4.4 divergence and curl of a Vector Field

Since \(\nabla \) has the character of a vector, we can consider the dot and cross product on which it appears as one factor. If \[F = F_1 \textbf {i} + F_2 \textbf {j} + F_3\textbf {k}\] is a vector whose components are differentiable functions of \(x, y, z\)

1.

The dot product \begin {align*} \nabla \cdot F & = \Bigg (\textbf {i}\frac {\partial }{\partial x} + \textbf {j}\frac {\partial }{\partial y} + \textbf {k}\frac {\partial }{\partial z}\Bigg )\cdot \Bigg (F_1\textbf {i} + F_2\textbf {j} + F_3\textbf {k}\Bigg )\\\\ \implies \hspace {0.5cm}\nabla \cdot F & = \frac {\partial F_1}{\partial x} + \frac {\partial F_2}{\partial y} + \frac {\partial F_3}{\partial z} \end {align*}

This is called the divergence of the vector \(F\) written as \(\dive \, F\).

2.

The cross product \begin {align*} \nabla \times F & = \Bigg (\textbf {i} \frac {\partial }{\partial x} + \textbf {j} \frac {\partial }{\partial y} + \textbf {k} \frac {\partial }{\partial z}\Bigg ) \times \Big ( F_1\textbf {i} + F_2\textbf {j} + F_3\textbf {k}\Big )\\\\ & = \Bigg ( \frac {\partial F_3}{\partial y}-\frac {\partial F_2}{\partial z}\Bigg )\textbf {i} - \Bigg (\frac {\partial F_3}{\partial x}-\frac {\partial F_1}{\partial z}\Bigg ) \textbf {j} + \Bigg (\frac {\partial F_2 }{\partial x}-\frac {\partial F_1}{\partial y}\Bigg ) \textbf {k} \end {align*}

is called the curl of \(F\) and written \(\curl \, F\).

Note that \[\curl F = \nabla \times F = \begin {vmatrix} \textbf {i} & \textbf {j} & \textbf {k}\\\\ \dfrac {\partial }{\partial x} & \dfrac {\partial }{\partial y} & \dfrac {\partial }{\partial z}\\\\ F_1 & F_2 & F_3\\ \end {vmatrix} \]

3.

If \(f(x,y,z)\) is a twice differentiable function, then \[\grad \, f = \nabla f = \frac {\partial f}{\partial x}\textbf {i} + \frac {\partial f}{\partial y}\textbf {j} + \frac {\partial f}{\partial z}\textbf {k}\] and \[\dive \big (\grad \, f\big ) = \frac {\partial ^2 f}{\partial x^2} + \frac {\partial ^2 f}{\partial y^2}+ \frac {\partial ^2f}{\partial ^2z} = \nabla ^2f\]

Here \(\displaystyle {\nabla ^2 = \frac {\partial ^2}{\partial x^2}+ \frac {\partial ^2}{\partial y^2} + \frac {\partial ^2}{\partial z^2}}\) is called the Laplacian operator.

Any function \(f\) which satisfies the equation \(\nabla ^2f=0\) is called a Harmonic function.

Properties

If \(\textbf {U}\) and \(\textbf {V}\) are vector point functions and \(f\) is a scalar function, then

1.

\(\nabla \cdot \big (\textbf {U}\pm \textbf {V}\big ) = \nabla \cdot \textbf {U} \pm \nabla \cdot \textbf {V}\) or \(\dive \big (\textbf {U}\pm \textbf {V}\big ) = \dive \big (\textbf {U}\big ) \pm \dive \big (\textbf {V}\big )\)

2.

\(\nabla \cdot \Big (f(\textbf {U})\Big ) = \Big (\nabla \cdot \textbf {U}\Big )f + \textbf {U}\cdot \nabla f\)

3.

\(\nabla \times (\textbf {U}+\textbf {V}) = \nabla \times \textbf {U} + \nabla \times \textbf {V}\)

4.

\(\curl (f\textbf {V}) = f\, \curl \textbf {V} + (\grad \, f)\times \textbf {V} = f(\nabla \times \textbf {V}\big ) + \big (\nabla f\big )\times \textbf {V}\)

To prove, let \(\textbf {U} = U_1\textbf {i} + U_2\textbf {j} + U_3\textbf {k}\) and \(\textbf {V} = V_1\textbf {i} + V_2\textbf {j} + V_3\textbf {k}\).

Let \(F(x,y,z)\) be a vector field. We say that \(F\) is irrotational if \(\curl \, F = \textbf {0}\) and \(F\) is incompressible if \(\dive F =0\).

Recall in \(\mathbb {R}^3\), a vector field \(F=P\textbf {i} + Q\textbf {j} + R\textbf {k}\) is the gradient of a function \(f\) if and only if \[\frac {\partial P}{\partial y} = \frac {\partial Q}{\partial x}\hspace {0.5cm} ,\hspace {1cm} \frac {\partial R}{\partial x} = \frac {\partial P}{\partial z}\hspace {0.5cm} , \hspace {1cm} \frac {\partial Q}{\partial z} = \frac {\partial R}{\partial y}\]

From the definition of \(\curl \, F\) , we get following result.

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