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I'm independently studying Colley's Vector Calculus and am on the section on line integrals. I understand that the line integral gives the amount of work done on a vector field for a predetermined path through the field, much like me walking across a river to the other side in a straight line, how much work do I need to do against the current. But what about a particle dropped into a vector field? I think this would be easy to compute if say the field were given by $$\mathbf{F}(x,y)=\mathbf{i}$$ But for something more dynamic like $$\mathbf{F}(x,y)=y\mathbf{i}-x\mathbf{j}$$ How does one go about finding the path $\mathbf{r}(t)$ that the particle traces out after say $t=n$ seconds? We can get the velocity vector and the acceleration vector, but would this be a prolbem for systems of differential equations and gradient? Intuitively it seems like the particle will follow the path of least resistance so perhaps the $-\nabla$ operator would help take care of that?

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Indeed, to find such $\mathbf{r}$ you have to solve the system of differential equations $\mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t))$. In general this can be hard to do in closed form, but there are many numerical methods. Euler's method is perhaps the simplest.

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that's great information. Yeah I'm working on the given path through vector fields and it dawned on me that this might be a difficult challenge considering how the particles movement changes over time in a more turbulent field, say. –  Eleven-Eleven Sep 18 '13 at 18:26

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