Undergrad Test for dependence/independence of multiple variables

[mishima]

I don't think this is related to linear dependence/independence of linear algebra. I am talking about multi-variable calculus.

I am getting confused with being able to quickly tell which variables depend on others when there are 4+ variables involved. For example, suppose I have w(r,s), r(u,v), and s(u,v). Then I am asked to find dv/du. If v is dependent on u, this is non-zero...but how can I tell?

If I rewrite v in terms of r and u to get v(r,u) then it seems clear v is dependent on u, but I am uncertain.

 

[member 587159]

What is v? On what does it depend? You just tell us what w,r,s depend on, but not what v depends on.

 

[mishima]

That's what I'm trying to figure out/understand. Are you saying just given w(r,s), r(u,v), and s(u,v) that there is no way to know if v depends on one of the other variables?

 

[Mark44]

Are you saying just given w(r,s), r(u,v), and s(u,v) that there is no way to know if v depends on one of the other variables?

From the given information, with r and s being functions of u and v, u and v would be the independent variables, with r and s being dependent on u and v.

The function w has r and s as independent variables, and w being dependent. But since both r and s depend on u and v, then ulitimately w is a function of u and v, so is dependent on those two variables. Making a tree diagram is helpful in these situations.

 

[mishima]

That makes sense, maybe a followup question to focus in on my confusion...

Does a variable being independent or dependent have anything to do with whether or not its derivative is zero? In other words, now that we have established that u and v are IV here, is dv/du necessarily 0? Thanks.

 

[Mark44]

Does a variable being independent or dependent have anything to do with whether or not its derivative is zero?

No, but see below. If we're talking about single-variable functions, such as y = f(t), if dy/dt = 0, all this means is that y is constant. For the functions you show in post #1, you have to use partial derivatives, as in the partial derivative of r with respect to one of its independent variables.

In other words, now that we have established that u and v are IV here, is dv/du necessarily 0?

That's sort of a different question. If u and v are unrelated (as in being independent) the du/dv = 0 and dv/du = 0.