日: 2014年3月21日

Limits, continuity and derivatives of vector functions

Limits, continuity and derivatives of vector functions follow rules similar to those for scalar functions already considered. The following statements show the analogy which exists.

  1. The vector function \bold{A}(u) is said to be continuous at u_0 if given any positive number \varepsilon, we can find some positive number \delta such that \left|\bold{A}(u) - \bold{A}(u_0)\right| < \varepsilon[/latex] whenever [latex]\left|u - u_0\right| < \delta[/latex]. This is equivalent to the statement [latex]\lim\limits_{u \rightarrow u_0}\bold{A}(u) = \bold{A}(u_0)[/latex]. </li> <li>The derivative of [latex]\bold{A}(u) is defined as
    \displaystyle \frac{d\bold{A}}{du} = \lim\limits_{\Delta{u} \rightarrow 0}\frac{\bold{A}(u + \Delta {u}) - \bold{A}(u)}{\Delta{u}}\cdots (7)
    provided this limit exists. In case \bold{A}(u) = A_1(u)\bold{i} + A_2(u)\bold{j} + A_3(u)\bold{k}; then
    \displaystyle \frac{d\bold{A}}{du} = \frac{dA_1}{du}\bold{i} + \frac{dA_2}{du}\bold{j} + \frac{dA_3}{du}\bold{k}
    Higher derivatives such as d^2\bold{A}/du^2, etc., can be similarly defined.
  2. If \bold{A}(x, y, z) = A_1(x, y, z)\bold{i} + A_2(x, y, z)\bold{j} + A_3(x, y, z)\bold{k}, then
    \displaystyle d\bold{A} = \frac{\partial\bold{A}}{\partial x}dx + \frac{\partial\bold{A}}{\partial y}dy + \frac{\partial\bold{A}}{\partial z}dz\cdots(8)
    is the differential of \bold{A}.
  3. Derivatives of products obey rules similar to those for scalar functions. However, when cross products are involved the order may be important. Some examples are:
    \displaystyle (a)\ \frac{d}{du}(\phi\bold{A}) = \phi\frac{d\bold{A}}{du} + \frac{d\phi}{du}\bold{A}
    \displaystyle (b)\ \frac{\partial}{\partial y}(\bold{A} \cdot \bold{B}) = \bold{A} \cdot \frac{\partial \bold{B}}{\partial y} + \frac{\partial\bold{A}}{\partial y} \cdot \bold{B}
    \displaystyle (c)\ \frac{\partial}{\partial z}(\bold{A} \times \bold{B}) = \bold{A} \times \frac{\partial\bold{B}}{\partial z} + \frac{\partial\bold{A}}{\partial z} \times \bold{B}

ベクトル関数の極限，連続と導関数

　ベクトル関数の極限，連続及び導関数は，スカラー関数のそれとよく似た規則に従います．以下の記述は存在する類似を示しています．

  1. ベクトル関数 \bold{A}(u) は u_0 において 連続 であると言われる．仮に任意の正の数 \varepsilon があってここで \left|u - u_0\right| < \delta[/latex] を満たす [latex]\left|\bold{A}(u) - \bold{A}(u_0)\right| < \varepsilon[/latex] が存在するようなある正の数 [latex]\delta[/latex] を見つけられるなら．このことは次の記述と等価である．[latex]\lim\limits_{u \rightarrow u_0}\bold{A}(u) = \bold{A}(u_0)[/latex]</li> <li>[latex]\bold{A}(u) の微分は次のように定義される．
    \displaystyle \frac{d\bold{A}}{du} = \lim\limits_{\Delta{u} \rightarrow 0}\frac{\bold{A}(u + \Delta {u}) - \bold{A}(u)}{\Delta{u}}\cdots (7)
    \bold{A}(u) = A_1(u)\bold{i} + A_2(u)\bold{j} + A_3(u)\bold{k} のような場合，
    \displaystyle \frac{d\bold{A}}{du} = \frac{dA_1}{du}\bold{i} + \frac{dA_2}{du}\bold{j} + \frac{dA_3}{du}\bold{k}
    　d^2\bold{A}/du^2 等のような高階の導関数も同様に定義される．
  2. 仮に \bold{A}(x, y, z) = A_1(x, y, z)\bold{i} + A_2(x, y, z)\bold{j} + A_3(x, y, z)\bold{k} ならば
    \displaystyle d\bold{A} = \frac{\partial\bold{A}}{\partial x}dx + \frac{\partial\bold{A}}{\partial y}dy + \frac{\partial\bold{A}}{\partial z}dz\cdots(8)
    は \bold{A} の 微分 である．
  3. 積の導関数はスカラー関数のそれの規則に従う．しかしながら，クロス積の従う順序は重要かもしれない．いくつかの例を挙げる．
    \displaystyle (a)\ \frac{d}{du}(\phi\bold{A}) = \phi\frac{d\bold{A}}{du} + \frac{d\phi}{du}\bold{A}
    \displaystyle (b)\ \frac{\partial}{\partial y}(\bold{A} \cdot \bold{B}) = \bold{A} \cdot \frac{\partial \bold{B}}{\partial y} + \frac{\partial\bold{A}}{\partial y} \cdot \bold{B}
    \displaystyle (c)\ \frac{\partial}{\partial z}(\bold{A} \times \bold{B}) = \bold{A} \times \frac{\partial\bold{B}}{\partial z} + \frac{\partial\bold{A}}{\partial z} \times \bold{B}