- #1
aise0603
- 6
- 0
This question is about if I have a path dependent function but a definite path, then can I integrate along the path to form a 2 dimensional field and then take partial derivatives along directions other than the path? And e.g. are points along path A connected to path B? (Note: all paths start at the origin and x and z can only be positive)
If I consider a non-conservative vector field
\vec{dt}= \frac{\vec{ds}}{V(z)}
where
ds = \sqrt{dx^{2}+dz^{2}}
and
V(z) is only a function of Z and not of X
so I specify the path:
p = \frac{sin \Theta}{V(z)} = \frac{dx}{V(z)\sqrt{dx^{2}+dz^{2}}} (eq. 1)
and integrate along the path to create the field:
t = \int_{0}^{z }\frac{dz}{\sqrt{1-p^{2}V(z)^{2}}} (eq. 2)
and also:
x = \int_{0}^{z }\frac{pV(z)dz}{\sqrt{1-p^{2}V(z)^{2}}} (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:
dt = \frac{\partial t}{\partial z}|_{p}dz + \frac{\partial t}{\partial p}|_{z}dp (eq. 4)
dx = \frac{\partial x}{\partial z}|_{p}dz + \frac{\partial x}{\partial p}|_{z}dp (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)?
If I consider a non-conservative vector field
\vec{dt}= \frac{\vec{ds}}{V(z)}
where
ds = \sqrt{dx^{2}+dz^{2}}
and
V(z) is only a function of Z and not of X
so I specify the path:
p = \frac{sin \Theta}{V(z)} = \frac{dx}{V(z)\sqrt{dx^{2}+dz^{2}}} (eq. 1)
and integrate along the path to create the field:
t = \int_{0}^{z }\frac{dz}{\sqrt{1-p^{2}V(z)^{2}}} (eq. 2)
and also:
x = \int_{0}^{z }\frac{pV(z)dz}{\sqrt{1-p^{2}V(z)^{2}}} (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:
dt = \frac{\partial t}{\partial z}|_{p}dz + \frac{\partial t}{\partial p}|_{z}dp (eq. 4)
dx = \frac{\partial x}{\partial z}|_{p}dz + \frac{\partial x}{\partial p}|_{z}dp (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)?