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:
- Why did those physicists use such a cumbersome, redundant way of writing what today we'd write with vector notation?
- 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:
- What were the criticisms against the introduction of "vector analysis"?