Notes on "A Brief on Tensor Analysis" by James Simmonds
This page is a supplement to the book "A Brief On Tensor Analysis" by James G. Simmonds, second edition, ISBN 0-387-94088-X and ISBN 3-540-94088-X, containing three sections, on notations used in the book, an errata list of mistakes in the book, and a critique of the mathematical arguments used in the book.
Notations
The book is extremely idiosyncratic in its use of symbols which are not used for a similar purpose in other textbooks, making it very difficult to drop into pages of the book. Explanations of the notation exist, but are scattered throughout the book's pages, without a single location to look up what is meant. The hapless reader must, therefore, not only have an excellent memory for Simmonds' idiosyncratic use of symbols, but also read the book strictly sequentially. The following list is an attempt to help readers of the book to follow it without having to search backwards. Some of the notations are more standard than others, but I've included all of the ones that I had to look up, since one often doesn't know whether the notation is Simmonds' own or the usual meaning.
The following table is ordered by the number of the page where the notation is introduced, or the first page where it is found in the cases that I could not find any definition.
| Page | Symbol | Meaning |
|---|---|---|
| 4 | $\mathbf{\bar{v}}$ | Simmonds uses a bar over the vector to indicate a unit vector in the $\mathbf v$ direction. Note that Simmonds also uses the more common symbol for a unit vector, $\mathbf{\hat{v}}$, for an arbitrary function of the coordinates, rather than a unit vector. |
| 5 | $\perp$ | Simmons uses this in the midst of the text as a substitute for the word "perpendicular" as well as for "orthogonal". For example in exercise 2.16 he rather suddenly substitutes the word "orthogonal" with the $\perp$ symbol, without bothering to explain what he means. |
| 7, 18 | $\sim$ | For reasons which are unclear, Simmonds uses this in place of an equals sign when expressing a vector or tensor in cartesian coordinates. |
| 16 | $\mathbf{Tv}$ | Simmonds variously uses $\mathbf{Tv}$ and $\mathbf{T\cdot v}$ to mean ${\mathbf T}({\mathbf v})$, the action of the tensor $\mathbf{T}$ on the vector $\mathbf{v}$. |
| 16 | $\mathbf{uv}$ | Simmonds uses this to mean the "dyad", the direct product of two vectors. |
| 27 | $(\mathbf{Ta})^\bullet$ | This doesn't seem to be explained in the text but from the context it presumably means the derivative of the vector produced by $\mathbf{Ta}$ with respect to $t$. |
| 37 | $\tilde{x}$ | Simmonds sometimes uses a tilde over the variable to indicate that the vector is in a changed basis. |
| 38, 54 | $3.41_2$ | Simmons uses a subscript after an equation number to indicate one equation amongst a group of equations. |
| 45 | $\mathbf{\dot{v}}$ | Simmonds uses this for the derivative with respect to the time, $t$, but he also uses it on pages 65 and 66 in exercise 3.4 for the derivative of a curve parameterised by the letter $t$, which is not necessarily the time. He also uses $t'$ to indicate the derivative of the tangent vector with respect to $s$, the arc length of the curve. |
| 51 | $v^{(r)}$ | Simmonds introduces a bracketed superscript to mean the "physical component" of a vector. |
| 52 | $g_{r,r}$ | The comma followed by $r$ indicates the partial derivative of the coordinate vector $g_r$ with respect to $r$. |
| 55 | $(u^j)$ | Simmonds sometimes uses $(u^j)$ as a way to abbreviate $(u^1, u^2, u^3)$ as function arguments. |
| 55 | $\hat{x}$ | I have not been able to find an explanation of the meaning of a hat symbol over a letter in the text anywhere, but Simmonds seems to use this to indicate a function which changes the coordinates. Note that a vertical bar over the variable is used for a unit vector. It is not really clear what the hat symbol is actually supposed to mean. For example on page 65, in the exercise at the bottom of the page, he writes an equation as $\mathbf{x}=\mathbf{\hat{x}}(...)$, and then refers to $\mathbf{x}$ being differentiable, rather than $\mathbf{\hat{x}}$ being differentiable. |
Errata
Here the number after the page number is the approximate line number of the error. Negative numbers indicate to count from the bottom of the page rather than the top. Equations are counted as lines.
| Page/line | Error | Correction | Notes |
|---|---|---|---|
| 19/11 | fom | from | |
| 20 | perspective | 3D | Presumably Simmonds means to draw a representation of the three dimensional vectors, not a "perspective" drawing, which doesn't make sense in this case, since there is no depth of the drawing which would make perspective (drawing things further away proportionately smaller) useful. |
| 26/2 | time | times | |
| 46, 49/-8, 15 | $\mathbf{o}$ | $\mathbf{0}$ | This appears more than once but I cannot find a definition. It's possible that somewhere or another Simmonds has defined what he means by $\mathbf{o}$, but from the context it means the same thing as $\mathbf{0}$, so I assume it was a typographic error. |
| 46/15 | discuss | discus | |
| 57/2 | $p$ | $\rho$ (rho) | The letter p has been used where the Greek letter rho ($\rho$) should have been in the first line of equations. |
| 65/8 | then | than | |
| 65/10 | third other tensor | third order tensor | |
| 65/-4 | $\mathbf{x}$ is differentiable | $\mathbf{\hat{x}}$ is differentiable | Simmonds gives an equation for a position $\mathbf{x}$ in terms of a function $\mathbf{\hat{x}}(t)$ but then repeatedly refers to the function as $\mathbf{x}$ rather than $\mathbf{\hat{x}}$ in the rest of the exercise. |
| 67/-9 | User | Use |
Mathematical critiques
In this section I address various places where the mathematics of the book seems to rest on shaky foundations.
Circular definition of dot product
On page 8, the "definition" of the dot product in equation (1.11) adds nothing to equation (1.8). Simmond's definition of the dot product in terms of squares of lengths and differences of vectors is a circular argument which doesn't add anything to what we already have in equation (1.8).
Further, the orientation of the angle $\theta$, which Simmonds devotes a special double-headed arrow notation to, is irrelevant if we are only considering the cosine, since $\cos(-\theta) = \cos\theta$ for all values of $\theta$. Since equation (1.11) is not used any further elsewhere in the book, the only reason to derive equation (1.11) seems to be to derive equation (1.13).
The argument presented by Simmonds at the top of page 9, that the dot product defined by (1.11) is invariant under change of cartesian coordinates is also spurious, since equation (1.8) contains only the vector lengths and an angle $\theta$ between the two vectors, and this angle is, like the lengths of the vectors, not only invariant under transformations of cartesian coordinates, but has not even been expressed in terms of cartesian coordinates.
Interpretation of Vector Addition
On page 10 and 11, in a subsection titled "Interpretation of Vector Addition", Simmonds argues that vector addition does not represent a physical reality, because when rotations are expressed as triplets of numbers, the sum of the components of two of these triplets does not correspond to the rotation produced, unless the two rotations are coaxial. He then goes on to make an analogy with the use of fractions to count people.
These arguments are silly, and surely very confusing for many readers. What Simmonds is actually saying is that when rotations are represented as triplets of numbers, these triplets cannot be added to form a meaningful result, and thus don't really form a meaningful linear space. Since the rotations don't form a usable vector space, there does not seem much point in describing them as "vectors".
Definition of the cross product
It's doubtful whether someone who has not encountered the cross product previously would be able to make head or tail of Simmonds' definition on page 11, which tries its best to be "coordinate free" but unhappily omits to point out, except in the figure, that that the cross product is a vector perpendicular to the two vectors in the cross product.
I cannot follow the argument at the top of page 12 about the volume of the paralleliped, complete with the italicised word right to add to the confusion, so I assume that the knowledge of this is a prerequisite for reading Simmonds' book.
Circular argument used in the derivation of the cartesian form of tensors
On page 19, Simmonds derives the conclusion that second-order tensors have a cartesian base formed of the dyads $\mathbf{e_i e_j}$, but he uses the result he claims to be deriving to derive the result itself, between equations (1.40) and (1.41).
If second-order tensors are a linear map from a three-dimensional vector space to another three-dimensional vector space, then by standard arguments from linear algebra the nine dyads form a basis for the map.
Tangents to the orbit
Simmonds defines "smooth" to mean $\mathbf{\hat{v}}$, the derivative
of $\mathbf{\hat{x}}$ with respect to $t$, is continuous, so he means
"has a continuous first derivative"
On page 47, Simmonds states "If $\mathbf{\hat{x}}$ is smooth, then $\mathbf{\hat{v}}$ is tangent to $C$ at $P(t)$". Although this is true, it is rather confusing. If $\mathbf{\hat{v}}$ is not continuous at $P(t)$, the orbit, $C$ is kinked there, so there is no meaningful tangent.