Test for dependence/independence of multiple variables

In summary, The conversation discusses the confusion with determining dependence and independence of variables in multi-variable calculus. It is mentioned that in a function with multiple variables, the independent variables are the ones that the others depend on. However, it is not necessary for a derivative to be zero for a variable to be independent or dependent.
  • #1
mishima
570
36
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.
 
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  • #2
What is v? On what does it depend? You just tell us what w,r,s depend on, but not what v depends on.
 
  • #3
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?
 
  • #4
mishima said:
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.
 
  • #5
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.
 
  • #6
mishima said:
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.

mishima said:
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.
 

Related to Test for dependence/independence of multiple variables

1. How do I test for dependence/independence of multiple variables?

To test for dependence/independence of multiple variables, you can use statistical methods such as correlation analysis, regression analysis, or ANOVA (Analysis of Variance). These methods can help determine the strength and direction of the relationship between variables and whether they are statistically significant.

2. What is correlation analysis and how does it test for dependence/independence of multiple variables?

Correlation analysis is a statistical method that measures the strength and direction of the relationship between two or more variables. It can help determine if there is a linear relationship between variables and if they are positively or negatively correlated. If there is a strong correlation between variables, it suggests a dependence between them.

3. Can regression analysis be used to test for dependence/independence of multiple variables?

Yes, regression analysis can be used to test for dependence/independence of multiple variables. It is a statistical method that examines the relationship between a dependent variable and one or more independent variables. It can help determine if there is a significant relationship between variables and if the relationship is positive or negative.

4. What is ANOVA and how does it test for dependence/independence of multiple variables?

ANOVA (Analysis of Variance) is a statistical method that compares the means of three or more groups to determine if there is a significant difference between them. It can be used to test for dependence/independence of multiple variables by comparing the means of different groups and determining if there is a significant relationship between variables.

5. What is the importance of testing for dependence/independence of multiple variables?

Testing for dependence/independence of multiple variables is important because it allows us to understand the relationships between variables and how they may influence each other. This information is crucial in making informed decisions and drawing meaningful conclusions from data. It also helps to identify any potential confounding variables that may impact the results of a study.

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