Back to the roots of vector and tensor calculus: Heaviside versus Gibbs

Abstract

In June 1888, Oliver Heaviside received by mail an officially unpublished pamphlet, which was written and printed by the American author Willard J. Gibbs around 1881–1884. This original document is preserved in the Dibner Library of the History of Science and Technology at the Smithsonian Institute in Washington DC. Heaviside studied Gibbs’s work very carefully and wrote some annotations in the margins of the booklet. He was a strong defender of Gibbs’s work on vector analysis against quaternionists, even if he criticised Gibbs’s notation system. The aim of our paper is to analyse Heaviside’s annotations and to investigate the role played by the American physicist in the development of Heaviside’s work.

Appendices

Appendix A: On Heaviside’s generalisations of the Divergence theorem: some technical details.

The Divergence theorem is usually formulated for continuously differentiable vector fields, its classical version, however, as suggested in Gibbs’s booklet, it holds for scalar and tensor fields also. Nowadays, we know that it can be generalised for tempered distributions, by introducing the correct measure in order to carefully give sense to the integrals involved (Šilhavý. 2005; van Dijk 2013). By allowing the function of the theorem to be a function of the unit vector of the surface itself, the theorem can be applied in its classical version if the surface ${\mathcal {S}}$ is an orientable differentiable manifold. Indeed, in this case the Gauss map $\mathbf {N}(\mathbf {x})$ can be globally defined and it represents the unit normal vector field of the theorem, pointing outward as usual. Hence, Heaviside’s request that the scalar, or vector, integrand be an odd function of $\mathbf {N}(\mathbf {x})$ is not needed. In the case of non-regular surfaces or in the case of a cubic surface, we would need the generalised version for distributions. In the following, first, we will not use any formulation of the Divergence theorem, but we briefly repeat Yavetz’s (1995) argument in order to emphasise how to avoid the use of distributions. Second, we show how Heaviside’s specific formula for the cube follows from Gibbs’s generalisation of the Divergence theorem, with the help of Heaviside’s additional hypothesis, by defining the derivatives in the distribution sense.

In the following, we consider scalar functions only. Let $F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right)$ be an arbitrary scalar function, on a closed surface ${\mathcal {S}} \in {\mathbb {R}}^3$. The closed surface projects into a region ${\mathcal {R}}$ in the $xy-$plane. We assume ${\mathcal {S}}$ is vertically simple.²²${\mathcal {S}}$ is then described by two equations, $z=\alpha (x,y)$ and $z=\beta (x,y)$, representing the upper and the lower part of the surface, respectively. The integral of F over the surface, namely:

$$\begin{aligned} \int _{{\mathcal {S}}} F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) {\mathrm{d}}S, \end{aligned}$$

(65)

can be rewritten as a volume integral. Indeed, there will always be some function $f\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right)$ such that:

$$\begin{aligned} \int _{{\mathcal {S}}} F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) {\mathrm{d}}S = \int _{{\mathcal {V}}} f\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) {\mathrm{d}}V. \end{aligned}$$

(66)

Like in the usual proof of the Divergence theorem, we start with the surface integral using the unit versors $\mathbf {i}$, $\mathbf {j}$ and $\mathbf {k}$. The l.h.s. of Eq. (66) is equivalent to the flux of the vector field $F\mathbf {N}$ across the surface, because $\mathbf {N}\cdot \mathbf {N} = 1$ by definition on the surface itself, and it can be rewritten as the sum of two integrals, one over the upper surface ${\mathcal {S}}_1$, on which $z = \alpha (x, y)$ and one over the lower one ${\mathcal {S}}_2$, on which $z = \beta (x, y)$, namely:

$$\begin{aligned}&\int _{{\mathcal {S}}} F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) {\mathrm{d}}S \nonumber \\&\quad = \int _{{\mathcal {S}}_1} F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) {\mathrm{d}}S + \int _{{\mathcal {S}}_2} F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) {\mathrm{d}}S \nonumber \\&\quad =\int _{{\mathcal {S}}_1} F\left( \alpha ,\,\mathbf {N}(\alpha )\right) \mathbf {N}(\alpha )\varvec{\cdot }d\mathbf {S}+\int _{{\mathcal {S}}_2} F\left( \beta ,\,\mathbf {N}(\beta )\right) \mathbf {N}(\beta )\varvec{\cdot }d\mathbf {S}\nonumber \\&\quad =\int _{{\mathcal {R}}} F\left( \alpha ,\,\mathbf {N}(\alpha )\right) \mathbf {N}(\alpha )\varvec{\cdot }\mathbf {n}_\alpha {\mathrm{d}}x{\mathrm{d}}y+\int _{{\mathcal {R}}} F\left( \beta ,\,\mathbf {N}(\beta )\right) \mathbf {N}(\beta )\varvec{\cdot }\mathbf {n}_\beta {\mathrm{d}}x{\mathrm{d}}y, \end{aligned}$$

(67)

where we have introduced the following notation $\mathbf {N}(\alpha ) = \mathbf {N}(x,y,\alpha (x,y))$, $d\mathbf {S}=\mathbf {N}{\mathrm{d}}S$, and we have used the outward un-normalised vectors $\displaystyle \mathbf {n}_\alpha = \left( \frac{\partial \alpha }{\partial x}, \frac{\partial \alpha }{\partial y}, 1\right)$ and $\displaystyle \mathbf {n}_\beta = \left( -\frac{\partial \beta }{\partial x}, -\frac{\partial \beta }{\partial y}, -1\right)$ and ${\mathcal {R}}$ is the projection of ${\mathcal {S}}_1$ and ${\mathcal {S}}_2$ onto the xy plane. Let us now suppose that the unit cube is centred at the origin of the coordinates, hence $\displaystyle \alpha (x,y)= z-\frac{1}{2}$ and $\displaystyle \beta (x,y)= z+\frac{1}{2}$. The normal vector $\mathbf {n}_\alpha$ is normalised and then $\mathbf {n}_\alpha = \mathbf {N}_\alpha$. The integral over the whole closed surface can be obtained by summing the integral over the three coupled opposite squares. The sum along the z-component is equal to Eq. (67), and by using $\mathbf {n}_\alpha = \mathbf {N}_\alpha$, the integral reads:

$$\begin{aligned} \int _{{\mathcal {R}}} \left[ F\left( \alpha ,\,\mathbf {k}\right) + F\left( \beta ,\,-\mathbf {k}\right) \right] {\mathrm{d}}x{\mathrm{d}}y = \int _{{\mathcal {R}}} \left[ F\left( \alpha ,\,\mathbf {k}\right) - F\left( \beta ,\,\mathbf {k}\right) \right] {\mathrm{d}}x{\mathrm{d}}y, \end{aligned}$$

(68)

where we used Heaviside’s additional hypothesis. In the last integral, $\mathbf {N}$ does not depend on $\mathbf {x}$ and therefore can be rewritten in the following equivalent form:

$$\begin{aligned} \int _{{\mathcal {R}}} \left[ F\left( \alpha ,\,\mathbf {k}\right) - F\left( \beta ,\,\mathbf {k}\right) \right] {\mathrm{d}}x{\mathrm{d}}y= & {} \int _{{\mathcal {R}}}{\mathrm{d}}x{\mathrm{d}}y\int _{\beta (x,y)}^{\alpha (x,y)} \frac{\partial }{\partial z} F\left( \mathbf {x},\,\mathbf {k}\right) {\mathrm{d}}z \\= & {} \int _{{\mathcal {V}}} \nabla _3 F\left( \mathbf {x},\,\mathbf {k}\right) {\mathrm{d}}V . \end{aligned}$$

As in Heaviside’s original formulation, the derivative has been reconstructed ‘by hand’ and no additional hypothesis on the differentiability of the unit normal vector field is needed. By summing the contribution of all surfaces, we get Heaviside’s formula:

$$\begin{aligned} \displaystyle f\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) = \nabla _1 F\left( \mathbf {x},\,\mathbf {i}\right) + \nabla _2 F\left( \mathbf {x},\,\mathbf {j}\right) + \nabla _3 F\left( \mathbf {x},\,\mathbf {k}\right) . \end{aligned}$$

(69)

In order to show that Heaviside’s formulation is a particular case of the most general version of the Divergence theorem, let us now suppose that $F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right)$ be differentiable in the distributional sense, where $\mathbf {N}(\mathbf {x})$ is the unit normal vector field of the cube. First, we rewrite the surface integral (65) as follows:

$$\begin{aligned} \int _{{\mathcal {S}}} F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) {\mathrm{d}}S = \int _{{\mathcal {S}}} F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) \mathbf {N}(\mathbf {x})\varvec{\cdot }d\mathbf {S}, \end{aligned}$$

(70)

because for the cube $\mathbf {N}(\mathbf {x})\cdot \mathbf {N}(\mathbf {x}) =1$ holds again. Second, we apply the generalisation of the Divergence theorem involving dyadics, Eq. (45), to Eq. (70), which reads:

$$\begin{aligned} \int _{{\mathcal {S}}} F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) \mathbf {N}(\mathbf {x})\varvec{\cdot }d\mathbf {S}= & {} \int _{{\mathcal {V}}} \nabla \varvec{\cdot }\left\{ F\left( \mathbf {x},\,\mathbf {N}(\mathbf {x})\right) \mathbf {N}(\mathbf {x})\right\} {\mathrm{d}}V\nonumber \\= & {} \int _{[0;1]^3} \left\{ \partial _x\left( FN_1\right) + \partial _y\left( FN_2\right) + \partial _z\left( FN_3\right) \right\} {\mathrm{d}}x{\mathrm{d}}y{\mathrm{d}}z, \nonumber \\ \end{aligned}$$

(71)

where we simplified the notation in the last line.²³ In order to show that Eq. (71) is equivalent to Heaviside’s Cartesian formula, we focus on the first term and rewrite it as follows:

$$\begin{aligned} \int _{[0;1]^2}{\mathrm{d}}y{\mathrm{d}}z\int _0^1 {\mathrm{d}}x \left\{ \partial _x\left( FN_1\right) \right\} = \int _{[0;1]^2}{\mathrm{d}}y{\mathrm{d}}z\int _0^1 {\mathrm{d}}x \left\{ F(1,y,z,\mathbf {i}) + F(0,y,z,-\mathbf {i}) \right\} , \end{aligned}$$

(72)

because $\mathbf {N}(1,y,z)=\mathbf {i}$ and $\mathbf {N}(0,y,z)=-\mathbf {i}$, when $(y,z)\in [0,1]^2$. If we use Heaviside’s additional hypothesis, i.e. $F(x,y,z,-\mathbf {i})=-F(x,y,z,\mathbf {i})$, Eq. (72) reads

$$\begin{aligned} \int _{[0;1]^2}{\mathrm{d}}y{\mathrm{d}}z\int _0^1 {\mathrm{d}}x \left\{ F(1,y,z,\mathbf {i}) - F(0,y,z,\mathbf {i}) \right\} = \int _{[0;1]^3} \left\{ \partial _x\left( F(x,y,z,\mathbf {i})\right) \right\} {\mathrm{d}}x{\mathrm{d}}y{\mathrm{d}}z. \end{aligned}$$

(73)

By repeating the argument for the second and the third terms of Eq. (71), we get Heaviside’s Cartesian formula.

Appendix B: Back cover transcription

See Fig. 17.

Fig. 17

Heaviside’s annotation on the back cover of the booklet

Heaviside used $\phi$ instead of Gibbs’s $\varPhi$. In the following, the word in the square bracket cannot be read clearly, because the writing has faded.

$$\begin{aligned} \text {Stress}.\text {What}&\text {is}\,[\text {the physical}]\,\text {meaning of curl}\; \phi \, ?\\ \\ \phi= & {} \mathbf {P}_1.\,\mathbf {i} + \mathbf {P}_2.\,\mathbf {j} + \mathbf {P}_3.\,\mathbf {k} \\ \nabla \phi= & {} F = \nabla \mathbf {P}_1.\,\mathbf {i} +\nabla \mathbf {P}_2.\,\mathbf {j} +\nabla \mathbf {P}_3.\,\mathbf {k} \\ V\nabla \phi= & {} G = V\nabla \mathbf {P}_1.\,\mathbf {i} +V\nabla \mathbf {P}_2.\,\mathbf {j} +V\nabla \mathbf {P}_3.\,\mathbf {k}\\&\phi \,\text {is stress tensor};\; F\,\text {is force vector}.\,\text {What is}\; G\, ?\\ \\ \\ G= & {} \left[ \mathbf {i}\left( \frac{{\mathrm{d}}P_{13}}{{\mathrm{d}}y}-\frac{{\mathrm{d}}P_{12}}{{\mathrm{d}}z}\right) +\mathbf {j}\left( \frac{{\mathrm{d}}P_{11}}{{\mathrm{d}}z}-\frac{{\mathrm{d}}P_{13}}{{\mathrm{d}}x}\right) + \mathbf {k}\left( \frac{{\mathrm{d}}P_{12}}{{\mathrm{d}}x}-\frac{{\mathrm{d}}P_{11}}{{\mathrm{d}}y}\right) \right] .\; \mathbf {i} \\&+\left[ \mathbf {i}\left( \frac{{\mathrm{d}}P_{23}}{{\mathrm{d}}y}-\frac{{\mathrm{d}}P_{22}}{{\mathrm{d}}z}\right) +\mathbf {j}\left( \frac{{\mathrm{d}}P_{21}}{{\mathrm{d}}z}-\frac{{\mathrm{d}}P_{23}}{{\mathrm{d}}x}\right) + \mathbf {k}\left( \frac{{\mathrm{d}}P_{22}}{{\mathrm{d}}x}-\frac{{\mathrm{d}}P_{21}}{{\mathrm{d}}y}\right) \right] .\; \mathbf {j} \\&+ \left[ \mathbf {i}\left( \frac{{\mathrm{d}}P_{33}}{{\mathrm{d}}y}-\frac{{\mathrm{d}}P_{32}}{{\mathrm{d}}z}\right) +\mathbf {j}\left( \frac{{\mathrm{d}}P_{31}}{{\mathrm{d}}z}-\frac{{\mathrm{d}}P_{33}}{{\mathrm{d}}x}\right) + \mathbf {k}\left( \frac{{\mathrm{d}}P_{32}}{{\mathrm{d}}x}-\frac{{\mathrm{d}}P_{31}}{{\mathrm{d}}y}\right) \right] .\; \mathbf {k} \\ \end{aligned}$$

$$\begin{aligned} V\mathbf {i}\nabla .\,\phi= & {} \left( \mathbf {k}\frac{\mathrm{d}}{{\mathrm{d}}y}-\mathbf {j}\frac{\mathrm{d}}{{\mathrm{d}}z}\right) \phi = \left( \mathbf {k}\frac{\mathrm{d}}{{\mathrm{d}}y}-\mathbf {j}\frac{\mathrm{d}}{{\mathrm{d}}z}\right) \left( \mathbf {P}_1.\,\mathbf {i} + \mathbf {P}_2.\,\mathbf {j} + \mathbf {P}_3.\,\mathbf {k} \right) \\= & {} \left( -\frac{{\mathrm{d}}P_{12}}{{\mathrm{d}}z}+\frac{{\mathrm{d}}P_{13}}{{\mathrm{d}}y}\right) .\; \mathbf {i}+\left( -\frac{{\mathrm{d}}P_{22}}{{\mathrm{d}}z}+\frac{{\mathrm{d}}P_{23}}{{\mathrm{d}}y}\right) .\; \mathbf {j} \\&+ \left( -\frac{{\mathrm{d}}P_{32}}{{\mathrm{d}}z}+\frac{{\mathrm{d}}P_{33}}{{\mathrm{d}}y}\right) .\; \mathbf {k}\\ G =V\mathbf {i}\nabla .\;\phi \;\;\text {etc}= & {} \mathbf {i}V\nabla \mathbf {P}_1 .\;\mathbf {i} + \mathbf {i}V\nabla \mathbf {P}_2 .\;\mathbf {j} + \mathbf {i}V\nabla \mathbf {P}_3 .\;\mathbf {k}\\ ^{\arrowvert }=\mathbf {i}V\nabla .\;\phi \;\;\,\,= & {} \mathbf {i}V\nabla .\;\phi \\ \frac{d\mathbf {P}_3}{{\mathrm{d}}y} - \frac{d\mathbf {P}_2}{{\mathrm{d}}z}\,\quad= & {} \quad \mathbf {i}\left( \frac{{\mathrm{d}}P_{13}}{{\mathrm{d}}y}-\frac{{\mathrm{d}}P_{12}}{{\mathrm{d}}z}\right) +\mathbf {j}\left( \frac{{\mathrm{d}}P_{32}}{{\mathrm{d}}y}-\frac{{\mathrm{d}}P_{22}}{{\mathrm{d}}z}\right) + \mathbf {k}\left( \frac{{\mathrm{d}}P_{33}}{{\mathrm{d}}y}-\frac{{\mathrm{d}}P_{23}}{{\mathrm{d}}z}\right) . \end{aligned}$$

Appendix C: Other annotations by Heaviside

In this appendix, we present the annotations we did not discuss in the main text.

Fig. 18

Misprint correction. In the third line, a minus sign has been added and the minus sign has been changed into a plus sign

On pages 6, 8 and 31, three corrected misprints are present. In Fig. 18, we reported one example. We suppose that these corrections were made by Gibbs, because they are present also in a different copy of the booklet, conserved by the University of Michigan library. (Gibbs 1881-4b).

On the top of page 11, Fig.  19, Heaviside translated Gibbs’s equations (5), (6), (7) and (8).

Fig. 19

Heaviside’s translation of Gibbs’s equations (5), (6), (7) and (8) on p. 11

In the left margin of page 12, Heaviside translated an equation regarding vector linear equations, as shown in Fig. 20.

Fig. 20

Heaviside’s translation on p. 12

In the last part of chapter 3 in Gibbs’s booklet, the author considered rotations and strains as applications of the dyadic’s tool. In particular, he remarked how rotations and strain are intimately related to the characteristics of the dyadic itself. In continuum mechanics the linear transformation associated with the dyadic maps the positions of the points of a rigid body subjected to a displacement into a new set of positions, after that stresses or strains have acted on the body itself. If the dyadic is reducible to the form ${\mathcal {R}}$, as given in Eq. (5), the body will suffer no change of form. The converse is also true and Gibbs observed that ‘the displacement of the body may be produced by a rotation about a certain axis’ [emphasis added] (Gibbs 1881-4a, p. 53). This statement clarifies why we chose the letter ${\mathcal {R}}$. In his booklet, Gibbs called ${\mathcal {R}}$ a versor. On the same page, Fig. 21, we found an annotation Heaviside wrote in pencil. Gibbs noticed that the conjugate of ${\mathcal {R}}$ produces the reverse rotation and from this fact it follows that a versor is the reciprocal of its conjugate. But the converse is not true: if a dyadic is the reciprocal of its conjugate it could correspond either to a versor or to a versor multiplied by $-1$. Heaviside’s annotation written in pencil shows that he named ‘reflector’ the negative of a versor, as shown in Fig. 21.

Fig. 21

Heaviside’s annotation written in pencil: ‘a reflector’ (p. 53)

On page 54, as shown in Fig. 22, there is a comment on Gibbs’s $\varPhi _\times$, i.e. Eq. (14). As already said, Heaviside had introduced it in 1886 by considering displacements of the points of a body due to stresses. He had pointed out that the reversed sign vector, namely $-\varPhi _\times$, ‘is required to balance the torque’ (Heaviside 1892a, p. 544). As we can see in Fig. 22, referring to the right-hand reference system of coordinates, Gibbs pointed out that the vector $-\varPhi _{\times }$ ‘will be parallel to the axis of rotation and equal in magnitude to twice the sine of the angle of rotation measured counter-clockwise as seen in the direction in which the vector points’ [emphasis added] (Gibbs 1881-4a, p. 54). Heaviside did not like the emphasised expression and commented: ‘awkward way it is clockwise if you look along the vector’ (Gibbs 1881-4a, p. 54). Of course, there is nothing substantial in Heaviside’s annotation, it is just a matter of taste if we choose to look along the vector or from the above.

Fig. 22

Heaviside’s annotation on Gibbs’s $\varPhi _{\times }$ (p. 54)

On pages 56, 57 and 58, there are three generic annotations, two written in pencil and one written in pen, which the author may have written in order to remind himself the topic of a particular formula. As shown in Fig. 23, Heaviside referred to the formula for $\theta _3$ by writing ‘Two rotations’ (Gibbs 1881-4a, p. 56), while on pages 57 and 58 he simply wrote the terms ‘Pure Strain’ and ‘Tonic’, respectively, close to the formula they referred to.

Fig. 23

Three generic annotations

On page 70, the versor appears, as shown in Fig. 24. Here, Gibbs discussed functions of linear operators and Heaviside wrote an annotation in pencil where he reminded himself the meaning of the exponential of the operator ${\mathcal {I}}\times \omega$, where ${\mathcal {I}}$ has been defined by Eq. (4) and $\omega$ is a vector.

Fig. 24

Heaviside’s annotation written in pencil: ‘Versor’ (p. 70)