Gradient, divergence and curl

Consider the vector operator $\nabla\ (del)$ defined by
$\displaystyle \nabla \equiv \bold{i}\frac{\partial}{\partial x} + \bold{j}\frac{\partial}{\partial y} + \bold{k}\frac{\partial}{\partial z}\cdots(13)$
Then if $\phi(x, y, z)$ and $\bold{A}(x, y, z)$ have continuous first partial derivatives in a region (a condition which is in many cases stronger than necessary), we can define the following.

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1 1. Gradient
2 2. Divergence
3 3. Curl

1. Gradient

The gradient of φ is defined by
$\displaystyle grad\phi = \nabla\phi = \left(\bold{i}\frac{\partial}{\partial x} + \bold{j}\frac{\partial}{\partial y} + \bold{k}\frac{\partial}{\partial z}\right)\phi\\ = \bold{i}\frac{\partial\phi}{\partial x} + \bold{j}\frac{\partial\phi}{\partial y} + \bold{k}\frac{\partial\phi}{\partial z}\\ = \frac{\partial\phi}{\partial x}\bold{i} + \frac{\partial\phi}{\partial y}\bold{j} + \frac{\partial\phi}{\partial z}\bold{k}\cdots(14)$
An interesting interpretation is that if $\phi(x, y, z) = c$ is the equation of a surface, then $\nabla\phi$ is a normal to this surface.

2. Divergence

The divergence of $\bold{A}$ is defined by
$\displaystyle div\bold{A} = \nabla\cdot\bold{A} = \left(\bold{i}\frac{\partial}{\partial x} + \bold{j}\frac{\partial}{\partial y} + \bold{k}\frac{\partial}{\partial z}\right)\cdot(A_1\bold{i} + A_2\bold{j} + A_3\bold{k})\\ = \frac{\partial A_1}{\partial x} + \frac{\partial A_2}{\partial y} + \frac{\partial A_3}{\partial z}\cdots(15)$
3. Curl

The curl of $\bold{A}$ is defined by
$\displaystyle curl\bold{A} = \nabla\times\bold{A} = \left(\bold{i}\frac{\partial}{\partial x} + \bold{j}\frac{\partial}{\partial y} + \bold{k}\frac{\partial}{\partial z}\right)\times(A_1\bold{i} + A_2\bold{j} + A_3\bold{k})\\ = \left|\begin{array}{ccc} \bold{i} & \bold{j} & \bold{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_1 & A_2 & A_3 \end{array}\right| \\ = \bold{i}\left|\begin{array}{cc} \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_2 & A_3 \end{array}\right| - \bold{j}\left|\begin{array}{cc} \frac{\partial}{\partial z} & \frac{\partial}{\partial z} \\ A_1 & A_3 \end{array}\right| + \bold{k}\left|\begin{array}{cc} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \\ A_1 & A_2 \end{array}\right|\\ = \left(\frac{\partial A_3}{\partial y} - \frac{\partial A_2}{\partial z}\right)\bold{i} + \left(\frac{\partial A_1}{\partial z} - \frac{\partial A_3}{\partial x}\right)\bold{j} + \left(\frac{\partial A_2}{\partial x} - \frac{\partial A_1}{\partial y}\right)\bold{k}\cdots(16)$
Note that in the expansion of the determinant, the operators $\partial/\partial x$ , $\partial/\partial y$ , $\partial/\partial z$ must precede $A_1$ , $A_2$ , $A_3$ .