日: 2014年3月26日

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.

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.

勾配，発散，回転

　以下で定義されるベクトル演算子 \nabla\ (del) を考えてみましょう．

\displaystyle \nabla \equiv \bold{i}\frac{\partial}{\partial x} + \bold{j}\frac{\partial}{\partial y} + \bold{k}\frac{\partial}{\partial z}\cdots(13)

　仮に \phi(x, y, z) および \bold{A}(x, y, z) が（多くの例において必要性よりも強い状態にある）ある地点において一階の偏微分を有する場合，以下のように定義できます．

1. 勾配

　 φ の 勾配 は以下の定義です．

\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)

　仮に \phi(x, y, z) = c が表面の方程式の場合， \nabla\phi はこの表面に対して垂直であることは興味深い解釈です．

2. 発散

　\bold{A} の 発散 は以下で定義されます．

\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. 回転

　\bold{A} の 回転 は以下で定義されます．

\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)

　行列式，演算子 \partial/\partial x, \partial/\partial y, \partial/\partial z においては必ず A_1, A_2, A_3 の前に置かねばならないことに注意してください．