In my course notes (Dynamics) I have the following written down:(adsbygoogle = window.adsbygoogle || []).push({});

(a) write as a set of implicit equations [tex]0=h(\dot{\vec{z}},\vec{z},\vec{\mu})[/tex]

(b) pick a reference flight condition (RFC) [tex]0=h(\dot{\vec{z*}},\vec{z*},\vec{\mu*})[/tex]

(c) Apply Taylor to linearize (a) about (b); Trick: Consider [tex]\dot{\vec{z*}}, \vec{z*}, \vec{\mu*}[/tex] to beindependentvariables.

The function [tex] h(\dot{\vec{z*}},\vec{z*},\vec{\mu*})[/tex] is found by finding the equations of motion. An example of a function h given in an example in class was:

[tex]0=-\frac{g}{L}sin(z_1)-\frac{b(z_2)}{mL}+\frac{\mu}{mL^2}-\dot{z_2}[/tex]

Based on this (or any) equation of motion, selection of a value of [tex]z_1,z_2,\mu[/tex] constrains a value of [tex]\dot{z_2}[/tex]. In this sense at least one value has to depend on the other three. In my mind, by calling thesefourvariables independent it means I can pick any value I want for them and input them into my function, regardless of the selection of the other values;however, it is obvious that any choice of those 4 variables isnotgoing to satisfy the fact that h must equal 0: h=0. So, I ask how is it that these values really are independent of eachother?

Upon searching for a taylor series expansion in multi-variables, [1], it is clear that the input arguments into the function h, areindependent.

So I tried to do more searching on what exactly are independent and dependent variables. It seems that an dependent variable is a function of an independent variable. This means that for each value of the independent variable there can be one and only one value for the dependent variable. So, does that mean if there are two possible solutions that satisfy the equation, there are no 'dependent/independent' variables, just, for a lack of a proper word, 'variables'? An example is the equation of a circle, [tex]x^2+y^2=R^2[/tex]. This is another implicit function, but I cannot pick any value of x and y and have the sum of their squares equal [tex]R^2[/tex]. Does it even make sense to use the word independent/dependent variables?

Any thoughts would be helpful.

[1] - http://www.chem.mtu.edu/~tbco/cm416/taylor.html" [Broken]

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# What exactly is a dependent variable?

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