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.

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.

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 $\bold{A}$ .

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:
$\displaystyle (a)\ \frac{d}{du}(\phi\bold{A}) = \phi\frac{d\bold{A}}{du} + \frac{d\phi}{du}\bold{A}$
$\displaystyle (b)\ \frac{\partial}{\partial y}(\bold{A} \cdot \bold{B}) = \bold{A} \cdot \frac{\partial \bold{B}}{\partial y} + \frac{\partial\bold{A}}{\partial y} \cdot \bold{B}$
$\displaystyle (c)\ \frac{\partial}{\partial z}(\bold{A} \times \bold{B}) = \bold{A} \times \frac{\partial\bold{B}}{\partial z} + \frac{\partial\bold{A}}{\partial z} \times \bold{B}$