Notational conventions are just that: conventions. Even closely related disciplines tend to have slightly different ways of saying the same things.

We summarize here several instances where our notation differs from standard mathematical usage as found in typical calculus texts, along with a brief discussion of our reasons for adopting an alternative.

• Points
  Mathematicians often refer to points as $P(x,y)$, a notation we fail to understand. Are these functions? A definition of the symbol $P$? We write this as $P=(x,y)$.

• Vectors
  Mathematicians usually write vectors in the form $\langle x,y \rangle$ or $\langle x,y,z \rangle$. There are several problems with this notation: It is different in different dimensions, it assumes that vectors will only ever be expanded in terms of the standard rectangular basis, it risks confusion with the quite similar notation used for points, and it makes it less obvious that vectors are a different “beast” (and can not, for example, be set equal to scalars).

  We always use an explicit basis, as in $x\,\ii+y\,\jj$, which could represent a vector in two or three (or more) dimensions depending on the context. In curvilinear coordinates, we write for instance $r\,\rhat$ — which suggests that $\{\xhat,\yhat,\zhat\}$ would be more appropriate names than $\{\ii,\jj,\kk\}$ for the rectangular basis vectors, a notation we are slowly adopting.

• Derivatives
  

  We strongly prefer Leibniz notation ($dy/dx$) for derivatives rather than the Newtonian use of prime ($f'$). First of all, in the presence of several possible variables, the prime fails to answer the question “with respect to what”. Furthermore primes are often used in other disciplines to denote a distinct quantity (e.g. the points $x$ and $x'$).

  Nonetheless, time derivatives are often written with a dot ($\dot{x}$), and primes are occasionally used for spatial derivatives when the context is clear (e.g. $\ddot{y}+y' '=0$).

• Partial Derivatives
  

  We strongly prefer explicit partial derivative notation ($\partial f/\partial x$) over the use of subscripts ($f_x$); subscripts are much more useful as a means to tie the components of a vector to the name of the vector itself ($\FF=F_x\,\ii+F_y\,\jj$).

• Parameterization
  

  We discourage the generic use of the variable $t$ for parameterized curves, unless the context truly involves time. The overuse of $t$ causes confusion in a context such as Ampère's Law for a steady current, where nothing is changing with time, and the notion of “moving” around a curve can be misleading. Since $u$, $v$ are commonly used to parameterize surfaces, we use $u$ as our generic parameter.

• Spherical Coordinates
  

  All American mathematicians refer to the radial coordinate as $\rho$, the angle from the North Pole as $\phi$, and the angle in the $xy$-plane as $\theta$. Everybody else refers to these angles as $\theta$ and $\phi$, respectively, and this usage is hardwired into the standard notation for spherical harmonics, $Y_{\ell m}(\theta,\phi)$. It doesn't help that many calculus texts list these angles in alphabetical, rather than geometrical order, resulting in a left-handed coordinate system. Furthermore, $\rho$ is used in other contexts for (volume) densities, making it difficult to write down the triple integral for the total charge in a given region.

  We write the spherical radial coordinate as $r$, the angle from the North Pole as $\theta$, and the angle in the $xy$-plane as $\phi$.

• Polar Coordinates
  

  The above usage for spherical coordinates can lead to confusion with polar coordinates. For this reason, we write the angle in polar coordinates as $\phi$, rather than $\theta$.

  (We are also considering the use of $s$ instead of $r$ for the polar radial coordinate, as is done by some physics authors. However, we prefer the use of $s$ for arclength — which further suggests using $\vec{s}$ rather than $\rr$ for the position vector, which in turn might mitigate the confusion caused by the many roles played by “$r$”, as in $\rr=r\,\rhat$.)