15 December 2015 Why tensors should be taught at undergraduate levels
Author Affiliations +
Proceedings Volume 9793, Education and Training in Optics and Photonics: ETOP 2015; 979319 (2015) https://doi.org/10.1117/12.2223104
Event: Education and Training in Optics and Photonics: ETOP 2015, 2015, Bordeaux, France
Many academic subjects that were taught previously in the framework of theoretical physics moved to engineering. These include courses in electromagnetics, statics and dynamics, heat and mass transfer, mechanics of solids, nuclear power, and courses that branch from these, like fiber optic communications thermodynamics. However, the mathematical foundation in engineering education has remained substantially unchanged during this transition period, typically peaking at the level of linear algebra, vector calculus and integral transforms. As a result many undergraduate engineering courses are built in such a way as to avoid tensor analysis and tensor calculus, as if such mathematical constructs are beyond the capacity of the undergraduate student to understand. We show that this not the case.
Beyle* and Maxum**: Why tensors should be taught at undergraduate levels

This paper is devoted to demonstrating examples of phenomena studied in engineering courses that are actually more easily explained based on tensor nature of participating factors. These examples are taken from optics/photonics applications by formulating their respective constitutive relationships with tensorial field quantities as a natural extension to what they already know in vector analysis. We show the tensorial relationships become quite clear and workable for the undergraduate student. Examples include birefringence, photoelasticity, nonlinear optics, physics and mechanics of composites and metamaterials.

With these examples we show why tensor algebra and tensor calculus should be taught as a part of vector algebra and vector calculus, thus obviating the necessity for students to have to un-learn material before the being ready to begin working in optics, photonics, photoelasticity mechanics of composites and metamaterials. This approach uses students’ time more efficiently.

To test these suppositions we used tensor concepts in teaching the following subjects:

  • Statics and Dynamics

  • Mechanics of Solids

  • Mechanics of Composite Materials and Structures

  • Engineering Electromagnetics,

  • Advanced Electromagnetics

  • Optical Communications

  • Fiber Optic Systems

  • Nonlinear Fiber Optics

  • Advanced Engineering Mathematics

Tensors were introduced under three points of view:

  • as guaranteed invariant objects in rotation of the system of coordinates having law transform in the form of a generalization of the corresponding law for vectors

  • as objects with rank r linearly connecting two objects with ranks p and q, where r=p+q ranks r=0 for scalar, r=1 for vector objects, and r=2 for dyadics, etc;

  • as a results of different multiplications of vectors (forming tensors of varying rank).

Let’s consider some examples of the modern analytical methods as a first exposure to tensors for undergraduate students.

Example 1. Constitutive Relationships between Electric and Magnetic Fields

Introductory courses dealing with electric and magnetic fields typically begin with homogeneous isotropic material media when the field intensities are at a sufficiently low level such that the medium remains linear with incremental increments in field strength. In these simplistic cases the flux density field vectors 00032_psisdg9793_979319_page_2_1.jpg and 00032_psisdg9793_979319_page_2_2.jpg are linearly related to their respective vector field intensities 00032_psisdg9793_979319_page_2_3.jpg and 00032_psisdg9793_979319_page_2_4.jpg by scalar permittivity and permeability constants ε0 and μ0, respectively. These are traditionally expressed in introductory EM courses by the constitutive expressions 00032_psisdg9793_979319_page_2_5.jpg and 00032_psisdg9793_979319_page_2_6.jpg, where we show vector fields with the single-headed “→” over bars. However, for anisotropic media the constitutive relations become


And combining these as

Figure 1.


where we show the permittivity and permeability as nine-component dyadics 00032_psisdg9793_979319_page_2_10.jpg and 00032_psisdg9793_979319_page_2_11.jpg with the double-headed arrow over bars to denote the “dual directional compoundedness” as described in Field Mathematics1, Section 3.1. This is often the first exposure for students that there are quantities that require “rank” values greater than one.

When vectors were first mentioned at the freshman-sophomore level, instead of defining vectors as quantities that have direction and magnitude in order to distinguish them from scalars that have magnitude only, the word “single” should be added2, namely “vectors are quantities that have magnitude and a single direction.” With emphasis on the word single, this implants the image on the student at an early stage that there are quantities that have multiple directionality and that such quantities are not three-component vectors. Thus 00032_psisdg9793_979319_page_2_10.jpg and 00032_psisdg9793_979319_page_2_11.jpg are nine-component dyadics described by


Here we point out to the undergraduate that the nine unit vector combinations 00032_psisdg9793_979319_page_3_1.jpg are bidirectional unitary dyadics and that without a dot or cross product operation between them one should think of them as a single entity which we refer to as a unitary dyadics. We explain in Reference 1, Section 3.1 that dual directional compoundedness in these cases refers to the direction of the applied field 00032_psisdg9793_979319_page_3_2.jpg or 00032_psisdg9793_979319_page_3_3.jpg and the resultant flux density fields 00032_psisdg9793_979319_page_3_4.jpg or 00032_psisdg9793_979319_page_3_5.jpg directions that in general differ when the medium is anisotropic.

In writing the bidirectional unitary dyadic 00032_psisdg9793_979319_page_3_1.jpg, with neither dot or cross product operations implied between the unit vectors 00032_psisdg9793_979319_page_3_6.jpg and 00032_psisdg9793_979319_page_3_7.jpg, the unitary dyadic is a single entity and thus we might prefer to write it as 00032_psisdg9793_979319_page_3_8.jpg to emphasize that situation. Thus,


In the second to last equality of (1-4) the explicit representation dyadics 00032_psisdg9793_979319_page_3_10.jpg where the nine components of the dyadics 00032_psisdg9793_979319_page_3_11.jpg, is represented by the scalar magnitude coefficients 00032_psisdg9793_979319_page_3_12.jpg in the bidirection 00032_psisdg9793_979319_page_3_8.jpg. In Field Mathematics3 the term “dyad” is coined to describe this unitary dyadic.

Notice that the last equality of (1-4) has a “2” inserted in the double-headed arrow over bar in order to emphasize another more explicit depiction of the dual directionality of the two-rank tensors (dyadics) 00032_psisdg9793_979319_page_3_13.jpg and 00032_psisdg9793_979319_page_3_14.jpg. This allows for later depicting higher-rank tensors beyond the dyadic with equal simplicity of notation, which we have found is readily understood by the undergraduate student.

Example 1a. Generalization to metamaterials

In this next example we generalize the prior example to include metamaterials where the constitutive expressions take the form 00032_psisdg9793_979319_page_3_15.jpg and 00032_psisdg9793_979319_page_3_16.jpg and where 00032_psisdg9793_979319_page_3_17.jpg and 00032_psisdg9793_979319_page_3_18.jpg are the cross field dyadics. As before we combine these expressions


which is a generalization of (1-3).

With these examples of dyadics—rank two tensors—provided to our undergraduate students, we next introduce third-rank tensors.

Example 2. Piezoelectric and magnetoelasticity effects

The piezoelectric effect4 is understood as the linear electromechanical interaction between the mechanical stress and the applied electric field intensity in crystalline materials with no inversion symmetry5. Whenever such a material is immersed in an electric field 00032_psisdg9793_979319_page_4_1.jpg, the material encounters an internal stress described by the dyadic 00032_psisdg9793_979319_page_4_2.jpg as


where 00032_psisdg9793_979319_page_4_4.jpg is the third rank piezoelectric stress triadic, which is a 27 component rank three tensor6. Its 27 components contain only 18 independent values for general anisotropic crystals. The piezoelectric effect is a reversible process in that there is an internal generation of an electric field when a mechanical strain is applied from an external force.

Magnetostriction7 is a property of ferromagnetic materials that causes them to change their shape or dimensions during the process of magnetization. The variation of material’s magnetization due to the applied magnetic field changes the magnetostrictive strain 00032_psisdg9793_979319_page_4_5.jpg until reaching its saturation value, 00032_psisdg9793_979319_page_4_6.jpg, described by


where 00032_psisdg9793_979319_page_4_8.jpg is the third-rank inverse magnetostrictive strain triadic, which is also centrosymmetric reducing its 27 components to 18 independent values for general isotropic ferromagnetic materials. Much like piezoelectric materials, Magnetostriction is also a reversible process in that applying tension or compression to the ferroelectric material created an internal magnetic field.

With these two examples we introduce rank-three tensors to the undergraduate explaining that the physics of piezoelectric and magnetostrictive effects is left to separate courses in coupled systems electromechanics and our purpose here is to get them ready to deal with even higher-ranked tensors without becoming overwhelmed with the tedious character of the expansions. We do however point out that the detailed expansion of the dyadic as detailed in Eq.(1-4) becomes three such expansions making up 27 components of the form


where the combination of the unit vectors 00032_psisdg9793_979319_page_4_10.jpg without dot or cross product operations implies tensor product operations denoted by ⊗ as 00032_psisdg9793_979319_page_4_11.jpg simplified by the triple-hatted unitary triadic uijk, which can be referred to as a triad8.

Example 3. Photoelasticity phenomena—Introducing the four-rank tensor

Photoelasticity phenomena is related to birefringence of the thin polymeric film put on the surface of the body subjected to some stresses. Birefringence of the film is related to the anisotropy of dielectric permittivity as a result of orientation of macromolecules differently stretched in different direction. Formally, the film dielectric permittivity dyadic 00032_psisdg9793_979319_page_5_1.jpg can be linearly linked to the stress 00032_psisdg9793_979319_page_5_2.jpg in the body with the four-rank photoelectric stress tensor 00032_psisdg9793_979319_page_5_3.jpg


where the four-rank tensor is expanded as


Although this tensor has in general 81 components it’s symmetry reduces that number to 36 because of the symmetry of both the stress dyadic and the permittivity dyadics. Still the notation is tedious and can be simplified as


where all of the summations, unit vectors and the unitary tensor with the four hats are suppressed for simplicity of notation, but understood as a necessary part of performing actual analyses of photoelastic phenomena where in this case a rank-four tensor was necessary.

Example 4. Coupled Systems Mechanics—Thermoelastic-Piezoelectric-Magnetostrictive Effects

The generalized linear law for coupled effects can be expressed as


where the strain 00032_psisdg9793_979319_page_5_8.jpg, stress 00032_psisdg9793_979319_page_5_2.jpg, electric and magnetic field intensities 00032_psisdg9793_979319_page_5_9.jpg and 00032_psisdg9793_979319_page_5_10.jpg are related by the thermal-strain dyadic 00032_psisdg9793_979319_page_5_11.jpg, the piezoelectric strain triadic 00032_psisdg9793_979319_page_5_12.jpg, the magnetostrictive strain triadic 00032_psisdg9793_979319_page_5_13.jpg, and the compliance tensor 00032_psisdg9793_979319_page_5_14.jpg.

In general Eq. (4-1) represents a system of nine linear equations and nine independent vaariables. In order to determine the inverse of (4-1), i.e. to obtain 00032_psisdg9793_979319_page_5_11.jpg as a function of 00032_psisdg9793_979319_page_5_15.jpg and 00032_psisdg9793_979319_page_5_8.jpg, we multiply each term of (4-1) by the inverse of the compliance tensor, and solve for 00032_psisdg9793_979319_page_5_11.jpg, which becomes


Therefore the inverse of (4-1) is expressed as


Where 00032_psisdg9793_979319_page_6_2.jpg is the thermoelastic dyadic given by


00032_psisdg9793_979319_page_6_4.jpg is the piezoelectric stress triadic given by


00032_psisdg9793_979319_page_6_6.jpg is the magnetostrictive stress triadic given by


and 00032_psisdg9793_979319_page_6_8.jpg is the stiffness tensor, which is the inverse of the compliance tensor


For the undergraduate student we explain why the first coefficient is a dyadic, the second and third coefficients are triadics and the fourth coefficient is a tensor of rank 4 (also known as a quadadic) by the explanation given in Maxum (Reference 1, Section 3.7).

In the cases where the electric and magnetic fields vanish, we have from (4-1)


and for zero stress the strain is 00032_psisdg9793_979319_page_6_11.jpg, which means that each of its nine components are given by the thermal-strain dyadic 00032_psisdg9793_979319_page_6_12.jpg components as a function of thermal change. Similarly, where the fields vanish, the stress is


and for zero strain the stress is 00032_psisdg9793_979319_page_6_14.jpg, which means that each of its components are given by the thermalstress dyadic 00032_psisdg9793_979319_page_6_15.jpg as a function of thermal change.

Example 5. Constitutive laws of nonlinear optics

The nonlinear constitutive relation between the electric flux density field 00032_psisdg9793_979319_page_7_1.jpg (also commonly referred to as the displacement vector field) and the electric field intensity field 00032_psisdg9793_979319_page_7_2.jpg is an extension of Eq. (1-1) into higher ranked permittivity tensors 00032_psisdg9793_979319_page_7_3.jpg, and the higher tensor products of the 00032_psisdg9793_979319_page_7_2.jpg field are 00032_psisdg9793_979319_page_7_4.jpg, and 00032_psisdg9793_979319_page_7_5.jpg, etc., as


At this point we introduce to the undergraduate the concept of the tensor product ⊗, pointing out that there are three types of vector-vector products, namely the dot product “·”, the cross product “×”, of which they are already familiar, and the tensor product as just introduced above.

Another approach to nonlinear optics is to expand the polarization vector 00032_psisdg9793_979319_page_7_7.jpg rather than 00032_psisdg9793_979319_page_7_1.jpg.

In linear optics of crystals the susceptibility dyadic 00032_psisdg9793_979319_page_7_8.jpg is used for linking the polarization density vector 00032_psisdg9793_979319_page_7_7.jpg with the electrical field intensity 00032_psisdg9793_979319_page_7_2.jpg as 00032_psisdg9793_979319_page_7_9.jpg. In nonlinear optics this is expressed in terms of susceptibility tensors of increasing ranks where the powers of 00032_psisdg9793_979319_page_7_2.jpg are defined by 00032_psisdg9793_979319_page_7_10.jpg and 00032_psisdg9793_979319_page_7_11.jpg, which are rank two and three tensors, respectively.

As is shown in Field Mathematics9

  • The tensor (direct) product of two tensors yields another tensor of rank equaling the sum of the ranks of the two tensors,

  • The cross product of two tensors yields a tensor of rank one less than the sum of the ranks of the two tensors.

  • The dot product of two tensors yields a tensor of rank two less than the sum of the ranks of the two tensors,

  • The multiple dot product of two tensors yields a tensor of rank equalling the sum of the two tensors less two for each multiple dot product.

Notice that in the last term of (5-1), 00032_psisdg9793_979319_page_7_12.jpg is rank 4 and 00032_psisdg9793_979319_page_7_13.jpg is rank 3, thus the sum of the ranks of the two tensors is seven (3+4=7). Then applying the fourth bullet above to the triple product we subtract 2 × 3 = 6 and obtain one for the rank of 00032_psisdg9793_979319_page_7_14.jpg which is a vector. Likewise in the second term we get 3 + 2 – (2 × 2) = 1, and the first term we get 2 + 1 – 2 = 1, resulting in a vector for each term, because every term of every equation must have the same rank10.

In isotropic material such as glass the polarization density and the electrical field intensity are collinear. In this case the constitutive relation for the nonlinear optical phenomena isotropic body is11


where susceptibilities of increasing rank are introduced. The generalization to the nonlinear case can be made formally in three ways:

(a) The susceptibility components in terms of the electric field intensity:


where E2 is the scalar 00032_psisdg9793_979319_page_8_3.jpg and 00032_psisdg9793_979319_page_8_4.jpg is a functional relationship.

(b) The susceptibility components in terms of the electric field intensity vector and susceptibility dyadic:


where 00032_psisdg9793_979319_page_8_6.jpg is the invariant of the susceptibility dyadic and the electric field intensity vector.

(c) The susceptibility tensors of increasing ranks:


The difference between these approaches is in the effects on the polynomial coefficients connecting each component of the vector 00032_psisdg9793_979319_page_8_7.jpg to each component of the vector 00032_psisdg9793_979319_page_8_8.jpg of the change in angle with respect to the axes of the crystal. In the relaxing media, the vector 00032_psisdg9793_979319_page_8_7.jpg depends not only on the vector 00032_psisdg9793_979319_page_8_8.jpg at time t, but also from the previous time history of the vector 00032_psisdg9793_979319_page_8_9.jpg. This would lead us to the concept of tensor-functionals, which is not a topic of current presentation for undergraduates. This and other such topics are being left to future work.


These examples show the importance of understanding the tensor nature of many objects and phenomena in modern physics and engineering, thus demonstrating why tensors need to be studied at undergraduate levels by first exposing dyadics (tensors of rank two) as shown in Example 1; then triadics (tensors of rank three) in Example 2; followed by Example 3 which introduces a tensor of rank four. In Example 4 we show how the undergraduate is introduced to coupled systems that use tensors of ranks 2, 3, and 4. And finally we show infinites series of tensors of ever increasing ranks in Example 5. This approach has been successfully done for many years at some advanced universities. However, the authors’ practice has shown that it can be successfully presented even in rural universities, such as Lamar University (Beaumont, TX) and Pittsburg State University (Pittsburg, KS) with students that come without the sophistication typically found at the ivy league schools.


[1] Maxum, Bernard, Field Mathematics for Electromagnetics, Photonics and Materials Science, SPIE Press, Fourth printing, 2007 {ISBN 081945523-7}, SPIE – The International Society for Optical Engineering, P.O. Box 10, Bellingham, WA, USA.

[2] Ibid., Section 2.3.

[3] Ibid., Section 3.3 and in particular pages 3-6 and 3-7 with Eq. (3.4-4).

[4] Piezoelectric effect from Wikipedia (http://en.wikipedia.org/wiki/Piezoelectricity#Mechanism).

[6] Maxum, op. cit, last paragraph of pg.3-13 and Table 3-1, pg. 3-22.

[7] Magnetostriction from Wikipedia (http://en.wikipedia.org/wiki/Magnetostriction).

[8] Maxum, op. cit., last paragraph of Pg. 3-13.

[9] Maxum, op. cit., Sections 3.4, 3.6, 3.7, 3.8 and Table 3-1.

[10] Maxum, op. cit.., Section 3.7.

[11] Agrawal, Govind P., Nonlinear Fiber Optics, Elsevier, Inc. 2007 (ISBN 13: 978-0-12-369516-1), Section 2.3.

© (2015) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Andrey Beyle, Bernard Maxum, "Why tensors should be taught at undergraduate levels", Proc. SPIE 9793, Education and Training in Optics and Photonics: ETOP 2015, 979319 (15 December 2015); doi: 10.1117/12.2223104; https://doi.org/10.1117/12.2223104


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