I am new to stack exchange, the only other place I have asked questions on thus far is stackoverflow, and though technically I am a student of computer science (a late bloomer, started my degree two years ago when I was 26), my sole hobby for the past three to four years has been mathematics, pure mathematics. I'm sorry to say but I am horrible at applying it to real life, but boy do I love crunching the numbers. My knowledge ranges up to Vector Analysis, right now I am just completing the exercise section on the Fundamental Theorem of Line Inegrals (Larson's Essential Calculus ETF). My question is quite simple actually:
Is there a specific notation to denote the potential function $\mathcal f$?
It is known that for a conservative vector field: $$\overrightarrow A = \nabla\mathcal f = M\hat{i}+N\hat{k}$$ where $M$ and $N$ are continous functions of $x$ and $y$ and, $$\frac{\partial M}{\partial y}= \frac{\partial N}{\partial x}$$ There is also the case that for a vector field in space to be conservative, $\mathbf{curl \overrightarrow A} = \mathbf{\overrightarrow 0}$, but there exist potential functions for the plane and for space.
I'm sort of a freak when it comes to notation, so though it may be a dumb question for my first, I'd love know. Thank you in advance!
While we have all sorts of simple notation for derived functions or fields ($f'$, $\nabla f$, ${\rm curl}\,{\bf F}$, etc.) there is no such thing for antiderivative. Note that even in the one-dimensional case we resort to "let $F$ be a primitive of $f$", or need the clumsy $\int f(x)\>dx$ to denote the infinite set of all primitives. A reason for this lack might be that the potential of a conservative field ${\bf F}$ is defined only up to an additive constant, while the various derived objects are uniquely determined by ${\bf F}$.