definition - Potential functions

Can someone please explain what a potential is? Example. velocity potential in ideal flows, acoustic potential (gradient of which gives the particle velocity in a sound wave). Whenever I see potential functions complex analysis is applied to compute integrals, complex functions for conformal mapping. I vaguely understand that potential functions are independent of path and in complex plane the integration around a contour is independent of path as well (the integration depends only on the end points, hence in a closed curve say region $Z$, if function, $f$ is analytic then integration of $f$ over $Z$ is 0). Can you please explain on this?

Answer

Potential is a special case of a more general construction in differential geometry. Let's start abstractly and we'll get to the potentials again at the end.

Differential forms

The framework of differential forms provides a basis for integration on arbitrary manifold. Differential $p$-forms are totally antisymmetric covariant $p$-tensors. What's special about them is that you can define exterior derivative taking $p$-form to $(p+1)$-form. Now, $0$-forms are just functions and $1$-forms are like usual vectors (if you have a metric on your manifold, as is the case e.g. in Euclidean space, you can freely move between forms (also called covectors) and vectors).

Now, exterior derivative on functions gives us a 1-form. Let's say ${\rm d}f = \alpha$. The important point about the operator $\rm d$ is that it is nilpotent ${\rm d}^2 = 0$. So this means that ${\rm d} \alpha = {\rm d}^2 f = 0$. Important point is that this relation can be reversed (at least on topologically trivial manifolds that don't have any holes in them, and so on -- see Poincaré lemma for more information. But as always, Euclidean space is fine): if you have some form $\alpha$ such that ${\rm d} \alpha = 0$ (we say that the form is closed) then there will exist another form $\beta$ such that $\alpha = {\rm d} \beta$ (we say that the form is exact). So this is you potential in a general setting.

Now to understand why potentials are useful in the first place we have to talk a bit about integration. It is possible to integrate $n$-forms on $n$-dimensional manifolds (the reason for this is that they have the similar transformation properties to Jacobian of usual integration substitution). So if you have some $p$-form $\alpha$ you can integrate it over some $p$-dimensional subset $U$ of the manifold and this is denoted by $\int_U \alpha$. The punchline is that if $\alpha$ has a potential $\alpha = {\rm d} \beta$ we can use Stokes' theorem that tells us

$$\int_U \alpha = \int_U {\rm d} \beta = \int_{\partial U} \beta$$ .

where $\partial U$ is a boundary of the given subset. So that we can transform some integrals into others which can often simplify calculations.

Physics

To connect again with your questions: potentials arise from closed forms. The closedness conditions can take various guises in standard vector formalism. The usual one is for conservative forces $\nabla \times {\mathbf F} = 0$. This can be translated to the language of the differential forms as the condition on $\mathbf F$ being closed and so we know that there must be another form, say $\phi$ such that ${\mathbf F} = \nabla \phi$ (notice the identity $\nabla \times \nabla \phi = 0$ -- this is our good old ${\rm d}^2 = 0$ in action again). Because of the Stoke's theorem we know that the usefulness of the concept of conservative forces stems from the fact that their integral over a closed path doesn't depend on the path (this is a trivial consequence of $U$ having no boundary in that case).

Another famous closed form is magnetic induction ${\mathbf B}$ because there are no monopoles (yet): $\nabla \cdot {\mathbf B} = 0$. This gives us

$${\mathbf B} = \nabla \times {\mathbf A}$$

where $\mathbf A$ is a vector potential. Again by using Stokes' theorem we can find that flow of $\mathbf B$ through any closed surface is zero.

Note: it might seems strange that we integrate vector $\mathbf B$ over a surface which is two-dimensional. This is not how we defined the integration for forms. But $\mathbf B$ is actually a two-form (you can see this from its relation to $\mathbf A$ which is a genuine one-form) and one is exploiting that in three-dimensional space one can identify these with one-forms. This is actually the usual identification of antisymmetric $3\times 3$ matrix with a pseudovector.

Complex analysis

The complex analysis is very similar (although slightly harder) setting. The complex plane can be regarded as a two-dimensional real manifold so that there are two linearly independent one-forms: holomorphic and antiholomorphic forms ${\rm d}z$ and ${\rm d} \bar z$ where $z = x + iy$ and $\bar z = x - iy$. It can be shown that holomorphic functions satisfy $\partial_{\bar z} f = 0$ and that any holomorphic form $h = f {\rm d}z$ is closed so this gives the Cauchy formula

$$\oint_{\gamma} f {\rm d}z = 0$$ as a special case of Stokes' theorem.

Alternatively, one can exploit the full power of the complex formalism by using both holomorphic and antiholomorphic functions to encode any information that can be found in a plane which is usually described by functions ${\mathbb R}^2 \to {\mathbb R}^2$ as functions ${\mathbb C} \to {\mathbb C}$. It is again possible to translate all the language of differential forms and potentials into the complex setting.