Find work done by the force field F in moving the particle from $(-1, 1)$ to $(3, 2)$
This sounds good till we are given that $\textbf{F} = \dfrac{2x}{y}\textbf{ i }- \dfrac{x^2}{y^2}\textbf{ j }$
Can someone explain how to understand this problem, does the word conservative field have a meaning if the field is not continuous everywhere
Best Answer
The vector field is continuous on its domain, which is $\mathbb{R}^2 - \{y = 0\}$. It's nevertheless quite reasonable to ask how not being defined on the $x$-axis affects a given problem. For integrations of functions along curves, it just means that the paths we consider must trace out curves contained entirely within the domain; in our situation, this just means we're making a statement about the work done by $\mathbf{F}$ along some curve from $(-1, 1)$ to $(3, 2)$ that doesn't cross the $x$-axis (and of course, conservative means that the work doesn't depend on which such path the particle takes).
Vector fields not being defined everywhere are quite common in practice, maybe most importantly as the force fields generated by point-sources where the force equation obeys an inverse-square law.