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-   -   path dependent function with a definite path (http://mymathforum.com/real-analysis/25526-path-dependent-function-definite-path.html)

aise5668 March 5th, 2012 07:03 AM

path dependent function with a definite path
 
This question is about if I have a path dependent function but a deinite path, then can I take partial derivatives? And are the points that are along path A connected to the points that are along path B.

If I consider a non-conservative vector field


I must specify a path, in this case the path is always:

(eq. 1)

and integrate along the path to create the field:

(eq. 2)

and also:

(eq. 3)

may I now treat t as path independent? In other words, may I now take the total derivative and evaluate at a constant x like this:

(eq. 4)

(eq. 5)

and then evaluate eq. 5 at a fixed x so that dx = 0

It seems to me that I can't do this because I think that I have defined a relationship between dx and dz in equation 1. Since t is path dependent, p must be in the definition of t so setting dx = 0 violates the constraint that p has set on the relationship between dx and dz. Is what I was thinking correct? Or am I allowed to take the total derivative of X and set dx = 0? Is the point t(x,z) connected to t(x,z+dz)? I am posting this question here and in the differentials section because I am not sure if this is a question about disconnected space or conservative fields.

aise5668 March 5th, 2012 04:27 PM

Re: path dependent function with a definite path
 
I should have mentioned that all paths go through the origin and x and z can only be positive.

outsos March 5th, 2012 06:30 PM

Re: path dependent function with a definite path
 
Quote:

Originally Posted by aise5668
I should have mentioned that all paths go through the origin and x and z can only be positive.


Is the vector field two dimentional or three dimentional ??

aise5668 March 5th, 2012 06:36 PM

Re: path dependent function with a definite path
 
It is 2 dimensional.


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