Divergence and Scalar Curl in Two Dimensions

Why Planar Fields Have Two Local Measures

A vector field in the plane assigns a two-component vector to each point. Two different local features can be useful: whether the field tends to carry more flow outward than inward near a point, and whether it tends to turn around that point. Divergence measures the first feature; scalar curl measures the second.

These quantities use different combinations of the same component derivatives. Keeping those combinations—and especially the order in the curl formula—straight is essential. The planar definition also fits directly into the three-dimensional curl convention: the scalar planar curl becomes the third component when the field is viewed as a three-dimensional field with zero vertical component.

Definition: Let $\mathbf F(x,y)=(P(x,y),Q(x,y))$, where $P$ and $Q$ have continuous first partial derivatives near the point under consideration. Its planar divergence and scalar curl are $$ \operatorname{div}\mathbf F=P_x+Q_y, \qquad \operatorname{curl}_{2D}\mathbf F=Q_x-P_y. $$ Here $P_x=\frac{\partial P}{\partial x}$, and the other subscripts have the same meaning. Scalar curl is a number, not a two-component vector.

Both quantities are evaluated at a point by substituting that point into the resulting derivative formulas. The differentiability condition ensures that these component derivatives exist and behave continuously nearby, as needed for the local interpretations below.

Computing Divergence and Scalar Curl

For divergence, differentiate the first component $P$ with respect to $x$, differentiate the second component $Q$ with respect to $y$, and add. For scalar curl, differentiate $Q$ with respect to $x$, differentiate $P$ with respect to $y$, and subtract the second result from the first:

\operatorname{div}\mathbf F=P_x+Q_y, \qquad \operatorname{curl}_{2D}\mathbf F=Q_x-P_y.

The placement of the components matters. In particular, planar curl is not $P_y-Q_x$; that expression has the opposite sign. The convention above corresponds to positive counterclockwise orientation in the plane and to the usual right-handed three-dimensional curl.

Worked Example: A Field with Both Divergence and Curl

Let $\mathbf F(x,y)=(x^2-y,\;x+3y)$. Find its divergence and scalar curl, then evaluate them at $(1,2)$.

Here $P=x^2-y$ and $Q=x+3y$. The required partial derivatives are $P_x=2x$, $Q_y=3$, $Q_x=1$, and $P_y=-1$. Therefore, \[ \operatorname{div}\mathbf F=2x+3, \qquad \operatorname{curl}_{2D}\mathbf F=1-(-1)=2. \] At $(1,2)$, the divergence is $2(1)+3=5$, and the scalar curl is $2$.

As a check, the field at that point is $\mathbf F(1,2)=(-1,7)$. The component derivatives used above are consistent with the formulas: the $x$-change in the first component is $2x=2$ there, and the $y$-change in the second is $3$, giving divergence $2+3=5$. For curl, $Q_x=1$ and $P_y=-1$, giving $1-(-1)=2$.

Worked Example: A Pure Rotational Pattern

Consider $\mathbf F(x,y)=(-3y,\;3x)$. Compute its divergence and scalar curl at an arbitrary point.

Set $P=-3y$ and $Q=3x$. Then $P_x=0$, $Q_y=0$, $Q_x=3$, and $P_y=-3$. Thus \[ \operatorname{div}\mathbf F=0+0=0, \qquad \operatorname{curl}_{2D}\mathbf F=3-(-3)=6. \] These values are the same at every point. The zero divergence says there is no local net source strength, while the positive scalar curl indicates a counterclockwise rotational tendency. The calculation also checks the sign convention: reversing the order to $P_y-Q_x$ would incorrectly give $-6$.

Worked Example: Values Depending on Position

Let $\mathbf F(x,y)=(xy,\;x^2-y^2)$. Find divergence and scalar curl, then evaluate them at $(2,-1)$.

With $P=xy$ and $Q=x^2-y^2$, the partial derivatives are $P_x=y$, $Q_y=-2y$, $Q_x=2x$, and $P_y=x$. Hence \[ \operatorname{div}\mathbf F=y-2y=-y, \qquad \operatorname{curl}_{2D}\mathbf F=2x-x=x. \] At $(2,-1)$, these become $\operatorname{div}\mathbf F=-(-1)=1$ and $\operatorname{curl}_{2D}\mathbf F=2$.

Direct substitution in the derivative expressions gives the same results: $P_x=-1$, $Q_y=2$, so divergence is $-1+2=1$; and $Q_x=4$, $P_y=2$, so scalar curl is $4-2=2$.

Connection to Three-Dimensional Curl

To compare planar and spatial notation, regard the planar field as a three-dimensional field that does not depend on $z$ and has zero $z$-component. Define $\widetilde{\mathbf F}(x,y,z)=(P(x,y),Q(x,y),0)$. In Cartesian coordinates the three-dimensional curl is \[ \nabla\times\widetilde{\mathbf F} = \left( \frac{\partial \widetilde F_3}{\partial y} -\frac{\partial \widetilde F_2}{\partial z}, \frac{\partial \widetilde F_1}{\partial z} -\frac{\partial \widetilde F_3}{\partial x}, \frac{\partial \widetilde F_2}{\partial x} -\frac{\partial \widetilde F_1}{\partial y} \right). \] The first two components vanish: $\widetilde F_3=0$, and $P,Q$ have no $z$-dependence. The third component is $Q_x-P_y$. Therefore:

Theorem: For $\widetilde{\mathbf F}(x,y,z)=(P(x,y),Q(x,y),0)$, $$ \nabla\times\widetilde{\mathbf F} = (0,0,\operatorname{curl}_{2D}\mathbf F) = (0,0,Q_x-P_y). $$ The planar scalar curl is exactly the vertical component of the three-dimensional curl of this extension.

The qualification about the extension is important. A general three-dimensional field may have nonzero components in all three directions and may vary with $z$; its curl is then a three-component vector. The scalar formula here describes a planar field embedded with zero vertical component, not an arbitrary spatial field.

Why Scalar Curl Measures Local Circulation

The sign and component order in the scalar curl formula can be justified by measuring circulation around a small counterclockwise rectangle. Let its lower-left corner be $(x,y)$, its horizontal side length be $h>0$, and its vertical side length be $k>0$. Circulation around a curve means the line integral $\int \mathbf F\cdot d\mathbf r$. For a planar field, its integrand along a path is $P\,dx+Q\,dy$.

Going counterclockwise, the bottom edge runs from $x$ to $x+h$, the right edge from $y$ to $y+k$, and the top and left edges run in the negative coordinate directions. The total circulation $C$ is therefore

\begin{aligned} C={}&\int_x^{x+h}P(s,y)\,ds +\int_y^{y+k}Q(x+h,t)\,dt\\ &-\int_x^{x+h}P(s,y+k)\,ds -\int_y^{y+k}Q(x,t)\,dt. \end{aligned}

Group the bottom and top contributions, and then the right and left contributions. By the fundamental theorem of calculus, their differences can be written as integrals of partial derivatives:

\begin{aligned} C &=\int_x^{x+h}\bigl(P(s,y)-P(s,y+k)\bigr)\,ds\\ &\quad+\int_y^{y+k}\bigl(Q(x+h,t)-Q(x,t)\bigr)\,dt\\ &=-\int_x^{x+h}\int_y^{y+k}P_y(s,t)\,dt\,ds +\int_y^{y+k}\int_x^{x+h}Q_x(s,t)\,ds\,dt. \end{aligned}

Dividing by the rectangle's area $hk$ shows that $C/(hk)$ is the average value of $Q_x-P_y$ over that rectangle. As $h$ and $k$ shrink to zero, continuity of the partial derivatives makes this average approach their value at $(x,y)$. Thus \[ \lim_{h,k\to0^+}\frac{C}{hk}=Q_x(x,y)-P_y(x,y). \] This explains both the formula and its sign: counterclockwise circulation per unit area approaches the scalar curl.

Interpreting the Two Quantities

Divergence and scalar curl describe distinct local tendencies. Divergence is the signed local source strength: positive divergence indicates net outward flow from a sufficiently small neighborhood, and negative divergence indicates net inward flow. Zero divergence means that this local measure of net outward flow vanishes; it does not by itself mean that the field is zero.

Scalar curl measures local rotational tendency with an orientation. Positive scalar curl corresponds to a counterclockwise tendency, and negative scalar curl to a clockwise tendency. A large magnitude indicates a stronger local circulation per unit area. These are local statements: the values at a point do not, on their own, describe the total flow or circulation across a large region.

A simple rotation field illustrates the interpretation. For $\mathbf F=(-\omega y,\omega x)$, where $\omega$ is constant, the formulas give divergence $0$ and scalar curl $2\omega$. The field turns counterclockwise when $\omega>0$, and clockwise when $\omega<0$. The factor of two arises because the two component derivatives contribute: $Q_x=\omega$ and $P_y=-\omega$.

Common Sign and Component Errors

A frequent mistake is to write planar curl as $P_y-Q_x$, reversing the convention. For the field $\mathbf F=(-y,x)$, that incorrect calculation gives \[ P_y-Q_x=-1-1=-2. \] But the defined scalar curl is \[ Q_x-P_y=1-(-1)=2. \] The positive answer agrees with the counterclockwise circulation of this field around the origin and with the third component of its three-dimensional curl.

Another error is to use $P_x+Q_y$ for curl. That sum is divergence, not curl. A useful check is to write the components beside each derivative before combining them: divergence pairs each component with its own coordinate derivative, while scalar curl differentiates the second component with respect to $x$ and subtracts the first component's derivative with respect to $y$.

Key takeaway: For $\mathbf F=(P,Q)$, divergence is $P_x+Q_y$, while scalar planar curl is $Q_x-P_y$. Under the zero-vertical-component extension, scalar curl is the third component of three-dimensional curl and is the limiting counterclockwise circulation per unit area.

Check Your Understanding

For each question, identify the relevant derivatives before simplifying.

For $\mathbf F(x,y)=(2x-y,\;x+4y)$, find divergence and scalar curl at $(1,0)$.
What three-dimensional field extension makes the third component of curl equal to the scalar planar curl?
For $\mathbf F=(-y,x)$, what is the scalar curl, and what sign of local rotation does it indicate?
Why does the counterclockwise rectangle calculation produce $Q_x-P_y$, rather than $P_y-Q_x$?
Can a field have zero divergence and nonzero scalar curl? Give the field from this tutorial that demonstrates this.