# The Divergence and Curl of a Vector Field In Two Dimensions

From The Divergence of a Vector Field and The Curl of a Vector Field pages we gave formulas for the divergence and for the curl of a vector field $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ on $\mathbb{R}^3$ given by the following formulas:

(1)

(2)

Now suppose that $\mathbf{F}(x, y) = P(x, y) \vec{i} + Q(x, y) \vec{j}$ is a vector field in $\mathbb{R}^2$. Then we define the divergence and curl of $\mathbb{F}$ as follows:

Definition: If $\mathbf{F}(x, y) = P(x, y)\vec{i} + Q(x, y) \vec{j}$ and $\frac{\partial P}{\partial x}$ and $\frac{\partial Q}{\partial y}$ both exist then the Divergence of $\mathbf{F}$ is the scalar field given by $\mathrm{div} (\mathbf{F}) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$. |

Definition: If $\mathbf{F}(x, y) = P(x,y)\vec{i} + Q(x, y) \vec{j}$ and $\frac{\partial Q}{\partial x}$ and $\frac{\partial P}{\partial y}$ both existence then the Curl of $\mathbf{F}$ is the vector field given by $\mathrm{curl} (\mathbf{F}) = \left ( \frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y} \right ) \vec{k}$. |

*It is important to note that the curl of $\mathbf{F}$ exists in three dimensional space despite $\mathbf{F}$ be a vector field on $\mathbb{R}^2$.*

## Example 1

**Find the divergence of the vector field $\mathbf{F}(x, y) = 2xy \vec{i} + 3 \cos y \vec{j}$.**

We can apply the formula above directly to get that:

(3)

## Example 2

**Find the divergence of the vector field $\mathbf{F}(x, y) = e^x y^2 \vec{i} + (x + 2y) \vec{j}$.**

We can apply the formula above directly to get that:

(4)

## Example 3

**Find the curl of the vector field $\mathbf{F}(x, y) = 2xy \vec{i} + 3 \cos y \vec{j}$.**

We can apply the formula above directly to get that:

(5)

## Example 4

**Find the curl of the vector field $\mathbf{F}(x, y) = e^x y^2 \vec{i} + (x + 2y) \vec{j}$.**

We can apply the formula above directly to get that:

(6)

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