1. Derivatives with respect to vectors and second-order tensors
    Derivatives of scalar valued functions of vectors  Derivatives of vector valued functions of vectors  Derivatives of scalar valued functions of second-order tensors  Derivatives of tensor valued functions of second-order tensors 

2. Gradient of a tensor field
     Cartesian coordinates    Curvilinear coordinates    Cylindrical polar coordinates  

3. Divergence of a tensor field
     Cartesian coordinates    Curvilinear coordinates    Cylindrical polar coordinates  

4. Curl of a tensor field
     Curl of a first-order tensor (vector) field    Curl of a second-order tensor field    Identities involving the curl of a tensor field  

5. Derivative of the determinant of a second-order tensor

6. Derivatives of the invariants of a second-order tensor

7. Derivative of the second-order identity tensor

8. Derivative of a second-order tensor with respect to itself

9. Derivative of the inverse of a second-order tensor

10. Integration by parts

11. See also

12. References

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.^[1]

The directional derivative provides a systematic way of finding these derivatives.^[2]

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

for all vectors u. The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction.

(1) If then

(2) If then

(3) If then

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

for all vectors u. The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u.

Properties:

(1) If then

(2) If then

(3) If then

Derivatives of scalar valued functions of second-order tensors

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

Derivatives of tensor valued functions of second-order tensors

Let be a second order 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

(4) If then

Gradient of a tensor field

The gradient, , of a tensor field 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 are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by (), then the gradient of the tensor field is given by

The vectors x and c can be written as and . Let y := x + αc. In that case the gradient 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 , a vector field v, and a second-order tensor field .

Curvilinear coordinates

If are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (), then the gradient of the tensor field is given by (see ^[3] for a proof.)

From this definition we have the following relations for the gradients of a scalar field , a vector field v, and a second-order tensor field .

where the Christoffel symbol 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 is defined using the recursive relation

where c is an arbitrary constant vector and v is a vector field. If 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 .

where tensor index notation for partial derivatives is used in the rightmost expressions. Note that last relation can be found in reference ^[4] under relation (1.14.13).

Note also that according to the same paper in the case of the second-order tensor field, we have:

Importantly, other written conventions for the divergence of a second-order tensor do exist. For example, in a Cartesian coordinate system the divergence of a second rank tensor can also be written as^[5]

The difference stems from whether the differentiation is performed with respect to the rows or columns of , and is conventional. This is demonstrated by an example. In a Cartesian coordinate system the second order tensor (matrix) is the gradient of a vector function .

The last equation is equivalent to the alternative definition / interpretation^[5]

Curvilinear coordinates

In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field are

Cylindrical polar coordinates

In cylindrical polar coordinates

Curl of a tensor field

The curl of an order-n > 1 tensor field 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 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

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, , is

This identity holds for tensor fields of all orders. For the important case of a second-order tensor, , this identity implies that

Derivative of the determinant of a second-order 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 A. In that case, the right hand side corresponds the cofactors of the matrix.

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 using (note the sign of λ)

Using this expansion we can write

Recall that the invariant is given by

Hence,

Invoking the arbitrariness of we then have

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 are

Derivative of the second-order identity tensor

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 .

Derivative of a second-order tensor with respect to itself

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 second-order 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

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 and are differentiable tensor fields of arbitrary order, is the unit outward normal to the domain over which the tensor fields are defined, represents a generalized tensor product operator, and is a generalized gradient operator. When 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 and are second order tensors, we have

In index notation,

See also

• Covariant derivative
• Ricci calculus

References

• Edward Aguilera
• Edward A. Kawānanakoa
• Edward A. Kelly
• Edward Akroyd
• Edward Akufo
• Edward Akufo-Addo
• Edward Alan John George
• Edward Albee
• Edward Albert
• Edward Albert Christian George Andrew Patrick David Windsor
• Edward Albert Feigenbaum
• Edward Albert, Jr.
• Edward Albert Sharpey-Schafer
• Edward Aldridge
• Edward Aldridge Jr.
• Edward Aldridge, Jr.
• Edward Aleksander Raczyński
• Edward A. LeLacheur Park
• Edward Alexander
• Edward Alexander Crowley
• Edward Alexander MacDowell
• Edward Alfred Alexander Baldwin, 4th Earl Baldwin of Bewdley
• Edward Allegheny Johnson
• Edward Allen
• Edward Alleyn