What were the criticisms against the introduction of "vector analysis"?

Question

Frequently, 19th century physicists—e.g., Helmholtz or Maxwell—did not use modern-day vector notation, which Gibbs contributed in large part to.

For example, Helmholtz in his famous paper on the conservation of energy writes

$$X=m\frac{du}{dt},\hspace{1em}Y=m\frac{dv}{dt},\hspace{1em}Z=m\frac{dw}{dt},$$

where $X,Y,Z$ are components of a force and $u=dx/dt$, $v=dy/dt$, $w=dz/dt$ are the components of the tangential velocity $q$. Instead, he could've written this much more concisely as:

$$\vec{F}=m\vec{a},$$

where $\vec{F}=(X,Y,Z)$ and $\vec{a}=(u,v,w)$.

Similarly, he could've written

$$d(q^2)=\frac{d(q^2)}{dx}dx+\frac{d(q^2)}{dy}dy+\frac{d(q^2)}{dz}dz$$

much more concisely as

$$d(q^2)=\nabla q^2\cdot d\vec{r}.$$

Thus:

1. Why did those physicists use such a cumbersome, redundant way of writing what today we'd write with vector notation?
2. Are there some expressions that cannot be expressed in vector notation but must be expressed in this seemingly cumbersome notation that predated vector notation?

In summary:

3. What were the criticisms against the introduction of "vector analysis"?

Answer 1

Pierre Duhem (1861-1916) gave in his The Aim & Structure of Physical Theory (1906) p. 77—commenting on what he considered a characteristic of, generally, English thinkers, i.e., their "ampleness of mind" (Pascal's esprit de géométrie, vs. the esprit de finesse characteristic of a "French [or German] mind")—a criticism of "vector analysis":

But only among the English is ampleness of mind found so frequently as an endemic, traditional habit; thus it is only among English men of science that symbolic algebras, the calculus of quaternions, and "vector analysis" are customary, most of the English treatises making use of these complex and shorthand languages. French and German mathematicians do not learn these languages readily; they never succeed in speaking them fluently or, above all, in thinking directly in the forms which constitute these languages. In order to follow a calculation based on the method of quaternions or of "vector analysis" they have to translate it into a version of classical algebra. One of the French mathematicians, Paul Morin, who had studied most profoundly the different kinds of symbolic calculi, once told me: "I am never sure of a result obtained by the method of quaternions until I have checked it by our old Cartesian algebra."

Answer 2

You can see :

• Michael Crowe, A History of Vector Analysis (1967) Ch.5.IV. Gibbs' Elements of Vector Analysis, page 155-on.

Josiah Willard Gibbs' Elements was privately printed in 1881 and 1884.

His pupil Edwin Bidwell Wilson compiled the textbook Vector Analysis, based on Gibbs' lectures, as Gibbs was at the time busy preparing his book on thermodynamics :

Vector Analysis, first published in 1901 and based on the lectures that Josiah Willard Gibbs had delivered on the subject at Yale University, did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus, as used by physicists and mathematicians. It went through seven editions (1913, 1916, 1922, 1925, 1929, 1931, and 1943).

Oliver Heaviside was the other co-founder of vector analysis :

Heaviside did much to develop and advocate vector methods and the vector calculus. Maxwell's formulation of electromagnetism consisted of 20 equations in 20 variables. Heaviside employed the curl and divergence operators of the vector calculus to reformulate 12 of these 20 equations into four equations in four variables, the form by which they have been known ever since.

See again Crowe's book : Ch.5.VII. Heaviside's Electrical Papers, page 163-on :

The first paper in which Heaviside introduced vector methods was his 1882-1883 paper "The Relations between Magnetic Force and Electric Current," published in the Electrician. The way in which vectors were introduced by Heaviside is somewhat surprising.

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J.C.Maxwell's A Treatise on Electricity and Magnetism was published in 1873, prior to Gibbs's pamphlet.

The same for Hermann von Helmholtz :

Helmholtz studied the phenomena of electrical oscillations from 1869 to 1871, and in a lecture delivered to the Naturhistorisch-medizinischen Vereins zu Heidelberg (Natural History and Medical Association of Heidelberg) on April 30, 1869 titled On Electrical Oscillations [...]. In 1871, Helmholtz moved from Heidelberg to Berlin to become a professor in physics. He became interested in electromagnetism and the Helmholtz equation is named for him.

Regarding Maxwell's attitude, see Crowe : Ch.5.V. James Clerk Maxwell: Critic of Quaternions, page 127-on :

Of the four books written by Maxwell after 1873 only one mentioned vectors; this is his elementary work on mechanics published in 1876 and entitled Matter and Motion. Herein Maxwell included a short section on the idea of and the addition and subtraction of vectors. Quaternions were not mentioned [page 138].