Differences in notation: are mathematicians and physicists irreconcilable?

Why the differences? --------------------------

The reasons for these differences may be pragmatic: following Euler's example, mathematicians use the letter i to denote the square root of –1, whereas physicists have chosen j, since the letter i is reserved for electric current. Mathematicians reserve j for the cube root of unity with positive imaginary part, namely $j=-{\displaystyle \frac{1}{2}}+i{\displaystyle \frac{\sqrt{3}}{2}}\mathrm{.}j=\; -\; \backslash dfrac\{1\}\{2\}\; +\; i\; \backslash dfrac\{\backslash sqrt\; 3\}\{2\}.$

Sometimes, however, these differences have deeper "roots" and reflect different ways of thinking about the underlying concepts. Mathematicians then find physicists lacking in rigor, with notation that sometimes conceals genuine difficulties; physicists, by contrast, think mathematicians indulge in pointless abstraction and are oblivious to physical reality.

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Notation for the derivative ----------------

When Newton introduced the notion of a fluxion, that is, a derivative (see the article "Newton's fluxions and infinitesimal calculus"), he marked the fluxion of the quantity under study with a dot above it. Since his approach was inspired by kinematics, the variable was—implicitly—time. Leibniz, by contrast, explicitly allowed one variable y to vary as a function of another variable x.

He considered the quotient of the difference in the y-values (denoted by dy) by the difference in the x-values (denoted by dx); naturally, he wrote it as dy / dx. When dy and dx became infinitesimal, this quotient became a number through a process that was somewhat mysterious at the time.

It was not until Lagrange that the "right" mathematical notation emerged. A century later, he had a firm grasp of the notion of a function and declared in 1797: "We shall call the function fx the antiderivative, in relation to the functions f’x, f’’x, etc. that are derived from it, and we shall call the latter derivative functions, in relation to the former."

Mathematicians adopted this notation, whereas physicists retained Leibniz's. For a mathematician, there is no need to specify the variable of differentiation: it is inherent in the definition of the function. In fact, a physicist is really writing down a physical quantity rather than a function: that quantity may depend on a variable y, which is itself a function of another variable x. The physicist therefore distinguishes df / dy from df / dx.

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Notation for the gradient --------------------

The gradient is a British invention: the Irish astronomer and mathematician William Hamilton introduced it in 1853, via his beloved quaternions, four-dimensional numbers with which he described three-dimensional geometry. James Maxwell, of electromagnetic-wave fame and as much a mathematician as a physicist, turned it into a vector and proposed denoting it by $\mathrm{\nabla }\backslash nabla$, an upside-down delta, and calling it atled ("delta" backwards). On the grounds that this symbol resembled a Phoenician harp called nabla in Ancient Greek, Peter Tait, an English physicist, proposed giving the operator that name.

But what exactly is the gradient? Let U be a two-dimensional subset of space, equipped with an orthonormal basis $(\overrightarrow{u},\overrightarrow{v})\mathrm{.}(\; \backslash overrightarrow\{u\},\; \backslash overrightarrow\{v\}).$ Let f be a scalar-valued function defined on U, and suppose that f has partial derivatives at a point (x, y) in U.

The vector ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}}(x,y)\overrightarrow{u}+{\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }y}}(x,y)\overrightarrow{v}\backslash dfrac\{\; \backslash partial\; f\}\{\backslash partial\; x\}\; (x,\; y)\; \backslash overrightarrow\{u\}\; +\; \backslash dfrac\{\; \backslash partial\; f\}\{\backslash partial\; y\}\; (x,\; y)\; \backslash overrightarrow\{v\}$ is then called the gradient of f at the point with coordinates (x, y). It points in the direction of greatest increase in f at that point. This is a "physicist's definition." Mathematicians object that it depends on the chosen basis. In fact, this vector, sometimes denoted by $\mathrm{\nabla }(x,y),\backslash nabla\; (x,\; y),$ is the same for every choice of orthonormal basis. For a physicist, the Euclidean structure—the notions of angle, distance, and so on—is inherent in the space. For a mathematician, the space is more general; mathematicians prefer to define the differential of f at the point (x, y): it is a linear functional, denoted by $\phi (x,y)\backslash varphi\; (x,\; y)$, and only a Euclidean structure on the space allows it to be associated with a vector, generally denoted by $\overrightarrow{{\text{grad}}_f}(x,y),\backslash overrightarrow\{\backslash text\{grad\}\; \_f\}\; (x,\; y),$ which is the unique vector satisfying the following condition for every vector: $\overrightarrow{u}:\overrightarrow{{\text{grad}}_f}(x,y)\cdot \overrightarrow{u}=\phi (x,y)\overrightarrow{u}\mathrm{.}\backslash overrightarrow\{u\}\; :\; \backslash overrightarrow\{\backslash text\{grad\}\; \_f\}\; (x,\; y)\; \backslash cdot\; \backslash overrightarrow\{u\}\; =\; \backslash varphi\; (x,\; y)\; \backslash overrightarrow\{u\}.$ Different conceptions, different notations: physicists, deeply attached to the melody of the Phoenician harp, denote by $\mathrm{\nabla }(\overrightarrow{u})\backslash nabla\; (\backslash overrightarrow\{u\}\; )$ what mathematicians write as $\overrightarrow{{\text{grad}}_f}(\overrightarrow{u})\mathrm{.}\backslash overrightarrow\{\backslash text\{grad\}\; \_f\}\; (\backslash overrightarrow\{u\}\; ).$