Two kinds of vector rates

This post builds on the previous one here.

Vector rates rates of change are of two kinds. An ordinary rate for the vector change of f relative to a unit of x is defined as:

$\frac{\Delta {\mathbf f}(x)}{\bar{\Delta} x}$

The reciprocal vector rate is the vector reciprocal of an ordinary rate with a vector change of g relative to a unit of x is defined as:

$\left (\frac{\Delta \mathbf{g}(x)}{\bar{\Delta} x} \right )^{-1} = \frac{\Delta x}{\Delta \mathbf{g}(x)}$

The reciprocal vector is defined as follows:

$\frac{1}{\mathbf{g}}:=\frac{\mathbf{g}}{\left \| g \right \|}\cdot \frac{1}{\left \| g \right \|}=\frac{\mathbf{g}}{\mathbf{g}\cdot \mathbf{g}}$

An ordinary vector rate has its independent variable in the denominator. The independent variable of a reciprocal vector rate is in the numerator. A reciprocal vector rate is the vector reciprocal of the inverse (or reverse) vector rate, which has the independent and dependent variables interchanged:

$\frac{\Delta \mathbf{y}(x)}{\bar{\Delta} x} \, vs.\, \frac{\Delta \mathbf{x}(y)}{\bar{\Delta} y}$

Ordinary vector rates are added arithmetically:

$\frac{\Delta \mathbf{f}_{1}}{\bar{\Delta} x}+ \frac{\Delta \mathbf{f}_{2}}{\bar{\Delta} x}= \frac{\Delta \mathbf{f}_{1}+\Delta \mathbf{f}_{2}}{\bar{\Delta} x}$

whereas reciprocal vector rates are added reciprocally (see post on reciprocal arithmetic):

$\left (\frac{\Delta \mathbf{g}_{1}}{\bar{\Delta} x} \right )^{-1} \boxplus \, \left (\frac{\Delta \mathbf{g}_{2}}{\bar{\Delta} x} \right )^{-1}= \left (\frac{\Delta \mathbf{g}_{1}+\Delta \mathbf{g}_{2}}{\bar{\Delta} x} \right )^{-1}$

Differential vector rates are also of two types. The vector derivative of f relative to x is defined as:

$\mathbf{f}'(x)=\frac{\mathrm{d} \mathbf{f}(x)}{\mathrm{d} x}= \lim_{h\rightarrow 0}\frac{\mathbf{f}(x+h)-\mathbf{f}(x)}{h}$

The reciprocal vector derivative of g relative to x is defined as:

$\grave{}\mathbf{g}(x)= \left (\frac{\mathrm{d} \mathbf{g}(x)}{\mathrm{d} x} \right )^{-1}= \left (\lim_{h\rightarrow 0}\frac{\mathbf{g}(x+h)-\mathbf{g}(x)}{h} \right )^{-1}$

This is similar to an ordinary vector derivative, but it is functionally related to the inverse vector function by the inverse function theorem.

Ordinary vector derivatives are added arithmetically:

$\mathbf{f}'_{1}(x)+\mathbf{f}'_{2}(x)= \frac{d\mathbf{f}_{1}}{\bar dx}+\frac{d\mathbf{f}_{2}}{\bar dx}= \frac{d(\mathbf{f}_{1}+\mathbf{f}_{2})}{\bar dx}$

whereas reciprocal vector derivatives are added reciprocally:

$\grave{}\mathbf{g}_{1}(x)+\, \grave{}\mathbf{g}_{2}(x)= \left (\frac{d\mathbf{g}_{1}}{\bar dx} \right )^{-1} \boxplus \, \left (\frac{d\mathbf{g}_{2}}{\bar dx} \right )^{-1}= \left (\frac{d(\mathbf{\mathbf{}g}_{1}+\mathbf{g}_{2})}{\bar dx} \right )^{-1}$

Ordinary vector derivatives are the rate of change of a function at a point. Reciprocal derivatives are the rate of change of the inverse function.

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