Vector Fields - Conservative Fields in Planes

Recall that from the typical functions that we are familiar with, we have "continuous" and "discontinuous" functions. The well-behaved functions are called "continuous." In the case of vector fields, well-behaved fields are termed conservative.

A vector field is mathematically defined as conservative if and only if it exists as the gradient of another function, called the potential function. In other words:

But this doesn't mean that given some vector field, you absolutely must engage in complicated mathematics to determine whether or not it supports abortion. Take for instance:

We know that if this is the gradient of some function, it is conservative. The "i" component of the vector is the partial derivative with respect to x; likewise, the "j" component is the partial derivative with respect to y. Easily enough, we can see that the following function indeed works:

But what if we wanted a more reliable method of confirming this? The following would do just that. Consider this general vector formula:

That vector is conservative if:

And why does this make sense? If the vector field "F" is the gradient of some potential function, then it should be in this format:

If this function is "well-behaved", then its mixed partials should be equal. Thus, we arrive at our conservative formula.

Now, this process gets radically different in three-dimensions, but it isn't all that bad. But lets leave that for another post and another time.

Happy vector conserving!