Tensor derivative (continuum mechanics)

The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.
The directional derivative provides a systematic way of finding these derivatives.
Derivatives with respect to vectors and second-order tensors.
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.
Derivatives of scalar valued functions of vectors.
Let "f"(v) be a real valued function of the vector v. Then the derivative of "f"(v) with respect to v (or at v) is the vector defined through its dot product with any vector u being
Derivatives of vector valued functions of vectors.
Let f(v) be a vector valued function of the vector v. Then the derivative of f(v) with respect to v (or at v) is the second order tensor defined through its dot product with any vector u being
Derivatives of scalar valued functions of second-order tensors.
Let formula_11 be a real valued function of the second order tensor formula_12. Then the derivative of formula_11 with respect to formula_12 (or at formula_12) in the direction formula_16 is the second order tensor defined as
Derivatives of tensor valued functions of second-order tensors.
Let formula_22 be a second order tensor valued function of the second order tensor formula_12. Then the derivative of formula_22 with respect to formula_12 (or at formula_12) in the direction formula_16 is the fourth order tensor defined as
Gradient of a tensor field.
The gradient, formula_35, of a tensor field formula_36 in the direction of an arbitrary constant vector c is defined as:
The gradient of a tensor field of order "n" is a tensor field of order "n"+1.
Cartesian coordinates.
If formula_38 are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by (formula_39), then the gradient of the tensor field formula_16 is given by
Since the basis vectors do not vary in a Cartesian coordinate system we have the following relations for the gradients of a scalar field formula_42, a vector field v, and a second-order tensor field formula_12.
Curvilinear coordinates.
If formula_45 are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (formula_46), then the gradient of the tensor field formula_16 is given by (see for a proof.)
From this definition we have the following relations for the gradients of a scalar field formula_42, a vector field v, and a second-order tensor field formula_12.
where the Christoffel symbol formula_52 is defined using
Cylindrical polar coordinates.
In cylindrical coordinates, the gradient is given by
Divergence of a tensor field.
The divergence of a tensor field formula_36 is defined using the recursive relation
where c is an arbitrary constant vector and v is a vector field. If formula_16 is a tensor field of order "n" > 1 then the divergence of the field is a tensor of order "n"− 1.
Cartesian coordinates.
In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field formula_12.
where tensor index notation for partial derivatives is used in the rightmost expressions. Note that
For a symmetric second-order tensor, the divergence is also often written as
The above expression is sometimes used as the definition of 
formula_62 in Cartesian component form (often also written as 
formula_63). Note that such a definition is not consistent with the rest of this article (see the section on curvilinear co-ordinates). 
The difference stems from whether the differentiation is performed with respect to the rows or columns of formula_12, and is conventional. This is demonstrated by an example. In a Cartesian coordinate system the second order tensor (matrix) formula_65 is the gradient of a vector function formula_66.
The last equation is equivalent to the alternative definition / interpretation
Curvilinear coordinates.
In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field formula_12 are
More generally, 
Cylindrical polar coordinates.
In cylindrical polar coordinates
Curl of a tensor field.
The curl of an order-"n" > 1 tensor field formula_36 is also defined using the recursive relation 
where c is an arbitrary constant vector and v is a vector field.
Curl of a first-order tensor (vector) field.
Consider a vector field v and an arbitrary constant vector c. In index notation, the cross product is given by
where formula_76 is the permutation symbol, otherwise known as the Levi-Civita symbol. Then,
Therefore,
Curl of a second-order tensor field.
For a second-order tensor formula_12 
Hence, using the definition of the curl of a first-order tensor field, 
Therefore, we have
Identities involving the curl of a tensor field.
The most commonly used identity involving the curl of a tensor field, formula_16, is
This identity holds for tensor fields of all orders. For the important case of a second-order tensor, formula_12, this identity implies that
Derivative of the determinant of a second-order tensor.
The derivative of the determinant of a second order tensor formula_87 is given by
In an orthonormal basis, the components of formula_87 can be written as a matrix A. In that case, the right hand side corresponds the cofactors of the matrix.
Derivatives of the invariants of a second-order tensor.
The principal invariants of a second order tensor are
The derivatives of these three invariants with respect to formula_87 are
Derivative of the second-order identity tensor.
Let formula_93 be the second order identity tensor. Then the derivative of this tensor with respect to a second order tensor formula_87 is given by
This is because formula_93 is independent of formula_87.
Derivative of a second-order tensor with respect to itself.
Let formula_87 be a second order tensor. Then
Therefore,
Here formula_101 is the fourth order identity tensor. In index 
notation with respect to an orthonormal basis
This result implies that
where
Therefore, if the tensor formula_87 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 second-order tensor.
Let formula_87 and formula_16 be two second order tensors, then 
In index notation with respect to an orthonormal basis
We also have
In index notation 
If the tensor formula_87 is symmetric then
Integration by parts.
Another important operation related to tensor derivatives in continuum mechanics is integration by parts. The formula for integration by parts can be written as
where formula_117 and formula_118 are differentiable tensor fields of arbitrary order, formula_119 is the unit outward normal to the domain over which the tensor fields are defined, formula_120 represents a generalized tensor product operator, and formula_121 is a generalized gradient operator. When formula_117 is equal to the identity tensor, we get the divergence theorem
We can express the formula for integration by parts in Cartesian index notation as
For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both formula_117 and formula_118 are second order tensors, we have
In index notation,