# 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.[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

${\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =Df(\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~f(\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}}$

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.

Properties:

(1) If ${\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )+f_{2}(\mathbf {v} )}$ then ${\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}+{\frac {\partial f_{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} }$

(2) If ${\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )~f_{2}(\mathbf {v} )}$ then ${\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)~f_{2}(\mathbf {v} )+f_{1}(\mathbf {v} )~\left({\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}$

(3) If ${\displaystyle f(\mathbf {v} )=f_{1}(f_{2}(\mathbf {v} ))}$ then ${\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial f_{1}}{\partial f_{2}}}~{\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} }$

### 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

${\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =D\mathbf {f} (\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~\mathbf {f} (\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}}$

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 ${\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )+\mathbf {f} _{2}(\mathbf {v} )}$ then ${\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}+{\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} }$
(2) If ${\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )\times \mathbf {f} _{2}(\mathbf {v} )}$ then ${\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)\times \mathbf {f} _{2}(\mathbf {v} )+\mathbf {f} _{1}(\mathbf {v} )\times \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}$
(3) If ${\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {f} _{2}(\mathbf {v} ))}$ then ${\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {f} _{2}}}\cdot \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}$

### Derivatives of scalar valued functions of second-order tensors

Let ${\displaystyle f({\boldsymbol {S}})}$ be a real valued function of the second order tensor ${\displaystyle {\boldsymbol {S}}}$. Then the derivative of ${\displaystyle f({\boldsymbol {S}})}$ with respect to ${\displaystyle {\boldsymbol {S}}}$ (or at ${\displaystyle {\boldsymbol {S}}}$) in the direction ${\displaystyle {\boldsymbol {T}}}$ is the second order tensor defined as

${\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=Df({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~f({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}}$

for all second order tensors ${\displaystyle {\boldsymbol {T}}}$.

Properties:
(1) If ${\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})+f_{2}({\boldsymbol {S}})}$ then ${\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}}$
(2) If ${\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})~f_{2}({\boldsymbol {S}})}$ then ${\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)~f_{2}({\boldsymbol {S}})+f_{1}({\boldsymbol {S}})~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}$
(3) If ${\displaystyle f({\boldsymbol {S}})=f_{1}(f_{2}({\boldsymbol {S}}))}$ then ${\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial f_{2}}}~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}$

### Derivatives of tensor valued functions of second-order tensors

Let ${\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}$ be a second order tensor valued function of the second order tensor ${\displaystyle {\boldsymbol {S}}}$. Then the derivative of ${\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}$ with respect to ${\displaystyle {\boldsymbol {S}}}$ (or at ${\displaystyle {\boldsymbol {S}}}$) in the direction ${\displaystyle {\boldsymbol {T}}}$ is the fourth order tensor defined as

${\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=D{\boldsymbol {F}}({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~{\boldsymbol {F}}({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}}$

for all second order tensors ${\displaystyle {\boldsymbol {T}}}$.

Properties:
(1) If ${\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})+{\boldsymbol {F}}_{2}({\boldsymbol {S}})}$ then ${\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}}$
(2) If ${\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})}$ then ${\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})+{\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot \left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}$
(3) If ${\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))}$ then ${\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}$
(4) If ${\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))}$ then ${\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}$

## Gradient of a tensor field

The gradient, ${\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}}$, of a tensor field ${\displaystyle {\boldsymbol {T}}(\mathbf {x} )}$ in the direction of an arbitrary constant vector c is defined as:

${\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}\cdot \mathbf {c} =\lim _{\alpha \rightarrow 0}\quad {\cfrac {d}{d\alpha }}~{\boldsymbol {T}}(\mathbf {x} +\alpha \mathbf {c} )}$

The gradient of a tensor field of order n is a tensor field of order n+1.

### Cartesian coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

If ${\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}$ are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by (${\displaystyle x_{1},x_{2},x_{3}}$), then the gradient of the tensor field ${\displaystyle {\boldsymbol {T}}}$ is given by

${\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}={\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}\otimes \mathbf {e} _{i}}$

Since the basis vectors do not vary in a Cartesian coordinate system we have the following relations for the gradients of a scalar field ${\displaystyle \phi }$, a vector field v, and a second-order tensor field ${\displaystyle {\boldsymbol {S}}}$.

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial x_{i}}}~\mathbf {e} _{i}=\phi _{,i}~\mathbf {e} _{i}\\{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial (v_{j}\mathbf {e} _{j})}{\partial x_{i}}}\otimes \mathbf {e} _{i}={\cfrac {\partial v_{j}}{\partial x_{i}}}~\mathbf {e} _{j}\otimes \mathbf {e} _{i}=v_{j,i}~\mathbf {e} _{j}\otimes \mathbf {e} _{i}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\cfrac {\partial (S_{jk}\mathbf {e} _{j}\otimes \mathbf {e} _{k})}{\partial x_{i}}}\otimes \mathbf {e} _{i}={\cfrac {\partial S_{jk}}{\partial x_{i}}}~\mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{i}=S_{jk,i}~\mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{i}\end{aligned}}}

### Curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

If ${\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}}$ are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (${\displaystyle \xi ^{1},\xi ^{2},\xi ^{3}}$), then the gradient of the tensor field ${\displaystyle {\boldsymbol {T}}}$ is given by (see [3] for a proof.)

${\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}={\cfrac {\partial {\boldsymbol {T}}}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}}$

From this definition we have the following relations for the gradients of a scalar field ${\displaystyle \phi }$, a vector field v, and a second-order tensor field ${\displaystyle {\boldsymbol {S}}}$.

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial \xi ^{i}}}~\mathbf {g} ^{i}\\{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial (v^{j}\mathbf {g} _{j})}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\cfrac {\partial v^{j}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{j}\right)~\mathbf {g} _{j}\otimes \mathbf {g} ^{i}=\left({\cfrac {\partial v_{j}}{\partial \xi ^{i}}}-v_{k}~\Gamma _{ij}^{k}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{i}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\cfrac {\partial (S_{jk}~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k})}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\cfrac {\partial S_{jk}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ij}^{l}-S_{jl}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k}\otimes \mathbf {g} ^{i}\end{aligned}}}

where the Christoffel symbol ${\displaystyle \Gamma _{ij}^{k}}$ is defined using

${\displaystyle \Gamma _{ij}^{k}~\mathbf {g} _{k}={\cfrac {\partial \mathbf {g} _{i}}{\partial \xi ^{j}}}\quad \implies \quad \Gamma _{ij}^{k}={\cfrac {\partial \mathbf {g} _{i}}{\partial \xi ^{j}}}\cdot \mathbf {g} ^{k}=-\mathbf {g} _{i}\cdot {\cfrac {\partial \mathbf {g} ^{k}}{\partial \xi ^{j}}}}$

#### Cylindrical polar coordinates

In cylindrical coordinates, the gradient is given by

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial r}}~\mathbf {e} _{r}+{\cfrac {1}{r}}~{\cfrac {\partial \phi }{\partial \theta }}~\mathbf {e} _{\theta }+{\cfrac {\partial \phi }{\partial z}}~\mathbf {e} _{z}\\{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{r}}{\partial \theta }}-v_{\theta }\right)~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{r}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\cfrac {\partial v_{\theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\&\quad +{\cfrac {\partial v_{\theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\cfrac {\partial v_{z}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}{\cfrac {\partial v_{z}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{z}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rr}}{\partial \theta }}-(S_{\theta r}+S_{r\theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rr}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\\&\quad +{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{r\theta }}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rz}}{\partial \theta }}-S_{\theta z}\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\\&\quad +{\frac {\partial S_{rz}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {\partial S_{\theta r}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta r}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\&\quad +{\frac {\partial S_{\theta \theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta \theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {\partial S_{\theta z}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}\\&\quad +{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta z}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\&\quad +{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {\partial S_{z\theta }}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{z\theta }}{\partial \theta }}+S_{zr}\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\&\quad +{\frac {\partial S_{zz}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}~{\frac {\partial S_{zz}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}}

## Divergence of a tensor field

The divergence of a tensor field ${\displaystyle {\boldsymbol {T}}(\mathbf {x} )}$ is defined using the recursive relation

${\displaystyle ({\boldsymbol {\nabla }}\cdot {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot (\mathbf {c} \cdot {\boldsymbol {T}}^{T})~;\qquad {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\text{tr}}({\boldsymbol {\nabla }}\mathbf {v} )}$

where c is an arbitrary constant vector and v is a vector field. If ${\displaystyle {\boldsymbol {T}}}$ is a tensor field of order n > 1 then the divergence of the field is a tensor of order n−1.

### Cartesian coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field ${\displaystyle {\boldsymbol {S}}}$.

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\cfrac {\partial v_{i}}{\partial x_{i}}}=v_{i,i}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\cfrac {\partial S_{ki}}{\partial x_{k}}}~\mathbf {e} _{i}=S_{ki,k}~\mathbf {e} _{i}\end{aligned}}}

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:

${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}\neq \operatorname {div} {\boldsymbol {S}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}^{\mathrm {T} }.}$

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]

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\cfrac {\partial S_{ki}}{\partial x_{i}}}~\mathbf {e} _{k}=S_{ki,i}~\mathbf {e} _{k}\end{aligned}}}

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

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \left({\boldsymbol {\nabla }}\mathbf {v} \right)={\boldsymbol {\nabla }}\cdot \left(v_{i,j}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{i,ji}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}=\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)_{,j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)\end{aligned}}}
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \left[\left({\boldsymbol {\nabla }}\mathbf {v} \right)^{\mathrm {T} }\right]={\boldsymbol {\nabla }}\cdot \left(v_{j,i}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{j,ii}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}v_{j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}\mathbf {v} \end{aligned}}}

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

{\displaystyle {\begin{aligned}\left({\boldsymbol {\nabla }}\cdot \right)_{alt}\left({\boldsymbol {\nabla }}\mathbf {v} \right)=\left({\boldsymbol {\nabla }}\cdot \right)_{alt}\left(v_{i,j}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{i,jj}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\cdot \mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}v_{i}~\mathbf {e} _{i}={\boldsymbol {\nabla }}^{2}\mathbf {v} \end{aligned}}}

### Curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field ${\displaystyle {\boldsymbol {S}}}$ are

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &=\left({\cfrac {\partial v^{i}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{i}\right)\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left({\cfrac {\partial S_{ik}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ii}^{l}-S_{il}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{k}\end{aligned}}}

#### Cylindrical polar coordinates

{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)+{\cfrac {\partial v_{z}}{\partial z}}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{z}\\&+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{z}\\&+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\end{aligned}}}

## Curl of a tensor field

The curl of an order-n > 1 tensor field ${\displaystyle {\boldsymbol {T}}(\mathbf {x} )}$ is also defined using the recursive relation

${\displaystyle ({\boldsymbol {\nabla }}\times {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {T}})~;\qquad ({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )}$

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

${\displaystyle \mathbf {v} \times \mathbf {c} =\varepsilon _{ijk}~v_{j}~c_{k}~\mathbf {e} _{i}}$

where ${\displaystyle \varepsilon _{ijk}}$ is the permutation symbol, otherwise known as the Levi-Civita symbol. Then,

${\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )=\varepsilon _{ijk}~v_{j,i}~c_{k}=(\varepsilon _{ijk}~v_{j,i}~\mathbf {e} _{k})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c} }$

Therefore,

${\displaystyle {\boldsymbol {\nabla }}\times \mathbf {v} =\varepsilon _{ijk}~v_{j,i}~\mathbf {e} _{k}}$

### Curl of a second-order tensor field

For a second-order tensor ${\displaystyle {\boldsymbol {S}}}$

${\displaystyle \mathbf {c} \cdot {\boldsymbol {S}}=c_{m}~S_{mj}~\mathbf {e} _{j}}$

Hence, using the definition of the curl of a first-order tensor field,

${\displaystyle {\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {S}})=\varepsilon _{ijk}~c_{m}~S_{mj,i}~\mathbf {e} _{k}=(\varepsilon _{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times {\boldsymbol {S}})\cdot \mathbf {c} }$

Therefore, we have

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {S}}=\varepsilon _{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m}}$

### Identities involving the curl of a tensor field

The most commonly used identity involving the curl of a tensor field, ${\displaystyle {\boldsymbol {T}}}$, is

${\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}{\boldsymbol {T}})={\boldsymbol {0}}}$

This identity holds for tensor fields of all orders. For the important case of a second-order tensor, ${\displaystyle {\boldsymbol {S}}}$, this identity implies that

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {S}}={\boldsymbol {0}}\quad \implies \quad S_{mi,j}-S_{mj,i}=0}$

## Derivative of the determinant of a second-order tensor

The derivative of the determinant of a second order tensor ${\displaystyle {\boldsymbol {A}}}$ is given by

${\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det({\boldsymbol {A}})=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}~.}$

In an orthonormal basis, the components of ${\displaystyle {\boldsymbol {A}}}$ 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

{\displaystyle {\begin{aligned}I_{1}({\boldsymbol {A}})&={\text{tr}}{\boldsymbol {A}}\\I_{2}({\boldsymbol {A}})&={\frac {1}{2}}\left[({\text{tr}}{\boldsymbol {A}})^{2}-{\text{tr}}{{\boldsymbol {A}}^{2}}\right]\\I_{3}({\boldsymbol {A}})&=\det({\boldsymbol {A}})\end{aligned}}}

The derivatives of these three invariants with respect to ${\displaystyle {\boldsymbol {A}}}$ are

{\displaystyle {\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T}\\{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T}~(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T})=({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}})^{T}\end{aligned}}}

## Derivative of the second-order identity tensor

Let ${\displaystyle {\boldsymbol {\mathit {1}}}}$ be the second order identity tensor. Then the derivative of this tensor with respect to a second order tensor ${\displaystyle {\boldsymbol {A}}}$ is given by

${\displaystyle {\frac {\partial {\boldsymbol {\mathit {1}}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathsf {0}}}:{\boldsymbol {T}}={\boldsymbol {\mathit {0}}}}$

This is because ${\displaystyle {\boldsymbol {\mathit {1}}}}$ is independent of ${\displaystyle {\boldsymbol {A}}}$.

## Derivative of a second-order tensor with respect to itself

Let ${\displaystyle {\boldsymbol {A}}}$ be a second order tensor. Then

${\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\left[{\frac {\partial }{\partial \alpha }}({\boldsymbol {A}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}={\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}}}$

Therefore,

${\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}={\boldsymbol {\mathsf {I}}}}$

Here ${\displaystyle {\boldsymbol {\mathsf {I}}}}$ is the fourth order identity tensor. In index notation with respect to an orthonormal basis

${\displaystyle {\boldsymbol {\mathsf {I}}}=\delta _{ik}~\delta _{jl}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}$

This result implies that

${\displaystyle {\frac {\partial {\boldsymbol {A}}^{T}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}^{T}:{\boldsymbol {T}}={\boldsymbol {T}}^{T}}$

where

${\displaystyle {\boldsymbol {\mathsf {I}}}^{T}=\delta _{jk}~\delta _{il}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}$

Therefore, if the tensor ${\displaystyle {\boldsymbol {A}}}$ is symmetric, then the derivative is also symmetric and we get

${\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}={\boldsymbol {\mathsf {I}}}^{(s)}={\frac {1}{2}}~({\boldsymbol {\mathsf {I}}}+{\boldsymbol {\mathsf {I}}}^{T})}$

where the symmetric fourth order identity tensor is

${\displaystyle {\boldsymbol {\mathsf {I}}}^{(s)}={\frac {1}{2}}~(\delta _{ik}~\delta _{jl}+\delta _{il}~\delta _{jk})~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}$

## Derivative of the inverse of a second-order tensor

Let ${\displaystyle {\boldsymbol {A}}}$ and ${\displaystyle {\boldsymbol {T}}}$ be two second order tensors, then

${\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-1}\right):{\boldsymbol {T}}=-{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\cdot {\boldsymbol {A}}^{-1}}$

In index notation with respect to an orthonormal basis

${\displaystyle {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}~T_{kl}=-A_{ik}^{-1}~T_{kl}~A_{lj}^{-1}\implies {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}=-A_{ik}^{-1}~A_{lj}^{-1}}$

We also have

${\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-T}\right):{\boldsymbol {T}}=-{\boldsymbol {A}}^{-T}\cdot {\boldsymbol {T}}^{T}\cdot {\boldsymbol {A}}^{-T}}$

In index notation

${\displaystyle {\frac {\partial A_{ji}^{-1}}{\partial A_{kl}}}~T_{kl}=-A_{jk}^{-1}~T_{lk}~A_{li}^{-1}\implies {\frac {\partial A_{ji}^{-1}}{\partial A_{kl}}}=-A_{li}^{-1}~A_{jk}^{-1}}$

If the tensor ${\displaystyle {\boldsymbol {A}}}$ is symmetric then

${\displaystyle {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}=-{\cfrac {1}{2}}\left(A_{ik}^{-1}~A_{jl}^{-1}+A_{il}^{-1}~A_{jk}^{-1}\right)}$

## Integration by parts

Domain ${\displaystyle \Omega }$, its boundary ${\displaystyle \Gamma }$ and the outward unit normal ${\displaystyle \mathbf {n} }$

Another important operation related to tensor derivatives in continuum mechanics is integration by parts. The formula for integration by parts can be written as

${\displaystyle \int _{\Omega }{\boldsymbol {F}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {G}}\,{\rm {d}}\Omega =\int _{\Gamma }\mathbf {n} \otimes ({\boldsymbol {F}}\otimes {\boldsymbol {G}})\,{\rm {d}}\Gamma -\int _{\Omega }{\boldsymbol {G}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {F}}\,{\rm {d}}\Omega }$

where ${\displaystyle {\boldsymbol {F}}}$ and ${\displaystyle {\boldsymbol {G}}}$ are differentiable tensor fields of arbitrary order, ${\displaystyle \mathbf {n} }$ is the unit outward normal to the domain over which the tensor fields are defined, ${\displaystyle \otimes }$ represents a generalized tensor product operator, and ${\displaystyle {\boldsymbol {\nabla }}}$ is a generalized gradient operator. When ${\displaystyle {\boldsymbol {F}}}$ is equal to the identity tensor, we get the divergence theorem

${\displaystyle \int _{\Omega }{\boldsymbol {\nabla }}{\boldsymbol {G}}\,{\rm {d}}\Omega =\int _{\Gamma }\mathbf {n} \otimes {\boldsymbol {G}}\,{\rm {d}}\Gamma \,.}$

We can express the formula for integration by parts in Cartesian index notation as

${\displaystyle \int _{\Omega }F_{ijk....}\,G_{lmn...,p}\,{\rm {d}}\Omega =\int _{\Gamma }n_{p}\,F_{ijk...}\,G_{lmn...}\,{\rm {d}}\Gamma -\int _{\Omega }G_{lmn...}\,F_{ijk...,p}\,{\rm {d}}\Omega \,.}$

For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both ${\displaystyle {\boldsymbol {F}}}$ and ${\displaystyle {\boldsymbol {G}}}$ are second order tensors, we have

${\displaystyle \int _{\Omega }{\boldsymbol {F}}\cdot ({\boldsymbol {\nabla }}\cdot {\boldsymbol {G}})\,{\rm {d}}\Omega =\int _{\Gamma }\mathbf {n} \cdot ({\boldsymbol {G}}\cdot {\boldsymbol {F}}^{T})\,{\rm {d}}\Gamma -\int _{\Omega }({\boldsymbol {\nabla }}{\boldsymbol {F}}):{\boldsymbol {G}}^{T}\,{\rm {d}}\Omega \,.}$

In index notation,

${\displaystyle \int _{\Omega }F_{ij}\,G_{pj,p}\,{\rm {d}}\Omega =\int _{\Gamma }n_{p}\,F_{ij}\,G_{pj}\,{\rm {d}}\Gamma -\int _{\Omega }G_{pj}\,F_{ij,p}\,{\rm {d}}\Omega \,.}$