Thederivatives ofscalars,vectors, and second-ordertensors with respect to second-order tensors are of considerable use incontinuum mechanics. These derivatives are used in the theories ofnonlinear elasticity andplasticity, particularly in the design ofalgorithms fornumerical simulations.^[1]

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

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.

Letf(v) be a real valued function of the vectorv. Then the derivative off(v) with respect tov (or atv) is thevector defined through itsdot product with any vectoru being

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

for all vectorsu. The above dot product yields a scalar, and ifu is aunit vector gives the directional derivative off atv, in theu direction.

Properties:

1. If${\displaystyle f(\mathbf{v})=f_1(\mathbf{v})+f_2(\mathbf{v})}$ then$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}=\left(\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }\mathbf{v}}+\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }\mathbf{v}}\right)\cdot \mathbf{u}}$$
2. If${\displaystyle f(\mathbf{v})=f_1(\mathbf{v})f_2(\mathbf{v})}$ then$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}=\left(\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}\right)f_2(\mathbf{v})+f_1(\mathbf{v})\left(\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}\right)}$$
3. If${\displaystyle f(\mathbf{v})=f_1(f_2(\mathbf{v}))}$ then$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}=\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{f}_2}\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}}$$

Letf(v) be a vector valued function of the vectorv. Then the derivative off(v) with respect tov (or atv) is the second order tensor defined through its dot product with any vectoru being

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

for all vectorsu. The above dot product yields a vector, and ifu is a unit vector gives the direction derivative off atv, in the directionalu.

Properties:

1. If${\displaystyle \mathbf{f}(\mathbf{v})={\mathbf{f}}_1(\mathbf{v})+{\mathbf{f}}_2(\mathbf{v})}$ then$${\displaystyle \frac{\mathrm{\partial }\mathbf{f}}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}=\left(\frac{\mathrm{\partial }{\mathbf{f}}_1}{\mathrm{\partial }\mathbf{v}}+\frac{\mathrm{\partial }{\mathbf{f}}_2}{\mathrm{\partial }\mathbf{v}}\right)\cdot \mathbf{u}}$$
2. If${\displaystyle \mathbf{f}(\mathbf{v})={\mathbf{f}}_1(\mathbf{v})\times {\mathbf{f}}_2(\mathbf{v})}$ then$${\displaystyle \frac{\mathrm{\partial }\mathbf{f}}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}=\left(\frac{\mathrm{\partial }{\mathbf{f}}_1}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}\right)\times {\mathbf{f}}_2(\mathbf{v})+{\mathbf{f}}_1(\mathbf{v})\times \left(\frac{\mathrm{\partial }{\mathbf{f}}_2}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}\right)}$$
3. If${\displaystyle \mathbf{f}(\mathbf{v})={\mathbf{f}}_1({\mathbf{f}}_2(\mathbf{v}))}$ then$${\displaystyle \frac{\mathrm{\partial }\mathbf{f}}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}=\frac{\mathrm{\partial }{\mathbf{f}}_1}{\mathrm{\partial }{\mathbf{f}}_2}\cdot \left(\frac{\mathrm{\partial }{\mathbf{f}}_2}{\mathrm{\partial }\mathbf{v}}\cdot \mathbf{u}\right)}$$

Let${\displaystyle f(\mathit{S})}$ be a real valued function of the second order tensor${\displaystyle \mathit{S}}$. Then the derivative of${\displaystyle f(\mathit{S})}$ with respect to${\displaystyle \mathit{S}}$ (or at${\displaystyle \mathit{S}}$) in the direction${\displaystyle \mathit{T}}$ is the second order tensor defined as$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathit{S}}:\mathit{T}=Df(\mathit{S})[\mathit{T}]={\left[\frac{d}{d\alpha }f(\mathit{S}+\alpha \mathit{T})\right]}_{\alpha =0}}$$for all second order tensors${\displaystyle \mathit{T}}$.

Properties:

1. If${\displaystyle f(\mathit{S})=f_1(\mathit{S})+f_2(\mathit{S})}$ then$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathit{S}}:\mathit{T}=\left(\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }\mathit{S}}+\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }\mathit{S}}\right):\mathit{T}}$$
2. If${\displaystyle f(\mathit{S})=f_1(\mathit{S})f_2(\mathit{S})}$ then$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathit{S}}:\mathit{T}=\left(\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }\mathit{S}}:\mathit{T}\right)f_2(\mathit{S})+f_1(\mathit{S})\left(\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }\mathit{S}}:\mathit{T}\right)}$$
3. If${\displaystyle f(\mathit{S})=f_1(f_2(\mathit{S}))}$ then$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathit{S}}:\mathit{T}=\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{f}_2}\left(\frac{\mathrm{\partial }{f}_2}{\mathrm{\partial }\mathit{S}}:\mathit{T}\right)}$$

Let${\displaystyle \mathit{F}(\mathit{S})}$ be a second order tensor valued function of the second order tensor${\displaystyle \mathit{S}}$. Then the derivative of${\displaystyle \mathit{F}(\mathit{S})}$ with respect to${\displaystyle \mathit{S}}$ (or at${\displaystyle \mathit{S}}$) in the direction${\displaystyle \mathit{T}}$ is the fourth order tensor defined as$${\displaystyle \frac{\mathrm{\partial }\mathit{F}}{\mathrm{\partial }\mathit{S}}:\mathit{T}=D\mathit{F}(\mathit{S})[\mathit{T}]={\left[\frac{d}{d\alpha }\mathit{F}(\mathit{S}+\alpha \mathit{T})\right]}_{\alpha =0}}$$for all second order tensors${\displaystyle \mathit{T}}$.

Properties:

1. If${\displaystyle \mathit{F}(\mathit{S})={\mathit{F}}_1(\mathit{S})+{\mathit{F}}_2(\mathit{S})}$ then$${\displaystyle \frac{\mathrm{\partial }\mathit{F}}{\mathrm{\partial }\mathit{S}}:\mathit{T}=\left(\frac{\mathrm{\partial }{\mathit{F}}_1}{\mathrm{\partial }\mathit{S}}+\frac{\mathrm{\partial }{\mathit{F}}_2}{\mathrm{\partial }\mathit{S}}\right):\mathit{T}}$$
2. If${\displaystyle \mathit{F}(\mathit{S})={\mathit{F}}_1(\mathit{S})\cdot {\mathit{F}}_2(\mathit{S})}$ then$${\displaystyle \frac{\mathrm{\partial }\mathit{F}}{\mathrm{\partial }\mathit{S}}:\mathit{T}=\left(\frac{\mathrm{\partial }{\mathit{F}}_1}{\mathrm{\partial }\mathit{S}}:\mathit{T}\right)\cdot {\mathit{F}}_2(\mathit{S})+{\mathit{F}}_1(\mathit{S})\cdot \left(\frac{\mathrm{\partial }{\mathit{F}}_2}{\mathrm{\partial }\mathit{S}}:\mathit{T}\right)}$$
3. If${\displaystyle \mathit{F}(\mathit{S})={\mathit{F}}_1({\mathit{F}}_2(\mathit{S}))}$ then$${\displaystyle \frac{\mathrm{\partial }\mathit{F}}{\mathrm{\partial }\mathit{S}}:\mathit{T}=\frac{\mathrm{\partial }{\mathit{F}}_1}{\mathrm{\partial }{\mathit{F}}_2}:\left(\frac{\mathrm{\partial }{\mathit{F}}_2}{\mathrm{\partial }\mathit{S}}:\mathit{T}\right)}$$
4. If${\displaystyle f(\mathit{S})=f_1({\mathit{F}}_2(\mathit{S}))}$ then$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathit{S}}:\mathit{T}=\frac{\mathrm{\partial }{f}_1}{\mathrm{\partial }{\mathit{F}}_2}:\left(\frac{\mathrm{\partial }{\mathit{F}}_2}{\mathrm{\partial }\mathit{S}}:\mathit{T}\right)}$$

Thegradient,${\displaystyle \mathbf{\nabla }\mathit{T}}$, of a tensor field${\displaystyle \mathit{T}(\mathbf{x})}$ in the direction of an arbitrary constant vectorc is defined as:$${\displaystyle \mathbf{\nabla }\mathit{T}\cdot \mathbf{c}=\underset{\alpha \to 0}{lim}\frac{d}{d\alpha }\mathit{T}(\mathbf{x}+\alpha \mathbf{c})}$$The gradient of atensor field of ordern is a tensor field of ordern+1.

If${\displaystyle {\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3}$ are the basis vectors in aCartesian coordinate system, with coordinates of points denoted by (${\displaystyle x_1,x_2,x_3}$), then the gradient of the tensor field${\displaystyle \mathit{T}}$ is given by$${\displaystyle \mathbf{\nabla }\mathit{T}=\frac{\mathrm{\partial }\mathit{T}}{\mathrm{\partial }{x}_i}\otimes {\mathbf{e}}_i}$$

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

If${\displaystyle {\mathbf{g}}^1,{\mathbf{g}}^2,{\mathbf{g}}^3}$ are thecontravariantbasis vectors in acurvilinear coordinate system, with coordinates of points denoted by (${\displaystyle {\xi }^1,{\xi }^2,{\xi }^3}$), then the gradient of the tensor field${\displaystyle \mathit{T}}$ is given by^[3]$${\displaystyle \mathbf{\nabla }\mathit{T}=\frac{\mathrm{\partial }\mathit{T}}{\mathrm{\partial }{\xi }^i}\otimes {\mathbf{g}}^i}$$

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

where theChristoffel symbol${\displaystyle {\mathrm{\Gamma }}_{ij}^k}$ is defined using$${\displaystyle {\mathrm{\Gamma }}_{ij}^k{\mathbf{g}}_k=\frac{\mathrm{\partial }{\mathbf{g}}_i}{\mathrm{\partial }{\xi }^j}⟹{\mathrm{\Gamma }}_{ij}^k=\frac{\mathrm{\partial }{\mathbf{g}}_i}{\mathrm{\partial }{\xi }^j}\cdot {\mathbf{g}}^k=-{\mathbf{g}}_i\cdot \frac{\mathrm{\partial }{\mathbf{g}}^k}{\mathrm{\partial }{\xi }^j}}$$

Incylindrical coordinates, the gradient is given by

Thedivergence of a tensor field${\displaystyle \mathit{T}(\mathbf{x})}$ is defined using the recursive relation$${\displaystyle (\mathbf{\nabla }\cdot \mathit{T})\cdot \mathbf{c}=\mathbf{\nabla }\cdot \left(\mathbf{c}\cdot {\mathit{T}}^{\mathsf{\text{T}}}\right);\mathbf{\nabla }\cdot \mathbf{v}=\text{tr}(\mathbf{\nabla }\mathbf{v})}$$

wherec is an arbitrary constant vector andv is a vector field. If${\displaystyle \mathit{T}}$ is a tensor field of ordern > 1 then the divergence of the field is a tensor of ordern− 1.

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

wheretensor index notation for partial derivatives is used in the rightmost expressions. Note that$${\displaystyle \mathbf{\nabla }\cdot \mathit{S}\ne \mathbf{\nabla }\cdot {\mathit{S}}^{\mathsf{\text{T}}}.}$$

For a symmetric second-order tensor, the divergence is also often written as^[4]

The above expression is sometimes used as the definition of${\displaystyle \mathbf{\nabla }\cdot \mathit{S}}$ in Cartesian component form (often also written as${\displaystyle \mathrm{div}\mathit{S}}$). 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${\displaystyle \mathit{S}}$, 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${\displaystyle \mathbf{v}}$.

The last equation is equivalent to the alternative definition / interpretation^[4]

$${\displaystyle \begin{array}{r}{\left(\mathbf{\nabla }\cdot \right)}_{\text{alt}}\left(\mathbf{\nabla }\mathbf{v}\right)={\left(\mathbf{\nabla }\cdot \right)}_{\text{alt}}\left(v_{i,j}{\mathbf{e}}_i\otimes {\mathbf{e}}_j\right)=v_{i,jj}{\mathbf{e}}_i\otimes {\mathbf{e}}_j\cdot {\mathbf{e}}_j={\mathbf{\nabla }}^2{v}_i{\mathbf{e}}_i={\mathbf{\nabla }}^2\mathbf{v}\end{array}}$$

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

More generally,

Incylindrical polar coordinates

Thecurl of an order-n > 1 tensor field${\displaystyle \mathit{T}(\mathbf{x})}$ is also defined using the recursive relation$${\displaystyle (\mathbf{\nabla }\times \mathit{T})\cdot \mathbf{c}=\mathbf{\nabla }\times (\mathbf{c}\cdot \mathit{T});(\mathbf{\nabla }\times \mathbf{v})\cdot \mathbf{c}=\mathbf{\nabla }\cdot (\mathbf{v}\times \mathbf{c})}$$wherec is an arbitrary constant vector andv is a vector field.

Consider a vector fieldv and an arbitrary constant vectorc. In index notation, the cross product is given by$${\displaystyle \mathbf{v}\times \mathbf{c}={\epsilon }_{ijk}{v}_j{c}_k{\mathbf{e}}_i}$$where${\displaystyle {\epsilon }_{ijk}}$ is thepermutation symbol, otherwise known as the Levi-Civita symbol. Then,$${\displaystyle \mathbf{\nabla }\cdot (\mathbf{v}\times \mathbf{c})={\epsilon }_{ijk}{v}_{j,i}{c}_k=({\epsilon }_{ijk}{v}_{j,i}{\mathbf{e}}_k)\cdot \mathbf{c}=(\mathbf{\nabla }\times \mathbf{v})\cdot \mathbf{c}}$$Therefore,$${\displaystyle \mathbf{\nabla }\times \mathbf{v}={\epsilon }_{ijk}{v}_{j,i}{\mathbf{e}}_k}$$

For a second-order tensor${\displaystyle \mathit{S}}$$${\displaystyle \mathbf{c}\cdot \mathit{S}=c_m{S}_{mj}{\mathbf{e}}_j}$$Hence, using the definition of the curl of a first-order tensor field,$${\displaystyle \mathbf{\nabla }\times (\mathbf{c}\cdot \mathit{S})={\epsilon }_{ijk}{c}_m{S}_{mj,i}{\mathbf{e}}_k=({\epsilon }_{ijk}{S}_{mj,i}{\mathbf{e}}_k\otimes {\mathbf{e}}_m)\cdot \mathbf{c}=(\mathbf{\nabla }\times \mathit{S})\cdot \mathbf{c}}$$Therefore, we have$${\displaystyle \mathbf{\nabla }\times \mathit{S}={\epsilon }_{ijk}{S}_{mj,i}{\mathbf{e}}_k\otimes {\mathbf{e}}_m}$$

The most commonly used identity involving the curl of a tensor field,${\displaystyle \mathit{T}}$, is$${\displaystyle \mathbf{\nabla }\times (\mathbf{\nabla }\mathit{T})=\mathbf{0}}$$This identity holds for tensor fields of all orders. For the important case of a second-order tensor,${\displaystyle \mathit{S}}$, this identity implies that$${\displaystyle \mathbf{\nabla }\times (\mathbf{\nabla }\mathit{S})=\mathbf{0}⟹S_{mi,j}-S_{mj,i}=0}$$

The derivative of the determinant of a second order tensor${\displaystyle \mathit{A}}$ is given by$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{A}}det(\mathit{A})=det(\mathit{A}){\left[{\mathit{A}}^{-1}\right]}^{\mathsf{\text{T}}}.}$$

In anorthonormal basis, the components of${\displaystyle \mathit{A}}$ can be written as a matrixA. In that case, the right hand side corresponds the cofactors of the matrix.

The principal invariants of a second order tensor are

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