17. Divergence, Curl and Potentials
The del operator will be heavily used in our study of divergence and curl and will lend further understanding of the gradient. So that is where we start.
a. The Del Operator
Question: What is the difference between \(\dfrac{df}{dx}\) and \(\dfrac{d}{dx}\)?
Answer: \(\dfrac{df}{dx}\) is the derivative of the function \(f(x)\). However, \(\dfrac{d}{dx}\) is a differential operator which is ready to differentiate any function but does not yet have a function to differentiate. If we have a formula for \(f(x)\), then we can find a formula for \(\dfrac{df}{dx}\). There is NO formula for \(\dfrac{d}{dx}\) by itself.
Question: Similarly, what is the difference between the gradient, \(\vec\nabla f\) and \(\vec\nabla\)?
Answer: The gradient, \(\vec\nabla f\), is the vector of the \(3\) partial derivatives of \(f\). However:
The del operator, denoted by the symbol \(\vec\nabla\), is the vector differential operator given in rectangular coordinates as: \[ \vec\nabla=\left\langle \dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \dfrac{\partial}{\partial z}\right\rangle =\left\langle \partial_x,\partial_y,\partial_z \right\rangle \] (The second formula is simply a more compact way to write the derivatives.) Del is nothing but a symbol which represents a vector of three differential operators which are ready to differentiate a scalar field (ordinary function) or the components of a vector field.
There are formulas for the del operator in curvilinear coordinates (such as polar, cylindrical or spherical), but they are beyond the scope of this course. In this course, we will always use the del operator in rectangular coordinates, and then convert the results to curvilinear coordinates if needed.
Note that del is not the gradient. However, the gradient
is one of the three common operations performed with the del operator.
These mimic
1. the scalar product of a scalar and a vector,
2. the dot product of two vectors and
3. the cross product of two vectors.
In each case, the del operator plays the roll of one of the vectors.

When the del operator acts on an ordinary function (scalar field), \(f(x,y,z)\), it produces the gradient: \[ \text{grad} f=\vec\nabla f =\left\langle \dfrac{\partial f}{\partial x}, \dfrac{\partial f}{\partial y}, \dfrac{\partial f}{\partial z}\right\rangle \]

When the del operator acts on a vector field, \(\vec F(x,y,z)\), by the dot product, it produces the divergence: \[\begin{aligned} \text{div} \vec F &=\vec\nabla\cdot\vec F =\left\langle \dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \dfrac{\partial}{\partial z}\right\rangle \cdot\left\langle F_1, F_2, F_3 \right\rangle \\[2pt] &=\dfrac{\partial F_1}{\partial x} + \dfrac{\partial F_2}{\partial y} + \dfrac{\partial F_3}{\partial z} =\partial_x F_1+\partial_y F_2+\partial_z F_3 \end{aligned}\]

When the del operator acts on a vector field, \(\vec F(x,y,z)\), by the cross product, it produces the curl: \[\begin{aligned} \text{curl} \vec F&=\vec\nabla\times\vec F =\begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ F_1 & F_2 & F_3 \end{vmatrix} \\ &=\left\langle \partial_y F_3\partial_z F_2, \partial_z F_1\partial_x F_3, \partial_x F_2\partial_y F_1 \right\rangle \end{aligned}\]
When we compute a usual scalar multiple, dot product or cross produce, we multiply the function and/or components. When we compute a gradient, divergence or curl, the components of del do not multiply the function or components, they differentiate the function or component.
The gradient was discussed in a previous chapter. The divergence and curl are discussed on the next few pages.
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