# 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:

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

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:

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

The reciprocal vector is defined as follows:

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

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:

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

Ordinary vector rates are added arithmetically:

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

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

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

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

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

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

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

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:

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

whereas reciprocal vector derivatives are added reciprocally:

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

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.