A sound understanding of tensors and tensor operation is essential if you want to read and understand modern papers on solid mechanics and finite element modeling of complex material behavior. This brief introduction gives you an overview of tensors and tensor notation. For more details you can read A Brief on Tensor Analysis by J. G. Simmonds, the appendix on vector and tensor notation from Dynamics of Polymeric Liquids  Volume 1 by R. B. Bird, R. C. Armstrong, and O. Hassager, and the monograph by R. M. Brannon. An introduction to tensors in continuum mechanics can be found in An Introduction to Continuum Mechanics by M. E. Gurtin. Most of the material in this page is based on these sources.
The following notation is usually used in the literature:
A force has a magnitude and a direction, can be added to another force, be multiplied by a scalar and so on. These properties make the force a vector.
Similarly, the displacement is a vector because it can be added to other displacements and satisfies the other properties of a vector.
However, a force cannot be added to a displacement to yield a physically meaningful quantity. So the physical spaces that these two quantities lie on must be different.
Recall that a constant force moving through a displacement does units of work. How do we compute this product when the spaces of and are different? If you try to compute the product on a graph, you will have to convert both quantities to a single basis and then compute the scalar product.
An alternative way of thinking about the operation is to think of as a linear operator that acts on to produce a scalar quantity (work). In the notation of sets we can write
A first order tensor is a linear operator that sends vectors to scalars.
Next, assume that the force acts at a point . The moment of the force about the origin is given by which is a vector. The vector product can be thought of as an linear operation too. In this case the effect of the operator is to convert a vector into another vector.
A second order tensor is a linear operator that sends vectors to vectors.
According to Simmonds, "the name tensor comes from elasticity theory where in a loaded elastic body the stress tensor acting on a unit vector normal to a plane through a point delivers the tension (i.e., the force per unit area) acting across the plane at that point."
Examples of second order tensors are the stress tensor, the deformation gradient tensor, the velocity gradient tensor, and so on.
Another type of tensor that we encounter frequently in mechanics is the fourth order tensor that takes strains to stresses. In elasticity, this is the stiffness tensor.
A fourth order tensor is a linear operator that sends second order tensors to second order tensors.
A secondorder tensor is a linear transformation from a vector space to . Thus, we can write
More often, we use the following notation:
I have used the "dot" notation in this handout. None of the above notations is obviously superior to the others and each is used widely.
Let and be two tensors. Then the sum is another tensor defined by
Let be a tensor and let be a scalar. Then the product is a tensor defined by
The zero tensor is the tensor which maps every vector into the zero vector.
The identity tensor takes every vector into itself.
The identity tensor is also often written as .
Let and be two tensors. Then the product is the tensor that is defined by
In general .
The transpose of a tensor is the unique tensor defined by
The following identities follow from the above definition:
A tensor is symmetric if
A tensor is skew if
Every tensor can be expressed uniquely as the sum of a symmetric tensor (the symmetric part of ) and a skew tensor (the skew part of ).
The tensor (or dyadic) product (also written ) of two vectors and is a tensor that assigns to each vector the vector .
Notice that all the above operations on tensors are remarkably similar to matrix operations.
The spectral theorem for tensors is widely used in mechanics. We will start off by definining eigenvalues and eigenvectors.
Let be a second order tensor. Let λ be a scalar and be a vector such that
Then λ is called an eigenvalue of and is an eigenvector .
A second order tensor has three eigenvalues and three eigenvectors, since the space is threedimentional. Some of the eigenvalues might be repeated. The number of times an eigenvalue is repeated is called multiplicity.
In mechanics, many second order tensors are symmetric and positive definite. Note the following important properties of such tensors:
For more on eigenvalues and eigenvectors see Applied linear operators and spectral methods.
Spectral theoremLet be a symmetric secondorder tensor. Then
This relation is called the spectral decomposition of . 
Let be second order tensor with . Then
Let be a second order tensor. Then the determinant of can be expressed as
The quantities are called the principal invariants of . Expressions of the principal invariants are given below.
Principal invariants of 
Note that λ is an eigenvalue of if and only if
The resulting equations is called the characteristic equation and is usually written in expanded form as
The CayleyHamilton theorem is a very useful result in continuum mechanics. It states that
CayleyHamilton theorem If is a second order tensor then it satisfies its own characteristic equation 
All the equations so far have made no mention of the coordinate system. When we use vectors and tensor in computations we have to express them in some coordinate system (basis) and use the components of the object in that basis for our computations.
Commonly used bases are the Cartesian coordinate frame, the cylindrical coordinate frame, and the spherical coordinate frame.
A Cartesian coordinate frame consists of an orthonormal basis together with a point called the origin. Since these vectors are mutually perpendicular, we have the following relations:
To make the above relations more compact, we introduce the Kronecker delta function
Then, instead of the nine equations in (1) we can write (in index notation)
Recall that the vector can be written as
In index notation, equation (2) can be written as
This convention is called the Einstein summation convention. If indices are repeated, we understand that to mean that there is a sum over the indices.
We can write the Cartesian components of a vector in the basis as
Similarly, the components of of a tensor are defined by
Using the definition of the tensor product, we can also write
Using the summation convention,
In this case, the bases of the tensor are and the components are .
From the definition of the components of tensor , we can also see that (using the summation convention)
Similarly, the dyadic product can be expressed as
We can also write a tensor in matrix notation as
Note that the Kronecker delta represents the components of the identity tensor in a Cartesian basis. Therefore, we can write
The inner product of two tensors and is an operation that generates a scalar. We define (summation implied)
The inner product can also be expressed using the trace :
Proof using the definition of the trace below :
The trace of a tensor is the scalar given by
(** Needs a proper definition **)
The magnitude of a tensor is defined by
Another tensor operation that is often seen is the tensor product of a tensor with a vector. Let be a tensor and let be a vector. Then the tensor cross product gives a tensor defined by
The permutation symbol is defined as
Let , and be three second order tensors. Then
Proof:
It is easiest to show these relations by using index notation with respect to an orthonormal basis. Then we can write
Similarly,
Recall that the vector differential operator (with respect to a Cartesian basis) is defined as
In this section we summarize some operations of on vectors and tensors.
The dyadic product (or ) is called the gradient of the vector field . Therefore, the quantity is a tensor given by
In the alternative dyadic notation,
'Warning: Some authors define the ij component of as .
Let be a tensor field. Then the divergence of the tensor field is a vector given by
To fix the definition of divergence of a general tensor field (possibly of higher order than 2), we use the relation
where is an arbitrary constant vector.
The Laplacian of a vector field is given by
Some important identities involving tensors are:
The following integral theorems are useful in continuum mechanics and finite elements.
If Ω is a region in space enclosed by a surface and is a tensor field, then
where is the unit outward normal to the surface.
If is a surface bounded by a closed curve , then
where is a tensor field, is the unit normal vector to in the direction of a righthanded screw motion along , and is a unit tangential vector in the direction of integration along .
Let Ω be a closed moving region of space enclosed by a surface . Let the velocity of any surface element be . Then if is a tensor function of position and time,
where is the outward unit normal to the surface .
We often have to find the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors. The directional directive provides a systematic way of finding these derivatives.
The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.
Let be a real valued function of the vector . Then the derivative of with respect to (or at ) in the direction is the vector defined as
for all vectors .
Properties:
1) If then
2) If then
3) If then
Let be a vector valued function of the vector . Then the derivative of with respect to (or at ) in the direction is the second order tensor defined as
for all vectors .
Properties:
1) If then
2) If then
3) If then
Let be a real valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the second order tensor defined as
for all second order tensors .
Properties:
1) If then
2) If then
3) If then
Let be a second oder tensor valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the fourth order tensor defined as
for all second order tensors .
Properties:
1) If then
2) If then
3) If then
3) If then
Derivative of the determinant of a tensor The derivative of the determinant of a second order tensor is given by In an orthonormal basis the components of can be written as a matrix . In that case, the right hand side corresponds the cofactors of the matrix. 
Proof:
Let be a second order tensor and let . Then, from the definition of the derivative of a scalar valued function of a tensor, we have
Recall that we can expand the determinant of a tensor in the form of a characteristic equation in terms of the invariants I_{1},I_{2},I_{ 3} using (note the sign of λ)
Using this expansion we can write
Recall that the invariant I_{1} is given by
Hence,
Invoking the arbitrariness of we then have
Derivatives of the principal invariants of a tensor The principal invariants of a second order tensor are The derivatives of these three invariants with respect to are 
Proof:
From the derivative of the determinant we know that
For the derivatives of the other two invariants, let us go back to the characteristic equation
Using the same approach as for the determinant of a tensor, we can show that
Now the left hand side can be expanded as
Hence
or,
Expanding the right hand side and separating terms on the left hand side gives
or,
If we define I_{0}: = 1 and I_{4}: = 0, we can write the above as
Collecting terms containing various powers of λ, we get
Then, invoking the arbitrariness of λ, we have
This implies that
Let be the second order identity tensor. Then the derivative of this tensor with respect to a second order tensor is given by
This is because is independent of .
Let be a second order tensor. Then
Therefore,
Here is the fourth order identity tensor. In index notation with respect to an orthonormal basis
This result implies that
where
Therefore, if the tensor is symmetric, then the derivative is also symmetric and we get
where the symmetric fourth order identity tensor is
Derivative of the inverse of a tensor Let and be two second order tensors, then In index notation with respect to an orthonormal basis We also have In index notation If the tensor is symmetric then 
Proof:
Recall that
Since , we can write
Using the product rule for second order tensors
we get
or,
Therefore,
The boldface notation that I've used is called the Gibbs notation. The index notation that I have used is also called Cartesian tensor notation.
