Are there good reasons for the absence of shear and gradients of vectors in the undergraduate curriculum, or have we simply been negligent by not explicitly introducing students to these concepts early on? In this paper, we (i) remind the reader that div and curl are not the entire story when it comes to vector derivatives and (ii) ask the reader to consider whether the missing information—shear or the more general vector gradient—should be included in the undergraduate curriculum. In an attempt to address this last point, we give a list of hypothetical responses to the question “Why no shear?,” along with some arguments both for and against teaching it. We leave it to the readers to choose among these reasons or to come up with one of their own when deciding whether or not to include shear and vector gradients in their undergraduate teaching.

Topics
Electromagnetism, Electrostatics, Lagrangian mechanics, Numerical linear algebra, Coordinate system, Vector calculus, Vector fields, Educational assessment, Teaching, Books
1.
H. M.
 
Schey
,
div, grad, curl, and all that, an informal text on vector calculus
, 3rd ed. (
W.W. Norton & Company
, New York,
1996
).
Google Scholar
 
2.
Geometric explanations of divergence and curl are also presented in the excellent texts by
E. M.
 
Purcell
,
Electricity and Magnetism
, Berkeley Physics Course, Vol.
II
, 2nd ed. (
McGraw-Hill
,
New York
,
1963
) and
D. J.
 
Griffiths
,
Introduction to Electrodynamics
, 3rd ed. (
Prentice-Hall Inc.
,
New Jersey
,
1999
).
Google Scholar
 
3.
The reader should note that our statement about the necessity of knowing the gradient of a vector field does not conflict with Helmholtz’s theorem (see e.g., Sec. 1.16 of Ref. 4) which states that any differentiable vector field is uniquely specified by giving its divergence and curl within a simply connected region of space and its normal component on the boundary. Helmholtz’s theorem is a uniqueness theorem about a global solution to a differential equation—namely, if we know the divergence and curl of a vector field everywhere in space, then we can integrate these expressions to find the field. The gradient ∇A of a vector field A, on the other hand, describes the local variation of the vector field as one moves from point to point.
4.
G. B.
 
Arfken
 and
H. J.
 
Weber
,
Mathematical Methods for Physicists
, 6th ed. (
Elsevier
,
Amsterdam
,
2005
).
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5.
The shortness of the reference list for this paper underscores the absence of shear and gradient of a vector in undergraduate physics. No appropriate reference could be found in the American Journal of Physics, the European Journal of Physics, the Physics Teacher, or using broader searches.
6.
We consistently use superscripts for coordinate indices and subscripts for component indices of vectors and tensors; these have nothing to do with covariant and contravariant components.
7.
εijk is a totally anti-symmetric tensor defined by: εijk=1 if ijk is an even permutation of 123, εijk=-1 if ijk is an odd permutation of 123, and εijk=0 otherwise.
8.
See, however, the unpublished text Applications of Classical Physics, Vol. 2, by R. D. Blandford and K. S. Thorne. In Part III, Sections 10.2 and 10.3, the authors discuss the gradient of a vector field and shear in the context of elastostatics.
9.
D. J.
 
Griffiths
,
Introduction to Electrodynamics
, 3rd ed. (
Prentice-Hall Inc.
,
New Jersey
,
1999
).
Google Scholar
 
10.
The same can be said about the vector identity for the curl-of-the-curl of a vector field
∇×(∇×A)=∇(∇·A)-∇2A
since the last term, which is the Laplacian of the vector field A, involves the vector gradient of A—i.e., ∇2A≡∇·(∇A) where the dot of the divergence is a sum over the second index of ∇A⁠. The fact that there is real potential for confusion here is illustrated by the discussion of this identity on page 51 of Ref. 4. The authors first state that the above curl-curl identity is “valid in Cartesian coordinates but not in curved coordinates,” which is an incorrect statement; the identity, being a vector identity, is valid in any coordinate system. Then they say that ∇·(∇A)⁠, although “not included in their list [of vector derivatives], may be defined by this equation,” which is a valid statement. Obviously, if we as teachers make conflicting statements such as these, it is no wonder why students have difficulty properly understanding these topics.
11.
R. P.
 
Feynman
,
The Feynman Lectures in Physics
(
Addison-Wesley, Reading, Massachusetts
,
1964
), Vol.
II
, Chap. 7.
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12.
It turns out that at each point the shear tensor can be represented by three vectors, its eigenvectors. These eigenvectors must be mutually orthogonal, so their orientation contains three numbers, equivalent to the three degrees of freedom in a general rotation in three dimensions. The other two pieces of information are carried by the two degrees of freedom in the three eigenvalues, which are required to sum to zero.
© 2012 American Association of Physics Teachers.
2012
American Association of Physics Teachers
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