The sign change follows directly from the factor \(x+y+z\).
As a check, for example, at \((1,1,1)\) the component derivatives contributing to divergence are \(2,2,2\), whose sum is \(6\); all cross-component derivatives contributing to curl are zero. At \((-1,-1,-1)\), those diagonal derivatives are \(-2,-2,-2\), whose sum is \(-6\), and the curl components are still all zero.
Worked Example: Zero Divergence with Nonzero Curl
Let \(\mathbf{F}(x,y,z)=\langle-y,x,0\rangle\), so \(P=-y\), \(Q=x\), and \(R=0\). The divergence derivatives are \(P_x=0\), \(Q_y=0\), and \(R_z=0\). Therefore, \(\nabla\cdot\mathbf{F}=0\) throughout space.
For curl, \(R_y-Q_z=0-0=0\), \(P_z-R_x=0-0=0\), and \(Q_x-P_y=1-(-1)=2\). Consequently,
At every point, this field has zero divergence and nonzero curl pointing in the positive \(z\)-direction. It has no first-order source or sink behavior, but it does have rotational behavior. To check the calculation, differentiating \(Q=x\) with respect to \(x\) gives \(1\), and differentiating \(P=-y\) with respect to \(y\) gives \(-1\); their difference is \(2\), as found.
Worked Example: Both Quantities Depend on Location
Let \(\mathbf{F}(x,y,z)=\langle xy,yz,zx\rangle\), with \(P=xy\), \(Q=yz\), and \(R=zx\). The diagonal derivatives are \(P_x=y\), \(Q_y=z\), and \(R_z=x\), giving
The curl components are \(R_y-Q_z=0-y=-y\), \(P_z-R_x=0-z=-z\), and \(Q_x-P_y=0-x=-x\). Thus \(\nabla\times\mathbf{F}=\langle-y,-z,-x\rangle\).
At \((1,2,3)\), divergence is \(1+2+3=6\), and curl is \(\langle-2,-3,-1\rangle\). This point is source-like and rotational. At the origin, the divergence and all three curl components are zero, so neither local measure is present at that point. The two conclusions at the origin do not extend to all of space: for instance, the values at \((1,2,3)\) are nonzero. As a check at \((1,2,3)\), the diagonal derivatives are \(2,3,1\), summing to \(6\), and the curl components evaluate to \(-2,-3,-1\), respectively.
A Reliable Classification Procedure
For a problem involving both quantities, keep the divergence and curl calculations separate until each is complete. Then evaluate them at the required point or describe their values across the required region.
Write \(\mathbf{F}=\langle P,Q,R\rangle\) so that the derivative formulas can be applied without confusing a component with its variable.
Find \(P_x\), \(Q_y\), and \(R_z\), then add them. Record whether the result is positive, negative, zero, or dependent on position.
Use the prescribed differences in order. Keep the result as a vector; a single nonzero component is enough for nonzero curl.
Substitute the point only after obtaining the formulas, unless the problem asks for classification throughout a domain. State source or sink behavior and rotational behavior separately.
Common Pitfall: Letting One Quantity Stand In for the Other
A frequent error is to infer “no rotation” from zero divergence, or “no source” from zero curl. These are different derivative combinations. The field \(\langle-y,x,0\rangle\) provides a direct correction: its divergence is \(0\), but its curl is \(\langle0,0,2\rangle\), not zero. It has zero local volume expansion and nonzero local rotational tendency.
Another error is to mistake a zero value at a point for a property throughout a region. For \(\langle xy,yz,zx\rangle\), both divergence and curl vanish at the origin, but at \((1,2,3)\) they are \(6\) and \(\langle-2,-3,-1\rangle\). Pointwise classification must not be silently extended to the whole field.
Why Mixed Classification Matters
In applications, a field’s local behavior may combine expansion or contraction with rotation. Reporting both quantities preserves information that either one alone would miss. A positive divergence does not prevent nonzero curl, and a nonzero curl does not determine the sign of divergence.
For any specific problem, the reliable evidence is the component calculation: divergence is the sum of the three same-variable derivatives, while curl consists of the three specified cross-derivative differences. Once those values are checked and evaluated at the correct location, the local classification follows without relying on a visual guess.
Check Your Understanding
For each question, show the relevant component derivatives before making a classification.
- For \(\mathbf{F}=\langle 2x,-y,3z\rangle\), find divergence and curl throughout space. Is the field source-like, sink-like, or neither at \((0,0,0)\), and does it have rotational behavior there?
- For \(\mathbf{F}=\langle -y,x,0\rangle\), explain why zero divergence does not imply zero curl.
- For \(\mathbf{F}=\langle x^2,y^2,z^2\rangle\), state where divergence is positive, negative, or zero, and give its curl everywhere.
- Can divergence and curl both be nonzero at the same point? Identify such a point from one worked example and give both values.
- What additional condition is needed to call a field solenoidal on a region, rather than to say its divergence is zero at one point?