# Introduction to Tensor Calculus and Continuum Mechanics

Staff Emeritus

Staff Emeritus
An Introduction to Tensors for Students of Physics and Engineering

http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM2002211716.pdf
Tensor analysis is the type of subject that can make even the best of students shudder. My own post-graduate instructor in the subject took away much of the fear by speaking of an implicit rhythm in the peculiar notation traditionally used, and helped us to see how this rhythm plays its way throughout the various formalisms. Prior to taking that class, I had spent many years “playing” on my own with tensors. I found the going to be tremendously difficult but was able, over time, to back out some physical and geometrical considerations that helped to make the subject a little more transparent. Today, it is sometimes hard not to think in terms of tensors and their associated concepts. This article, prompted and greatly enhanced by Marlos Jacob, whom I’ve met only by e-mail, is an attempt to record those early notions concerning tensors. It is intended to serve as a bridge from the point where most undergraduate students “leave off” in their studies of mathematics to the place where most texts on tensor analysis begin. A basic knowledge of vectors, matrices, and physics is assumed. A semi-intuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and higher vector products. The reader must be prepared to do some mathematics and to think. For those students who wish to go beyond this humble start, I can only recommend my professor’s wisdom: find the rhythm in the mathematics and you will fare pretty well.  :tongue2: Thanks alot, this is a great resource :)
Tensors are alot clearer to me now.

Thanks! now I dont feel so tense about tensors.

tiny-tim
Homework Helper
Welcome to PF!

Hi mdique! Welcome to PF! Thanks! now I dont feel so tense about tensors.

oh, are you that guy who felt vexed about vectors? Thats so funny, I never heard the "vexed by vectors" before. :rofl: nearly choked on my lunch.

Great find! I have been searching for a resource like this for the past 3 weeks! Thank you!

wow, great introduction to continuum mechanics, thanks for the post!

Cheers for the future of science & engineering

I wrote a short introduction to tensors with a more abstract approach (not the usual approach of mathematicians either that complicates things and forgets the means to build complex formulas with the tensors they define) : I define an abstract and general concept of "tensorial expression" (without specifying a syntax, that can be done with indices or diagrams as well) that can involve any number of independent vector spaces (instead of just one space and its dual) and how its value is defined in those spaces without any choice of coordinate system (but the method to manage coordinate systems is also mentioned).
Now it is only the basic definitions but I made them short and clear, and someday I of course intend to continue with more developments and uses of this formalism for geometry and physics...

It's amazing, I read an entire book on tensors this Christmas, but I never got this insight! Kudos Hi guys,

I am coming from the computer science background and I did not find any of the tutorials you mentioned useful. First of all, why should I read hundred of pages about linear algebra to understand the concept of tensors? Second, even after reading all these pages about basis transformations, dot product, cross product, .... you do not have any idea about what dyadic are and for what exactly I need a tensor. In the end they give as example a second order tensor which is a matrix. Then why do I need a tensor if I can use matrices? Nobody explains to you why exactly did people introduce the tensors.
I think if somebody could illustrate you the equations of the elasticity tensor (which is a tensor of order 4) and then introduce tensors to illustrate you how simple it would be to express the material law in terms of tensors, then maybe you can get the idea. But until now, I heard of people giving you only the second order tensors example (which are simply matrices). The same for dyadics, everybody just introduce dyadics as a product or as a vector pair. But what the hell are they? why a pair, is there an example? geometric interpretation???

Hi and welcome to the forum!

I think that it depends on the subject you are studying and the math background of the students. The stress tensor is usually introduced in the 1st-2nd bachelor year, often just a semester or two after you learn matrix algebra. This means that, for engineering purposes, it is usually a good idea to build on existing knowledge instead of continuously introducing new concepts (my students are already struggling with matrices as it is).

In hindsight, I also think that it would have indeed been better to have been taught tensors in a more qualitative manner, instead of a manner focused on calculations, but I am 1 out of 150 people in the class. This means that most people would simply have had a harder time with the concept. We should keep in mind that in engineering we learn tensors as means to an end, not as general mathematical education.

Of course, when I took a course in grids for my master's degree, we tackled Riemannian geometry but even there it wasn't too deeply. On the other hand, a friend in applied mathematics had a whole semester of tensor calculus, so again, it depends on the focus of the class Yes, it is quite cumbersome and I think that either there is no good introduction to it or I am missing some nice books. But after weeks of reading of different tutorials, I am still stucked in the definition. Still no idea about what dyads exactly are and what they represent, still no examples in relativity or elasticity with tensors of order three or four, still no idea why tensors where introduced...

Let me give this a crack:

There are two kinds of tensors. There are tensors that represent linear operators on vectors, and they give you the ability to talk about rotations, dilations, reflections, and so on. Examples of these are rotation matrices, metric tensors, moment of inertia tensors, and so on.

The other kind of tensor is a generalization of vectors to higher-dimensional objects: to represent planes, volumes, and above. These are usually where dyads are introduced. When two vectors are multiplied into a dyad, there are two parts that result: a symmetric part, representing a scalar, and an antisymmetric part, representing a plane. By judiciously choosing whether to keep or reject symmetric or antisymmetric parts, we can build up planes, volumes, and so on--whatever is appropriate for the theory in question.

A full-on dyadic tensor, then, is just the sum of a scalar and a plane. It may sound strange, but it's no more unusual than saying a complex number is the sum of a real number and an imaginary number.

I do think that in general tensors get very divorced from their geometric interpretation and that this is a serious source of confusion. When are you going to need tensors? When the objects you deal with are no longer vectors but something of higher dimensionality iike planes. I'm not familiar with this elasticity tensor--I think I could divine its geometric interpretation by looking at how it's defined, perhaps--but let me give another example of a big tensor that requires understanding: the Riemann tensor.

The Riemann tensor is rank 4, and the interpretation is as follows: for $R_{\alpha \beta \gamma \delta}$, you're looking at a linear operator on the plane spanned by $e_\gamma, e_\delta$ and extracting the component that lies in the plane of $e_\alpha, e_\beta$. What does this operator do? It measures extra terms introduced by the covariant derivatives in the $\gamma \delta$ plane, and we look at various combinations of $\alpha \beta$ to see what those components are in the six planes that span 3+1D spacetime.

Tensors give us the ability to talk about objects that aren't vectors but can be built from vectors--like planes--and to talk about operators on vectors and those higher dimensional objects.

It's funny, I actually had a similar discussion with a colleague yesterday. Don't forget that tensors make use of the idea of an invariant quantity. This is actually what differentiates a matrix from a 2nd order tensor; that the latter describes a quantity that is coordinate invariant. The fact that we can define this, and more importantly test it, was a great leap forward in many aspects of science, including relativity (although Einstein used tensors after deriving special relativity). From my viewpoint, for common engineering applications, tensors are what made computational mechanics possible. When you know that the maximum stress you calculated in a complex geometry is coordinate invariant, tensors are a great language to communicate the message. Of course they are invaluable in disciplines such as differential topology, but this is a little outside the scope of this thread.

Thank you for the nice discussion about dyads. Could you go into more details, maybe with an example in relativity? Riemann is somehow strange to me..
In continuum mechanics they usually say that a stress tensor for example can be introduced as sigma*dyad_1(dyad_product)dyad_2, where sigma is the stress (a scalar) and dyad_1 and dyad_2 the Force vector and Area vector. So basically what they are saying is that to describe the stress you need two vectors: the applied forse and the area where the force is applied.