Tensor derivative (continuum mechanics)

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 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

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

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 $f(\backslash mathbf\{v\})\; =\; f\_1(\backslash mathbf\{v\})\; +\; f\_2(\backslash mathbf\{v\})$ then $$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash mathbf\{v\}\}\backslash cdot\backslash mathbf\{u\}\; =\; \backslash left(\backslash frac\{\backslash partial\; f\_1\}\{\backslash partial\; \backslash mathbf\{v\}\}\; +\; \backslash frac\{\backslash partial\; f\_2\}\{\backslash partial\; \backslash mathbf\{v\}\}\backslash right)\backslash cdot\backslash mathbf\{u\}$$
2. If $f(\backslash mathbf\{v\})\; =\; f\_1(\backslash mathbf\{v\})~\; f\_2(\backslash mathbf\{v\})$ then $$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash mathbf\{v\}\}\backslash cdot\backslash mathbf\{u\}\; =\; \backslash left(\backslash frac\{\backslash partial\; f\_1\}\{\backslash partial\; \backslash mathbf\{v\}\}\; \backslash cdot\; \backslash mathbf\{u\}\; \backslash right)~f\_2(\backslash mathbf\{v\})\; +\; f\_1(\backslash mathbf\{v\})~\backslash left(\backslash frac\{\backslash partial\; f\_2\}\{\backslash partial\; \backslash mathbf\{v\}\}\backslash cdot\backslash mathbf\{u\}\; \backslash right)$$
3. If $f(\backslash mathbf\{v\})\; =\; f\_1(f\_2(\backslash mathbf\{v\}))$ then $$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash mathbf\{v\}\}\backslash cdot\backslash mathbf\{u\}\; =\; \backslash frac\{\backslash partial\; f\_1\}\{\backslash partial\; f\_2\}~\backslash frac\{\backslash partial\; f\_2\}\{\backslash partial\; \backslash mathbf\{v\}\}\backslash cdot\backslash mathbf\{u\}$$

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

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

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

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

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

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

Properties:

1. If $f(\backslash boldsymbol\{S\})\; =\; f\_1(\backslash boldsymbol\{S\})\; +\; f\_2(\backslash boldsymbol\{S\})$ then $$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; =\; \backslash left(\backslash frac\{\backslash partial\; f\_1\}\{\backslash partial\; \backslash boldsymbol\{S\}\}\; +\; \backslash frac\{\backslash partial\; f\_2\}\{\backslash partial\; \backslash boldsymbol\{S\}\}\backslash right):\backslash boldsymbol\{T\}$$
2. If $f(\backslash boldsymbol\{S\})\; =\; f\_1(\backslash boldsymbol\{S\})~\; f\_2(\backslash boldsymbol\{S\})$ then $$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; =\; \backslash left(\backslash frac\{\backslash partial\; f\_1\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\backslash right)~f\_2(\backslash boldsymbol\{S\})\; +\; f\_1(\backslash boldsymbol\{S\})~\backslash left(\backslash frac\{\backslash partial\; f\_2\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; \backslash right)$$
3. If $f(\backslash boldsymbol\{S\})\; =\; f\_1(f\_2(\backslash boldsymbol\{S\}))$ then $$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; =\; \backslash frac\{\backslash partial\; f\_1\}\{\backslash partial\; f\_2\}~\backslash left(\backslash frac\{\backslash partial\; f\_2\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; \backslash right)$$

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

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

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

Properties:

1. If $\backslash boldsymbol\{F\}(\backslash boldsymbol\{S\})\; =\; \backslash boldsymbol\{F\}\_1(\backslash boldsymbol\{S\})\; +\; \backslash boldsymbol\{F\}\_2(\backslash boldsymbol\{S\})$ then $$\backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; =\; \backslash left(\backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\_1\}\{\backslash partial\; \backslash boldsymbol\{S\}\}\; +\; \backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\_2\}\{\backslash partial\; \backslash boldsymbol\{S\}\}\backslash right):\backslash boldsymbol\{T\}$$
2. If $\backslash boldsymbol\{F\}(\backslash boldsymbol\{S\})\; =\; \backslash boldsymbol\{F\}\_1(\backslash boldsymbol\{S\})\backslash cdot\backslash boldsymbol\{F\}\_2(\backslash boldsymbol\{S\})$ then $$\backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; =\; \backslash left(\backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\_1\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\backslash right)\backslash cdot\backslash boldsymbol\{F\}\_2(\backslash boldsymbol\{S\})\; +\; \backslash boldsymbol\{F\}\_1\; (\backslash boldsymbol\{S\})\; \backslash cdot\backslash left(\backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\_2\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; \backslash right)$$
3. If $\backslash boldsymbol\{F\}(\backslash boldsymbol\{S\})\; =\; \backslash boldsymbol\{F\}\_1(\backslash boldsymbol\{F\}\_2(\backslash boldsymbol\{S\}))$ then $$\backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; =\; \backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\_1\}\{\backslash partial\; \backslash boldsymbol\{F\}\_2\}:\backslash left(\backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\_2\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; \backslash right)$$
4. If $f(\backslash boldsymbol\{S\})\; =\; f\_1(\backslash boldsymbol\{F\}\_2(\backslash boldsymbol\{S\}))$ then $$\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; =\; \backslash frac\{\backslash partial\; f\_1\}\{\backslash partial\; \backslash boldsymbol\{F\}\_2\}:\backslash left(\backslash frac\{\backslash partial\; \backslash boldsymbol\{F\}\_2\}\{\backslash partial\; \backslash boldsymbol\{S\}\}:\backslash boldsymbol\{T\}\; \backslash right)$$

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

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

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

{{Einstein_summation_convention}}

If $\backslash mathbf\{e\}\_1,\backslash mathbf\{e\}\_2,\backslash mathbf\{e\}\_3$ are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by ($x\_1,\; x\_2,\; x\_3$), then the gradient of the tensor field $\backslash boldsymbol\{T\}$ is given by

$$\backslash boldsymbol\{\backslash nabla\}\backslash boldsymbol\{T\}\; =\; \backslash cfrac\{\backslash partial\{\backslash boldsymbol\{T\}\}\}\{\backslash partial\; x\_i\}\; \backslash otimes\; \backslash mathbf\{e\}\_i$$

{{math proof | proof = The vectors x and c can be written as $\backslash mathbf\{x\}\; =\; x\_i~\backslash mathbf\{e\}\_i$ and $\backslash mathbf\{c\}\; =\; c\_i~\backslash mathbf\{e\}\_i$. Let y := x + αc. In that case the gradient is given by

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\boldsymbol{T}\cdot\mathbf{c} & = \left.\cfrac{d}{d\alpha}~\boldsymbol{T}(x_1+\alpha c_1, x_2 + \alpha c_2, x_3 + \alpha c_3)\right|_{\alpha=0} \equiv \left.\cfrac{d}{d\alpha}~\boldsymbol{T}(y_1, y_2, y_3)\right|_{\alpha=0} \\

& = \left [\cfrac{\partial{\boldsymbol{T}}}{\partial y_1}~\cfrac{\partial y_1}{\partial \alpha} + \cfrac{\partial{\boldsymbol{T}}}{\partial y_2}~\cfrac{\partial y_2}{\partial \alpha} +

\cfrac{\partial{\boldsymbol{T}}}{\partial y_3}~\cfrac{\partial y_3}{\partial \alpha} \right]_{\alpha=0} =

\left [\cfrac{\partial{\boldsymbol{T}}}{\partial y_1}~c_1 + \cfrac{\partial{\boldsymbol{T}}}{\partial y_2}~c_2 +

\cfrac{\partial{\boldsymbol{T}}}{\partial y_3}~c_3 \right]_{\alpha=0} \\

& = \cfrac{\partial{\boldsymbol{T}}}{\partial x_1}~c_1 + \cfrac{\partial{\boldsymbol{T}}}{\partial x_2}~c_2 +

\cfrac{\partial{\boldsymbol{T}}}{\partial x_3}~c_3 \equiv \cfrac{\partial{\boldsymbol{T}}}{\partial x_i}~c_i = \cfrac{\partial{\boldsymbol{T}}}{\partial x_i}~(\mathbf{e}_i\cdot\mathbf{c})

= \left[\cfrac{\partial{\boldsymbol{T}}}{\partial x_i} \otimes \mathbf{e}_i\right]\cdot\mathbf{c} \qquad \square

\end{align} }}

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

$$\backslash begin\{align\}$$

\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{align}

{{main|Tensors in curvilinear coordinates}}

{{Einstein_summation_convention}}

If $\backslash mathbf\{g\}^1,\backslash mathbf\{g\}^2,\backslash mathbf\{g\}^3$ are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by ($\backslash xi^1,\; \backslash xi^2,\; \backslash xi^3$), then the gradient of the tensor field $\backslash boldsymbol\{T\}$ is given by{{ cite book | first = R. W. | last = Ogden | year = 2000 | title = Nonlinear Elastic Deformations | publisher = Dover | isbn = 978-0-486-696485 }}

\boldsymbol{\nabla}\boldsymbol{T} = \frac{\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 $\backslash phi$, a vector field v, and a second-order tensor field $\backslash boldsymbol\{S\}$.

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\phi & = \frac{\partial\phi}{\partial\xi^i}~\mathbf{g}^i \\[1.2ex]

\boldsymbol{\nabla}\mathbf{v} & = \frac{\partial\left(v^j \mathbf{g}_j\right)}{\partial\xi^i}\otimes\mathbf{g}^i \\

&= \left(\frac{\partial v^j}{\partial\xi^i} + v^k~\Gamma_{ik}^j\right)~\mathbf{g}_j\otimes\mathbf{g}^i

= \left(\frac{\partial v_j}{\partial\xi^i} - v_k~\Gamma_{ij}^k\right)~\mathbf{g}^j\otimes\mathbf{g}^i \\[1.2ex]

\boldsymbol{\nabla}\boldsymbol{S} & = \frac{\partial\left(S_{jk}~\mathbf{g}^j\otimes\mathbf{g}^k\right)}{\partial\xi^i}\otimes\mathbf{g}^i \\

&= \left(\frac{\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{align}

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

\Gamma_{ij}^k~\mathbf{g}_k = \frac{\partial\mathbf{g}_i}{\partial\xi^j} \quad \implies \quad

\Gamma_{ij}^k = \frac{\partial\mathbf{g}_i}{\partial\xi^j}\cdot\mathbf{g}^k = -\mathbf{g}_i\cdot\frac{\partial \mathbf{g}^k}{\partial\xi^j}

In cylindrical coordinates, the gradient is given by

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\phi ={}\quad

&\frac{\partial\phi}{\partial r}~\mathbf{e}_r

+ \frac{1}{r}~\frac{\partial \phi}{\partial \theta}~\mathbf{e}_\theta

+ \frac{\partial\phi}{\partial z}~\mathbf{e}_z \\

\end{align}

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\mathbf{v} ={}\quad

&\frac{\partial v_r}{\partial r}~\mathbf{e}_r \otimes \mathbf{e}_r

+ \frac{1}{r}\left(\frac{\partial v_r}{\partial\theta} - v_\theta\right)~\mathbf{e}_r \otimes \mathbf{e}_\theta

+ \frac{\partial v_r}{\partial z}~\mathbf{e}_r \otimes \mathbf{e}_z \\

{}+{} &\frac{\partial v_\theta}{\partial r}~\mathbf{e}_\theta \otimes \mathbf{e}_r

+ \frac{1}{r}\left(\frac{\partial v_\theta}{\partial\theta} + v_r\right)~\mathbf{e}_\theta \otimes \mathbf{e}_\theta

+ \frac{\partial v_\theta}{\partial z}~\mathbf{e}_\theta \otimes \mathbf{e}_z \\

{}+{} &\frac{\partial v_z}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r

+ \frac{1}{r}\frac{\partial v_z}{\partial\theta}~\mathbf{e}_z \otimes\mathbf{e}_\theta

+ \frac{\partial v_z}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z \\

\end{align}

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\boldsymbol{S} ={}\quad

&\frac{\partial S_{rr}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_r

+ \frac{\partial S_{rr}}{\partial z}~\mathbf{e}_r \otimes \mathbf{e}_r \otimes \mathbf{e}_z

+ \frac{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_{r\theta}}{\partial r}~\mathbf{e}_r \otimes \mathbf{e}_\theta \otimes \mathbf{e}_r

+ \frac{\partial S_{r\theta}}{\partial z}~\mathbf{e}_r \otimes \mathbf{e}_\theta \otimes \mathbf{e}_z

+ \frac{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_{rz}}{\partial r}~\mathbf{e}_r \otimes \mathbf{e}_z \otimes \mathbf{e}_r

+ \frac{\partial S_{rz}}{\partial z}~\mathbf{e}_r \otimes \mathbf{e}_z \otimes \mathbf{e}_z

+ \frac{1}{r}\left[\frac{\partial S_{rz}}{\partial \theta} - S_{\theta z}\right]~\mathbf{e}_r \otimes \mathbf{e}_z \otimes \mathbf{e}_\theta \\

{}+{} &\frac{\partial S_{\theta r}}{\partial r}~\mathbf{e}_\theta \otimes \mathbf{e}_r \otimes \mathbf{e}_r

+ \frac{\partial S_{\theta r}}{\partial z}~\mathbf{e}_\theta \otimes \mathbf{e}_r \otimes \mathbf{e}_z

+ \frac{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\theta}}{\partial r}~\mathbf{e}_\theta \otimes \mathbf{e}_\theta \otimes \mathbf{e}_r

+ \frac{\partial S_{\theta\theta}}{\partial z}~\mathbf{e}_\theta \otimes \mathbf{e}_\theta \otimes \mathbf{e}_z

+ \frac{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 z}}{\partial r}~\mathbf{e}_\theta \otimes \mathbf{e}_z \otimes \mathbf{e}_r

+ \frac{\partial S_{\theta z}}{\partial z}~\mathbf{e}_\theta \otimes \mathbf{e}_z \otimes \mathbf{e}_z

+ \frac{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_{zr}}{\partial r}~\mathbf{e}_z \otimes \mathbf{e}_r \otimes \mathbf{e}_r

+ \frac{\partial S_{zr}}{\partial z}~\mathbf{e}_z \otimes \mathbf{e}_r \otimes \mathbf{e}_z

+ \frac{1}{r}\left[\frac{\partial S_{zr}}{\partial \theta} - S_{z\theta}\right]~\mathbf{e}_z \otimes \mathbf{e}_r \otimes \mathbf{e}_\theta \\

{}+{} &\frac{\partial S_{z\theta}}{\partial r}~\mathbf{e}_z \otimes \mathbf{e}_\theta \otimes \mathbf{e}_r

+ \frac{\partial S_{z\theta}}{\partial z}~\mathbf{e}_z \otimes \mathbf{e}_\theta \otimes \mathbf{e}_z

+ \frac{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_{zz}}{\partial r}~\mathbf{e}_z \otimes \mathbf{e}_z \otimes \mathbf{e}_r

+ \frac{\partial S_{zz}}{\partial z}~\mathbf{e}_z \otimes \mathbf{e}_z \otimes \mathbf{e}_z

+ \frac{1}{r}~\frac{\partial S_{zz}}{\partial\theta}~ \mathbf{e}_z \otimes \mathbf{e}_z \otimes \mathbf{e}_\theta

\end{align}

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

(\boldsymbol{\nabla}\cdot\boldsymbol{T})\cdot\mathbf{c} =

\boldsymbol{\nabla}\cdot\left(\mathbf{c}\cdot\boldsymbol{T}^\textsf{T}\right) ~;\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 $\backslash boldsymbol\{T\}$ is a tensor field of order n > 1 then the divergence of the field is a tensor of order n− 1.

{{Einstein_summation_convention}}

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

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\cdot\mathbf{v} &= \frac{\partial v_i}{\partial x_i} = v_{i,i} \\

\boldsymbol{\nabla}\cdot\boldsymbol{S} &= \frac{\partial S_{ik}}{\partial x_i}~\mathbf{e}_k = S_{ik, i}~\mathbf{e}_k

\end{align}

where tensor index notation for partial derivatives is used in the rightmost expressions. Note that

$$\backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash boldsymbol\{S\}\; \backslash neq\; \backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash boldsymbol\{S\}^\backslash textsf\{T\}.$$

For a symmetric second-order tensor, the divergence is also often written as{{ cite book | last1 = Hjelmstad | first1 = Keith | title = Fundamentals of Structural Mechanics | date = 2004 | publisher = Springer Science & Business Media | isbn = 978-0-387-233307 | page = 45 }}

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\cdot\boldsymbol{S} &= \cfrac{\partial S_{ki}}{\partial x_i}~\mathbf{e}_k = S_{ki,i}~\mathbf{e}_k

\end{align}

The above expression is sometimes used as the definition of

$\backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash boldsymbol\{S\}$ in Cartesian component form (often also written as

$\backslash operatorname\{div\}\backslash boldsymbol\{S\}$). 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 $\backslash boldsymbol\{S\}$, and is conventional. This is demonstrated by an example. In a Cartesian coordinate system the second order tensor (matrix) $\backslash mathbf\{S\}$ is the gradient of a vector function $\backslash mathbf\{v\}$.

$$\backslash begin\{align\}$$

\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) \\

\boldsymbol{\nabla} \cdot \left[ \left( \boldsymbol{\nabla} \mathbf{v} \right)^\textsf{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{align}

The last equation is equivalent to the alternative definition / interpretation

$$\backslash begin\{align\}$$

\left( \boldsymbol{\nabla} \cdot \right)_\text{alt} \left( \boldsymbol{\nabla} \mathbf{v} \right) =

\left( \boldsymbol{\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 =

\boldsymbol{\nabla}^2 v_i ~\mathbf{e}_i =

\boldsymbol{\nabla}^2 \mathbf{v}

\end{align}

{{main|Tensors in curvilinear coordinates}}

{{Einstein_summation_convention}}

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

$$\backslash begin\{align\}$$

\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{align}

More generally,

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\cdot\boldsymbol{S} & = \left[\cfrac{\partial S_{ij}}{\partial q^k} - \Gamma^l_{ki}~S_{lj} - \Gamma^l_{kj}~S_{il}\right]~g^{ik}~\mathbf{b}^j \\[8pt]

& = \left[\cfrac{\partial S^{ij}}{\partial q^i} + \Gamma^i_{il}~S^{lj} + \Gamma^j_{il}~S^{il}\right]~\mathbf{b}_j \\[8pt]

& = \left[\cfrac{\partial S^i_{~j}}{\partial q^i} + \Gamma^i_{il}~S^l_{~j} - \Gamma^l_{ij}~S^i_{~l}\right]~\mathbf{b}^j \\[8pt]

& = \left[\cfrac{\partial S_i^{~j}}{\partial q^k} - \Gamma^l_{ik}~S_l^{~j} + \Gamma^j_{kl}~S_i^{~l}\right]~g^{ik}~\mathbf{b}_j

\end{align}

In cylindrical polar coordinates

$$\backslash begin\{align\}$$

\boldsymbol{\nabla}\cdot\mathbf{v} =\quad

&\frac{\partial v_r}{\partial r}

+ \frac{1}{r}\left(\frac{\partial v_\theta}{\partial\theta} + v_r \right)

+ \frac{\partial v_z}{\partial z}\\

\boldsymbol{\nabla}\cdot\boldsymbol{S} =\quad

&\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 \\

{}+{} &\frac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial \theta}

+ (S_{rr} - S_{\theta\theta})\right]~\mathbf{e}_r

+ \frac{1}{r}\left[\frac{\partial S_{\theta\theta}}{\partial\theta}

+ (S_{r\theta} + S_{\theta r})\right]~\mathbf{e}_\theta

+ \frac{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{align}

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

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

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

Consider a vector field v and an arbitrary constant vector c. In index notation, the cross product is given by

$$\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{c\}\; =\; \backslash varepsilon\_\{ijk\}~v\_j~c\_k~\backslash mathbf\{e\}\_i$$

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

\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,

$$\backslash boldsymbol\{\backslash nabla\}\backslash times\backslash mathbf\{v\}\; =\; \backslash varepsilon\_\{ijk\}~v\_\{j,i\}~\backslash mathbf\{e\}\_k$$

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

$$\backslash mathbf\{c\}\backslash cdot\backslash boldsymbol\{S\}\; =\; c\_m~S\_\{mj\}~\backslash mathbf\{e\}\_j$$

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

$$\backslash boldsymbol\{\backslash nabla\}\backslash times(\backslash mathbf\{c\}\backslash cdot\backslash boldsymbol\{S\})\; =\; \backslash varepsilon\_\{ijk\}~c\_m~S\_\{mj,i\}~\backslash mathbf\{e\}\_k\; =\; (\backslash varepsilon\_\{ijk\}~S\_\{mj,i\}~\backslash mathbf\{e\}\_k\backslash otimes\backslash mathbf\{e\}\_m)\backslash cdot\backslash mathbf\{c\}\; =\; (\backslash boldsymbol\{\backslash nabla\}\backslash times\backslash boldsymbol\{S\})\; \backslash cdot\; \backslash mathbf\{c\}$$

Therefore, we have

$$\backslash boldsymbol\{\backslash nabla\}\backslash times\backslash boldsymbol\{S\}\; =\; \backslash varepsilon\_\{ijk\}~S\_\{mj,i\}~\backslash mathbf\{e\}\_k\backslash otimes\backslash mathbf\{e\}\_m$$

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

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

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

$$\backslash boldsymbol\{\backslash nabla\}\backslash times(\backslash boldsymbol\{\backslash nabla\}\backslash boldsymbol\{S\})\; =\; \backslash boldsymbol\{0\}\; \backslash quad\; \backslash implies\; \backslash quad\; S\_\{mi,j\}\; -\; S\_\{mj,i\}\; =\; 0$$

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

\frac{\partial}{\partial\boldsymbol{A}}\det(\boldsymbol{A}) = \det(\boldsymbol{A})~\left[\boldsymbol{A}^{-1}\right]^\textsf{T} ~.

In an orthonormal basis, the components of $\backslash boldsymbol\{A\}$ can be written as a matrix A. In that case, the right hand side corresponds the cofactors of the matrix.

{{math proof| proof = Let $\backslash boldsymbol\{A\}$ be a second order tensor and let $f(\backslash boldsymbol\{A\})\; =\; \backslash det(\backslash boldsymbol\{A\})$. Then, from the definition of the derivative of a scalar valued function of a tensor, we have

$$\backslash begin\{align\}$$

\frac{\partial f}{\partial\boldsymbol{A}}:\boldsymbol{T} & = \left.\cfrac{d}{d\alpha} \det(\boldsymbol{A} + \alpha~\boldsymbol{T}) \right|_{\alpha=0} \\

& = \left.\cfrac{d}{d\alpha} \det\left[\alpha~\boldsymbol{A}\left(\cfrac{1}{\alpha}~\boldsymbol{\mathit{I}} + \boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right) \right] \right|_{\alpha=0} \\

& = \left.\cfrac{d}{d\alpha} \left[\alpha^3~\det(\boldsymbol{A})~\det\left(\cfrac{1}{\alpha}~\boldsymbol{\mathit{I}} + \boldsymbol{A}^{-1} \cdot \boldsymbol{T}\right)\right]\right|_{\alpha=0}.

\end{align}

The determinant of a tensor can be expressed in the form of a characteristic equation in terms of the invariants $I\_1,I\_2,I\_3$ using

$$\backslash det(\backslash lambda~\backslash boldsymbol\{\backslash mathit\{I\}\}\; +\; \backslash boldsymbol\{A\})\; =\; \backslash lambda^3\; +\; I\_1(\backslash boldsymbol\{A\})~\backslash lambda^2\; +\; I\_2(\backslash boldsymbol\{A\})~\backslash lambda\; +\; I\_3(\backslash boldsymbol\{A\}).$$

Using this expansion we can write

$$\backslash begin\{align\}$$

\frac{\partial f}{\partial\boldsymbol{A}}: \boldsymbol{T}

& = \left.\cfrac{d}{d\alpha} \left[\alpha^3~\det(\boldsymbol{A})~\left(

\cfrac{1}{\alpha^3} +

I_1\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)~\cfrac{1}{\alpha^2} +

I_2\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)~\cfrac{1}{\alpha} +

I_3\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)

\right) \right] \right|_{\alpha=0} \\

& = \left.\det(\boldsymbol{A})~\cfrac{d}{d\alpha} \left[

1 + I_1\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)~\alpha +

I_2\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)~\alpha^2 +

I_3\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)~\alpha^3

\right] \right|_{\alpha=0} \\

& = \left.\det(\boldsymbol{A})~\left[

I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}) +

2~I_2\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)~\alpha +

3~I_3\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)~\alpha^2

\right] \right|_{\alpha=0} \\

& = \det(\boldsymbol{A})~I_1\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right) ~.

\end{align}

Recall that the invariant $I\_1$ is given by

$$I\_1(\backslash boldsymbol\{A\})\; =\; \backslash text\{tr\}\{\backslash boldsymbol\{A\}\}.$$

Hence,

\frac{\partial f}{\partial\boldsymbol{A}}: \boldsymbol{T} =

\det(\boldsymbol{A})~\text{tr}\left(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right) =

\det(\boldsymbol{A})~\left[\boldsymbol{A}^{-1}\right]^\textsf{T} : \boldsymbol{T}.

Invoking the arbitrariness of $\backslash boldsymbol\{T\}$ we then have

\frac{\partial f}{\partial\boldsymbol{A}} = \det(\boldsymbol{A})~\left[\boldsymbol{A}^{-1}\right]^\textsf{T} ~.

}}

The principal invariants of a second order tensor are

\begin{align}

I_1(\boldsymbol{A}) & = \text{tr}{\boldsymbol{A}} \\

I_2(\boldsymbol{A}) & = \tfrac{1}{2} \left[ (\text{tr}{\boldsymbol{A}})^2 - \text{tr}{\boldsymbol{A}^2} \right] \\

I_3(\boldsymbol{A}) & = \det(\boldsymbol{A})

\end{align}

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

$$\backslash begin\{align\}$$

\frac{\partial I_1}{\partial\boldsymbol{A}} & = \boldsymbol{\mathit{1}} \\[3pt]

\frac{\partial I_2}{\partial\boldsymbol{A}} & = I_1 \, \boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T} \\[3pt]

\frac{\partial I_3}{\partial\boldsymbol{A}} & = \det(\boldsymbol{A})~\left[\boldsymbol{A}^{-1}\right]^\textsf{T} \\

&= I_2~\boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T}~\left(I_1~\boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T}\right)

= \left(\boldsymbol{A}^2 - I_1~\boldsymbol{A} + I_2~\boldsymbol{\mathit{1}}\right)^\textsf{T}

\end{align}

{{math proof | proof = From the derivative of the determinant we know that

\frac{\partial I_3}{\partial \boldsymbol{A}} = \det(\boldsymbol{A})~\left[\boldsymbol{A}^{-1}\right]^\textsf{T} ~.

For the derivatives of the other two invariants, let us go back to the characteristic equation

\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A}) =

\lambda^3 + I_1(\boldsymbol{A})~\lambda^2 + I_2(\boldsymbol{A})~\lambda + I_3(\boldsymbol{A}) ~.

Using the same approach as for the determinant of a tensor, we can show that

\frac{\partial }{\partial \boldsymbol{A}}\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A}) =

\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})~\left[(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})^{-1}\right]^\textsf{T} ~.

Now the left hand side can be expanded as

\begin{align}

\frac{\partial}{\partial \boldsymbol{A}}\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A}) & =

\frac{\partial}{\partial \boldsymbol{A}}\left[

\lambda^3 + I_1(\boldsymbol{A})~\lambda^2 + I_2(\boldsymbol{A})~\lambda + I_3(\boldsymbol{A}) \right] \\

& =

\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda +

\frac{\partial I_3}{\partial \boldsymbol{A}}~.

\end{align}

Hence

\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda +

\frac{\partial I_3}{\partial \boldsymbol{A}} =

\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})~\left[(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})^{-1}\right]^\textsf{T}

or,

(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})^\textsf{T}\cdot\left[

\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda +

\frac{\partial I_3}{\partial \boldsymbol{A}}\right] =

\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})~\boldsymbol{\mathit{1}} ~.

Expanding the right hand side and separating terms on the left hand side gives

\left(\lambda~\boldsymbol{\mathit{1}} +\boldsymbol{A}^\textsf{T}\right)\cdot\left[

\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda +

\frac{\partial I_3}{\partial \boldsymbol{A}}\right] =

\left[\lambda^3 + I_1~\lambda^2 + I_2~\lambda + I_3\right]

\boldsymbol{\mathit{1}}

or,

\begin{align}

\left[\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^3 \right.&

\left.+ \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda^2 +

\frac{\partial I_3}{\partial \boldsymbol{A}}~\lambda\right]\boldsymbol{\mathit{1}} +

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 +

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda +

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_3}{\partial \boldsymbol{A}} \\

& =

\left[\lambda^3 + I_1~\lambda^2 + I_2~\lambda + I_3\right]

\boldsymbol{\mathit{1}} ~.

\end{align}

If we define $I\_0\; :=\; 1$ and $I\_4\; :=\; 0$, we can write the above as

\begin{align}

\left[\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^3 \right.&

\left.+ \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda^2 +

\frac{\partial I_3}{\partial \boldsymbol{A}}~\lambda + \frac{\partial I_4}{\partial \boldsymbol{A}}\right]\boldsymbol{\mathit{1}} +

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_0}{\partial \boldsymbol{A}}~\lambda^3 +

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 +

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda +

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_3}{\partial \boldsymbol{A}} \\

&=

\left[I_0~\lambda^3 + I_1~\lambda^2 + I_2~\lambda + I_3\right]

\boldsymbol{\mathit{1}} ~.

\end{align}

Collecting terms containing various powers of λ, we get

\begin{align}

\lambda^3&\left(I_0~\boldsymbol{\mathit{1}} - \frac{\partial I_1}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_0}{\partial \boldsymbol{A}}\right) +

\lambda^2\left(I_1~\boldsymbol{\mathit{1}} - \frac{\partial I_2}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_1}{\partial \boldsymbol{A}}\right) + \\

&\qquad \qquad\lambda\left(I_2~\boldsymbol{\mathit{1}} - \frac{\partial I_3}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_2}{\partial \boldsymbol{A}}\right) +

\left(I_3~\boldsymbol{\mathit{1}} - \frac{\partial I_4}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -

\boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_3}{\partial \boldsymbol{A}}\right) = 0 ~.

\end{align}

Then, invoking the arbitrariness of λ, we have

$$\backslash begin\{align\}$$

I_0~\boldsymbol{\mathit{1}} - \frac{\partial I_1}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_0}{\partial \boldsymbol{A}} & = 0 \\

I_1~\boldsymbol{\mathit{1}} - \frac{\partial I_2}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - I_2~\boldsymbol{\mathit{1}} - \frac{\partial I_3}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_2}{\partial \boldsymbol{A}} & = 0 \\

I_3~\boldsymbol{\mathit{1}} - \frac{\partial I_4}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T}\cdot\frac{\partial I_3}{\partial \boldsymbol{A}} & = 0 ~.

\end{align}

This implies that

$$\backslash begin\{align\}$$

\frac{\partial I_1}{\partial \boldsymbol{A}} &= \boldsymbol{\mathit{1}} \\

\frac{\partial I_2}{\partial \boldsymbol{A}} & = I_1~\boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T} \\

\frac{\partial I_3}{\partial \boldsymbol{A}} & = I_2~\boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T}~\left(I_1~\boldsymbol{\mathit{1}} - \boldsymbol{A}^\textsf{T}\right) = \left(\boldsymbol{A}^2 -I_1~\boldsymbol{A} + I_2~\boldsymbol{\mathit{1}}\right)^\textsf{T}

\end{align}

}}

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

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

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

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

\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,

$$\backslash frac\{\backslash partial\; \backslash boldsymbol\{A\}\}\{\backslash partial\; \backslash boldsymbol\{A\}\}\; =\; \backslash boldsymbol\{\backslash mathsf\{I\}\}$$

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

\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

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

where

$$\backslash boldsymbol\{\backslash mathsf\{I\}\}^\backslash textsf\{T\}\; =\; \backslash delta\_\{jk\}~\backslash delta\_\{il\}~\backslash mathbf\{e\}\_i\backslash otimes\backslash mathbf\{e\}\_j\backslash otimes\backslash mathbf\{e\}\_k\backslash otimes\backslash mathbf\{e\}\_l$$

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

\frac{\partial \boldsymbol{A}}{\partial \boldsymbol{A}} = \boldsymbol{\mathsf{I}}^{(s)}

= \frac{1}{2}~\left(\boldsymbol{\mathsf{I}} + \boldsymbol{\mathsf{I}}^\textsf{T}\right)

where the symmetric fourth order identity tensor is

\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

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

\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

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

We also have

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

In index notation

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

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

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

{{math proof | proof = Recall that

$$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash mathit\{1\}\}\}\{\backslash partial\; \backslash boldsymbol\{A\}\}:\backslash boldsymbol\{T\}\; =\; \backslash boldsymbol\{\backslash mathit\{0\}\}$$

Since $\backslash boldsymbol\{A\}^\{-1\}\backslash cdot\backslash boldsymbol\{A\}\; =\; \backslash boldsymbol\{\backslash mathit\{1\}\}$, we can write

$$\backslash frac\{\backslash partial\; \}\{\backslash partial\; \backslash boldsymbol\{A\}\}\backslash left(\backslash boldsymbol\{A\}^\{-1\}\backslash cdot\backslash boldsymbol\{A\}\backslash right):\backslash boldsymbol\{T\}\; =\; \backslash boldsymbol\{\backslash mathit\{0\}\}$$

Using the product rule for second order tensors

\frac{\partial }{\partial \boldsymbol{S}}[\boldsymbol{F}_1(\boldsymbol{S})\cdot\boldsymbol{F}_2(\boldsymbol{S})]:\boldsymbol{T} =

\left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)\cdot\boldsymbol{F}_2 +

\boldsymbol{F}_1\cdot\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)

we get

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

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

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

= \boldsymbol{\mathit{0}}

or,

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

Therefore,

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

}}

File:StressMeasures.png

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

\int_{\Omega} \boldsymbol{F}\otimes\boldsymbol{\nabla}\boldsymbol{G}\,d\Omega = \int_{\Gamma} \mathbf{n} \otimes (\boldsymbol{F}\otimes\boldsymbol{G})\,d\Gamma - \int_{\Omega} \boldsymbol{G}\otimes\boldsymbol{\nabla}\boldsymbol{F}\,d\Omega

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

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

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

\int_{\Omega} F_{ijk....}\,G_{lmn...,p}\,d\Omega = \int_{\Gamma} n_p\,F_{ijk...}\,G_{lmn...}\,d\Gamma - \int_{\Omega} G_{lmn...}\,F_{ijk...,p}\,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 $\backslash boldsymbol\{F\}$ and $\backslash boldsymbol\{G\}$ are second order tensors, we have

$$\backslash int\_\{\backslash Omega\}\; \backslash boldsymbol\{F\}\backslash cdot(\backslash boldsymbol\{\backslash nabla\}\backslash cdot\backslash boldsymbol\{G\})\backslash ,d\backslash Omega\; =\; \backslash int\_\{\backslash Gamma\}\; \backslash mathbf\{n\}\backslash cdot\backslash left(\backslash boldsymbol\{G\}\backslash cdot\backslash boldsymbol\{F\}^\backslash textsf\{T\}\backslash right)\backslash ,d\backslash Gamma\; -\; \backslash int\_\{\backslash Omega\}\; (\backslash boldsymbol\{\backslash nabla\}\backslash boldsymbol\{F\}):\backslash boldsymbol\{G\}^\backslash textsf\{T\}\backslash ,d\backslash Omega\; \backslash ,.$$ evelina

In index notation,

$$\backslash int\_\{\backslash Omega\}\; F\_\{ij\}\backslash ,G\_\{pj,p\}\backslash ,d\backslash Omega\; =\; \backslash int\_\{\backslash Gamma\}\; n\_p\backslash ,F\_\{ij\}\backslash ,G\_\{pj\}\backslash ,d\backslash Gamma\; -\; \backslash int\_\{\backslash Omega\}\; G\_\{pj\}\backslash ,F\_\{ij,p\}\backslash ,d\backslash Omega\; \backslash ,.$$

• Covariant derivative
• Ricci calculus

{{Reflist}}

Category:Solid mechanics

Category:Mechanics