16.5 - Curl and Divergence

Two important operations on vector fields are curl and divergence. We will define them in terms of the nabla operator $\nabla$:

$$\nabla =⟨\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}⟩$$

Note the difference between $\nabla$ and $\nabla f$: $\nabla$ merely contains the operations of differentiation, and we define:

$$\nabla f=⟨\frac{\partial f}{\partial x},\frac{\partial f}{partaily},\frac{\partial f}{partailz}⟩$$

Curl and Divergence are defined as:

Defintion

$$Curl:curl(\vec{F})=\nabla \times \vec{F}$$ $$Divergence:div(\vec{F})=\nabla \cdot \vec{F}$$

Curl is a vector quantity, and divergence is a scalar quantity. If we write $\vec{F}=⟨P(x,y,z),Q(x,y,z),R(x,y,z)⟩$, then the curl written in component form is:

$$\begin{array}{rl}curl(\vec{F})=\nabla \times \vec{F}& =\left|\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|\\ & =i\left(\frac{\partial }{\partial y}R-\frac{\partial }{\partial z}Q\right)-j\left(\frac{\partial }{\partial x}R-\frac{\partial }{\partial z}P\right)+k\left(\frac{\partial }{\partial x}Q-\frac{\partial }{\partial y}P\right)\\ & =⟨R_y-Q_z,P_z-R_x,Q_x-P_y⟩\end{array}$$

Observe

The z-component of curl is $Q_x-P_y$, a quantity we have seen several times now.

In component form, the divergence of $\vec{F}$ is:

$$\begin{array}{rl}div(\vec{F})& =\nabla \cdot \vec{F}=⟨\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}⟩\cdot ⟨P,Q,R⟩\\ & =P_x+Q_y+R_z\end{array}$$

We will discuss the interpretation of the curl and the divergence later in the lecture. For now, we will see how to compute them, and investigate some of their properties.

---

Example

Find the curl and divergence of $\vec{F}=⟨x^{2}y,2y^{2}z,3z⟩$

Solution

Use the formulae from the beggining

$$\begin{array}{rl}curl\vec{F}& =\nabla \times \vec{F}=\left|\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ x^{2}y& 2y^3& 3z\end{array}\right|\\ & =⟨(3z{)}_y-(2y^{3}z{)}_z,(x^{2}y{)}_z-(3z{)}_x,(2y^{3}z{)}_x-(x^{2}y{)}_y⟩\\ & =⟨-2y^3,0,-x^2⟩\\ & \\ & \\ div\vec{F}& =\nabla \cdot \vec{F}=(x^{2}y{)}_x+(2y^{3}z{)}_y+(3z{)}_z\\ & =2xy+6y^{2}z+3\end{array}$$

---

Properties of Divergence and Curl

The divergence and curl operators have some interesting and useful properties.