# Derivada tensorial (mecánica de medios continuos)

Las derivadas de escalares, vectores y tensores de segundo orden con respecto a los tensores de segundo orden son de considerable utilidad en la mecánica de medios continuos. Estos derivadas se utilizan en las teorías de la elasticidad no lineal y de la plasticidad, particularmente en el diseño de algoritmos de simulación.
La derivada direccional proporciona una forma sistemática de determinar estas derivadas.

## Derivadas con respecto a vectores y tensores de segundo orden

A continuación se dan las definiciones de derivadas direccionales para diversas situaciones. Se supone que las funciones son lo suficientemente suaves como para poder tomar derivadas.

### Derivadas de funciones escalares de vectores

Sea f(v) una función con valor real del vector v. Entonces la derivada de f(v) con respecto a v (o en v) es el vector definido mediante es producto escalar con cualquier vector u'
    ${\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =Df(\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~f(\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}$
para todos los vectores u. El producto escalar anterior produce un escalar, y si u es un vector unitario, da la derivada direccional de f en v, en la dirección u.

### Coordenadas curvilíneas

Si $\mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}$ son los contravariant base (álgebra) en un sistema curvilinear coordinate, con las coordenadas de los puntos indicadas por ($\xi ^{1},\xi ^{2},\xi ^{3}$), entonces el gradiente del campo tensorial ${\boldsymbol {T}}$ viene dado por (consulte para obtener una prueba).
    ${\boldsymbol {\nabla }}{\boldsymbol {T}}={\frac {\partial {\boldsymbol {T}}}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}$
De esta definición tenemos las siguientes relaciones para los gradientes de un campo escalar $\phi$, un campo vectorial v y un campo tensorial de segundo orden ${\boldsymbol {S}}$.
    ${\begin{aligned}{\boldsymbol {\nabla }}\phi &={\frac {\partial \phi }{\partial \xi ^{i}}}~\mathbf {g} ^{i}\\{\boldsymbol {\nabla }}\mathbf {v} &={\frac {\partial \left(v^{j}\mathbf {g} _{j}\right)}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\frac {\partial v^{j}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{j}\right)~\mathbf {g} _{j}\otimes \mathbf {g} ^{i}=\left({\frac {\partial v_{j}}{\partial \xi ^{i}}}-v_{k}~\Gamma _{ij}^{k}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{i}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial \left(S_{jk}~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k}\right)}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\frac {\partial S_{jk}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ij}^{l}-S_{jl}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k}\otimes \mathbf {g} ^{i}\end{aligned}}$
donde Símbolos de Christoffel $\Gamma _{ij}^{k}$ se define usando
    $\Gamma _{ij}^{k}~\mathbf {g} _{k}={\frac {\partial \mathbf {g} _{i}}{\partial \xi ^{j}}}\quad \implies \quad \Gamma _{ij}^{k}={\frac {\partial \mathbf {g} _{i}}{\partial \xi ^{j}}}\cdot \mathbf {g} ^{k}=-\mathbf {g} _{i}\cdot {\frac {\partial \mathbf {g} ^{k}}{\partial \xi ^{j}}}$

#### Coordenadas polares cilíndricas

En coordenadas cilíndricas, el gradiente viene dado por
    ${\begin{aligned}{\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}\\{\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}\\{\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{aligned}}$

## Divergencia de un campo tensorial

El divergencia (matemática) de un campo tensor ${\boldsymbol {T}}(\mathbf {x} )$ se define usando la relación recursiva
    $({\boldsymbol {\nabla }}\cdot {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot \left(\mathbf {c} \cdot {\boldsymbol {T}}^{\textsf {T}}\right)~;\qquad {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\text{tr}}({\boldsymbol {\nabla }}\mathbf {v} )$
donde c es un vector constante arbitrario y v es un campo vectorial. Si ${\boldsymbol {T}}$ es un campo tensorial de orden n > 1, entonces la divergencia del campo es un tensor de orden n- 1.

### Coordenadas cartesianas

En un sistema de coordenadas cartesiano tenemos las siguientes relaciones para un campo vectorial v' y un campo tensorial de segundo orden ${\boldsymbol {S}}$.
    ${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\frac {\partial v_{i}}{\partial x_{i}}}=v_{i,i}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\frac {\partial S_{ik}}{\partial x_{i}}}~\mathbf {e} _{k}=S_{ik,i}~\mathbf {e} _{k}\end{aligned}}$
donde tensor index notation para derivadas parciales se utiliza en las expresiones situadas más a la derecha. Tenga en cuenta que
    ${\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}\neq {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}^{\textsf {T}}.$
Para un tensor simétrico de segundo orden, la divergencia también suele escribirse como
    ${\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\cfrac {\partial S_{ki}}{\partial x_{i}}}~\mathbf {e} _{k}=S_{ki,i}~\mathbf {e} _{k}\end{aligned}}$
La expresión anterior se utiliza a veces como definición de ${\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}$ en forma de componente cartesiano (a menudo también escrito como $\operatorname {div} {\boldsymbol {S}}$). Tenga en cuenta que dicha definición no es coherente con el resto de este artículo (consulte la sección sobre coordenadas curvilíneas).
La diferencia surge de si la diferenciación se realiza respecto de las filas o columnas de ${\boldsymbol {S}}$, y es convencional. Esto se demuestra con un ejemplo. En un sistema de coordenadas cartesiano, el tensor (matriz) de segundo orden $\mathbf {S}$ es el gradiente de una función vectorial $\mathbf {v}$.
    ${\begin{aligned}{\boldsymbol {\nabla }}\cdot \left({\boldsymbol {\nabla }}\mathbf {v} \right)&={\boldsymbol {\nabla }}\cdot \left(v_{i,j}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{i,ji}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}=\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)_{,j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)\\{\boldsymbol {\nabla }}\cdot \left[\left({\boldsymbol {\nabla }}\mathbf {v} \right)^{\textsf {T}}\right]&={\boldsymbol {\nabla }}\cdot \left(v_{j,i}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{j,ii}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}v_{j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}\mathbf {v} \end{aligned}}$
La última ecuación es equivalente a la definición/interpretación alternativa
    ${\begin{aligned}\left({\boldsymbol {\nabla }}\cdot \right)_{\text{alt}}\left({\boldsymbol {\nabla }}\mathbf {v} \right)=\left({\boldsymbol {\nabla }}\cdot \right)_{\text{alt}}\left(v_{i,j}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{i,jj}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\cdot \mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}v_{i}~\mathbf {e} _{i}={\boldsymbol {\nabla }}^{2}\mathbf {v} \end{aligned}}$

### Coordenadas curvilíneas

En coordenadas curvilíneas, las divergencias de un campo vectorial v' y un campo tensorial de segundo orden ${\boldsymbol {S}}$ son
    ${\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &=\left({\cfrac {\partial v^{i}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{i}\right)\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left({\cfrac {\partial S_{ik}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ii}^{l}-S_{il}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{k}\end{aligned}}$
Más generalmente,
    ${\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left[{\cfrac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~g^{ik}~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S^{ij}}{\partial q^{i}}}+\Gamma _{il}^{i}~S^{lj}+\Gamma _{il}^{j}~S^{il}\right]~\mathbf {b} _{j}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{i}}}+\Gamma _{il}^{i}~S_{~j}^{l}-\Gamma _{ij}^{l}~S_{~l}^{i}\right]~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~g^{ik}~\mathbf {b} _{j}\end{aligned}}$

#### Coordenadas polares cilíndricas

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

## Rotacional de un campo tensorial

El curl de un campo tensor de orden-n > 1 ${\boldsymbol {T}}(\mathbf {x} )$ también se define usando la relación recursiva
    $({\boldsymbol {\nabla }}\times {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {T}})~;\qquad ({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )$
donde c es un vector constante arbitrario y v es un campo vectorial.

### Rotacional de un campo tensorial (vectorial) de primer orden

Considere un campo vectorial v y un vector constante arbitrario c. En notación de índice, el producto cruzado viene dado por
    $\mathbf {v} \times \mathbf {c} =\varepsilon _{ijk}~v_{j}~c_{k}~\mathbf {e} _{i}$
donde $\varepsilon _{ijk}$ es símbolo de Levi-Civita, también conocido como símbolo de Levi-Civita. Entonces,
    ${\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )=\varepsilon _{ijk}~v_{j,i}~c_{k}=(\varepsilon _{ijk}~v_{j,i}~\mathbf {e} _{k})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c}$
Por lo tanto,
    ${\boldsymbol {\nabla }}\times \mathbf {v} =\varepsilon _{ijk}~v_{j,i}~\mathbf {e} _{k}$

### Rotacional de un campo tensorial de segundo orden

Para un tensor de segundo orden ${\boldsymbol {S}}$
    $\mathbf {c} \cdot {\boldsymbol {S}}=c_{m}~S_{mj}~\mathbf {e} _{j}$
Por tanto, utilizando la definición de rizo de un campo tensorial de primer orden,
    ${\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {S}})=\varepsilon _{ijk}~c_{m}~S_{mj,i}~\mathbf {e} _{k}=(\varepsilon _{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times {\boldsymbol {S}})\cdot \mathbf {c}$
Por lo tanto, tenemos
    ${\boldsymbol {\nabla }}\times {\boldsymbol {S}}=\varepsilon _{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m}$

### Identidades que involucran la curvatura de un campo tensorial

La identidad más comúnmente utilizada que involucra la curvatura de un campo tensorial, ${\boldsymbol {T}}$, es
    ${\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}{\boldsymbol {T}})={\boldsymbol {0}}$
Esta identidad es válida para campos tensoriales de todos los órdenes. Para el caso importante de un tensor de segundo orden, ${\boldsymbol {S}}$, esta identidad implica que
    ${\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}{\boldsymbol {S}})={\boldsymbol {0}}\quad \implies \quad S_{mi,j}-S_{mj,i}=0$

## Derivada del determinante de un tensor de segundo orden

La derivada del determinante de un tensor de segundo orden ${\boldsymbol {A}}$ viene dada por
    ${\frac {\partial }{\partial {\boldsymbol {A}}}}\det({\boldsymbol {A}})=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}~.$
En términos ortonormales, los componentes de ${\boldsymbol {A}}$ se pueden escribir como una matriz A. En ese caso, el lado derecho corresponde a los cofactores de la matriz.
* Demostración
* Sea ${\boldsymbol {A}}$ un tensor de segundo orden y sea $f({\boldsymbol {A}})=\det({\boldsymbol {A}})$. Entonces, a partir de la definición de la derivada de una función escalar de un tensor, tenemos
* ${\begin{aligned}{\frac {\partial f}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}&=\left.{\cfrac {d}{d\alpha }}\det({\boldsymbol {A}}+\alpha ~{\boldsymbol {T}})\right|_{\alpha =0}\\&=\left.{\cfrac {d}{d\alpha }}\det \left[\alpha ~{\boldsymbol {A}}\left({\cfrac {1}{\alpha }}~{\boldsymbol {\mathit {I}}}+{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)\right]\right|_{\alpha =0}\\&=\left.{\cfrac {d}{d\alpha }}\left[\alpha ^{3}~\det({\boldsymbol {A}})~\det \left({\cfrac {1}{\alpha }}~{\boldsymbol {\mathit {I}}}+{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)\right]\right|_{\alpha =0}.\end{aligned}}$
* El determinante de un tensor se puede expresar en forma de ecuación característica en términos de los invariantes $I_{1},I_{2},I_{3}$ usando
* $\det(\lambda ~{\boldsymbol {\mathit {I}}}+{\boldsymbol {A}})=\lambda ^{3}+I_{1}({\boldsymbol {A}})~\lambda ^{2}+I_{2}({\boldsymbol {A}})~\lambda +I_{3}({\boldsymbol {A}}).$
* Usando esta expansión podemos escribir
* ${\begin{aligned}{\frac {\partial f}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}&=\left.{\cfrac {d}{d\alpha }}\left[\alpha ^{3}~\det({\boldsymbol {A}})~\left({\cfrac {1}{\alpha ^{3}}}+I_{1}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~{\cfrac {1}{\alpha ^{2}}}+I_{2}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~{\cfrac {1}{\alpha }}+I_{3}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)\right)\right]\right|_{\alpha =0}\\&=\left.\det({\boldsymbol {A}})~{\cfrac {d}{d\alpha }}\left[1+I_{1}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha +I_{2}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha ^{2}+I_{3}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha ^{3}\right]\right|_{\alpha =0}\\&=\left.\det({\boldsymbol {A}})~\left[I_{1}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})+2~I_{2}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha +3~I_{3}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha ^{2}\right]\right|_{\alpha =0}\\&=\det({\boldsymbol {A}})~I_{1}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~.\end{aligned}}$
* Recuerde que el invariante $I_{1}$ viene dado por
* $I_{1}({\boldsymbol {A}})={\text{tr}}{\boldsymbol {A}}.$
* Por eso,
* ${\frac {\partial f}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\det({\boldsymbol {A}})~{\text{tr}}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}:{\boldsymbol {T}}.$
* Invocando la arbitrariedad de ${\boldsymbol {T}}$ tenemos entonces
* ${\frac {\partial f}{\partial {\boldsymbol {A}}}}=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}~.$

## Derivadas de las invariantes de un tensor de segundo orden

Los principales invariantes de un tensor de segundo orden son
    ${\begin{aligned}I_{1}({\boldsymbol {A}})&={\text{tr}}{\boldsymbol {A}}\\I_{2}({\boldsymbol {A}})&={\frac {1}{2}}\left[({\text{tr}}{\boldsymbol {A}})^{2}-{\text{tr}}{{\boldsymbol {A}}^{2}}\right]\\I_{3}({\boldsymbol {A}})&=\det({\boldsymbol {A}})\end{aligned}}$
Las derivadas de estos tres invariantes con respecto a ${\boldsymbol {A}}$ son
    ${\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\[3pt]{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\\[3pt]{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}~\left(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\right)=\left({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}}\right)^{\textsf {T}}\end{aligned}}$
* Demostración
* De la derivada del determinante sabemos que
* ${\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}=(A)~[A^{-}1]^{T}~.$
* Para las derivadas de las otras dos invariantes, volvamos a la ecuación característica
* $\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})=\lambda ^{3}+I_{1}({\boldsymbol {A}})~\lambda ^{2}+I_{2}({\boldsymbol {A}})~\lambda +I_{3}({\boldsymbol {A}})~.$
* Utilizando el mismo enfoque que para el determinante de un tensor, podemos demostrar que
* ${\frac {\partial }{\partial {\boldsymbol {A}}}}\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~\left[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}\right]^{\textsf {T}}~.$
* Ahora el lado izquierdo se puede expandir como
* ${\begin{aligned}{\frac {\partial }{\partial {\boldsymbol {A}}}}\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})&={\frac {\partial }{\partial {\boldsymbol {A}}}}\left[\lambda ^{3}+I_{1}({\boldsymbol {A}})~\lambda ^{2}+I_{2}({\boldsymbol {A}})~\lambda +I_{3}({\boldsymbol {A}})\right]\\&={\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~.\end{aligned}}$
* Por eso
* ${\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~\left[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}\right]^{\textsf {T}}$
* o,
* $(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{\textsf {T}}\cdot \left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right]=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~{\boldsymbol {\mathit {1}}}~.$
* Expandir el lado derecho y separar términos en el lado izquierdo da
* $\left(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\right)\cdot \left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right]=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}$
* o,
* ${\begin{aligned}\left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}\right.&\left.+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~\lambda \right]{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\\&=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}~.\end{aligned}}$
* Si definimos $I_{0}:=1$ y $I_{4}:=0$, podemos escribir lo anterior como
* ${\begin{aligned}\left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}\right.&\left.+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{4}}{\partial {\boldsymbol {A}}}}\right]{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\\&=\left[I_{0}~\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}~.\end{aligned}}$
* Reuniendo términos que contienen varias potencias de λ, obtenemos
* ${\begin{aligned}\lambda ^{3}&\left(I_{0}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}\right)+\lambda ^{2}\left(I_{1}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}\right)+\\&\qquad \qquad \lambda \left(I_{2}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}\right)+\left(I_{3}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{4}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right)=0~.\end{aligned}}$
* Entonces, invocando la arbitrariedad de λ, tenemos
* ${\begin{aligned}I_{0}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}&=0\\I_{1}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-I_{2}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=0\\I_{3}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{4}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=0~.\end{aligned}}$
* Esto implica que
* ${\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\\{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}~\left(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\right)=\left({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}}\right)^{\textsf {T}}\end{aligned}}$

## Derivada del tensor de identidad de segundo orden

Sea ${\boldsymbol {\mathit {1}}}$ el tensor de identidad de segundo orden. Entonces la derivada de este tensor con respecto a un tensor de segundo orden ${\boldsymbol {A}}$ viene dada por
    ${\frac {\partial {\boldsymbol {\mathit {1}}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathsf {0}}}:{\boldsymbol {T}}={\boldsymbol {\mathit {0}}}$
Esto se debe a que ${\boldsymbol {\mathit {1}}}$ es independiente de ${\boldsymbol {A}}$.

## Derivada de un tensor de segundo orden con respecto a sí mismo

Sea ${\boldsymbol {A}}$ un tensor de segundo orden. Entonces
    ${\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\left[{\frac {\partial }{\partial \alpha }}({\boldsymbol {A}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}={\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}}$
Por lo tanto,
    ${\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}={\boldsymbol {\mathsf {I}}}$
Aquí ${\boldsymbol {\mathsf {I}}}$ es el tensor de identidad de cuarto orden. En notación de índice con respecto a una base ortonormal
    ${\boldsymbol {\mathsf {I}}}=\delta _{ik}~\delta _{jl}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}$
Este resultado implica que
    ${\frac {\partial {\boldsymbol {A}}^{\textsf {T}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}^{\textsf {T}}:{\boldsymbol {T}}={\boldsymbol {T}}^{\textsf {T}}$
dónde
    ${\boldsymbol {\mathsf {I}}}^{\textsf {T}}=\delta _{jk}~\delta _{il}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}$
Por lo tanto, si el tensor ${\boldsymbol {A}}$ es simétrico, entonces la derivada también es simétrica y obtenemos
    ${\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}={\boldsymbol {\mathsf {I}}}^{(s)}={\frac {1}{2}}~\left({\boldsymbol {\mathsf {I}}}+{\boldsymbol {\mathsf {I}}}^{\textsf {T}}\right)$
donde el tensor de identidad simétrico de cuarto orden es
    ${\boldsymbol {\mathsf {I}}}^{(s)}={\frac {1}{2}}~(\delta _{ik}~\delta _{jl}+\delta _{il}~\delta _{jk})~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}$

## Derivada de la inversa de un tensor de segundo orden

Sean ${\boldsymbol {A}}$ y ${\boldsymbol {T}}$ dos tensores de segundo orden, entonces
    ${\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-1}\right):{\boldsymbol {T}}=-{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\cdot {\boldsymbol {A}}^{-1}$
En notación de índice con respecto a una base ortonormal
    ${\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}~T_{kl}=-A_{ik}^{-1}~T_{kl}~A_{lj}^{-1}\implies {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}=-A_{ik}^{-1}~A_{lj}^{-1}$
También tenemos
    ${\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-{\textsf {T}}}\right):{\boldsymbol {T}}=-{\boldsymbol {A}}^{-{\textsf {T}}}\cdot {\boldsymbol {T}}^{\textsf {T}}\cdot {\boldsymbol {A}}^{-{\textsf {T}}}$
En notación de índice
    ${\frac {\partial A_{ji}^{-1}}{\partial A_{kl}}}~T_{kl}=-A_{jk}^{-1}~T_{lk}~A_{li}^{-1}\implies {\frac {\partial A_{ji}^{-1}}{\partial A_{kl}}}=-A_{li}^{-1}~A_{jk}^{-1}$
Si el tensor ${\boldsymbol {A}}$ es simétrico entonces
    ${\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}=-{\cfrac {1}{2}}\left(A_{ik}^{-1}~A_{jl}^{-1}+A_{il}^{-1}~A_{jk}^{-1}\right)$
* Demostración
* Recordar que
* ${\frac {\partial {\boldsymbol {\mathit {1}}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathit {0}}}$
* Desde ${\boldsymbol {A}}^{-1}\cdot {\boldsymbol {A}}={\boldsymbol {\mathit {1}}}$, podemos escribir
* ${\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {A}}\right):{\boldsymbol {T}}={\boldsymbol {\mathit {0}}}$
* Usando la regla del producto para tensores de segundo orden
* ${\frac {\partial }{\partial {\boldsymbol {S}}}}[{\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})]:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}_{2}+{\boldsymbol {F}}_{1}\cdot \left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)$
* obtenemos
* ${\frac {\partial }{\partial {\boldsymbol {A}}}}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {A}}):{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {A}}^{-1}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {A}}+{\boldsymbol {A}}^{-1}\cdot \left({\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}\right)={\boldsymbol {\mathit {0}}}$
* o,
* $\left({\frac {\partial {\boldsymbol {A}}^{-1}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {A}}=-{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}$
* Por lo tanto,
* ${\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-1}\right):{\boldsymbol {T}}=-{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\cdot {\boldsymbol {A}}^{-1}$

## Integración por partes

Otra operación importante relacionada con las derivadas tensoriales en la mecánica continua es la integración por partes. La fórmula de integración por partes se puede escribir como
    $\int _{\Omega }{\boldsymbol {F}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {G}}\,d\Omega =\int _{\Gamma }\mathbf {n} \otimes ({\boldsymbol {F}}\otimes {\boldsymbol {G}})\,d\Gamma -\int _{\Omega }{\boldsymbol {G}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {F}}\,d\Omega$
donde ${\boldsymbol {F}}$ y ${\boldsymbol {G}}$ son campos tensoriales diferenciables de orden arbitrario, $\mathbf {n}$ es la unidad normal hacia afuera al dominio sobre el cual se definen los campos tensoriales, $\otimes$ representa un operador de producto tensorial generalizado y ${\boldsymbol {\nabla }}$ es un operador de gradiente generalizado. Cuando ${\boldsymbol {F}}$ es igual al tensor de identidad, obtenemos teorema de la divergencia
    $\int _{\Omega }{\boldsymbol {\nabla }}{\boldsymbol {G}}\,d\Omega =\int _{\Gamma }\mathbf {n} \otimes {\boldsymbol {G}}\,d\Gamma \,.$
Podemos expresar la fórmula de integración por partes en notación de índice cartesiano como
    $\int _{\Omega }F_{ijk....}\,G_{lmn...,p}\,d\Omega =\int _{\Gamma }n_{p}\,F_{ijk...}\,G_{lmn...}\,d\Gamma -\int _{\Omega }G_{lmn...}\,F_{ijk...,p}\,d\Omega \,.$
Para el caso especial donde la operación del producto tensorial es una contracción de un índice y la operación del gradiente es una divergencia, y tanto ${\boldsymbol {F}}$ como ${\boldsymbol {G}}$ son tensores de segundo orden, tenemos
    $\int _{\Omega }{\boldsymbol {F}}\cdot ({\boldsymbol {\nabla }}\cdot {\boldsymbol {G}})\,d\Omega =\int _{\Gamma }\mathbf {n} \cdot \left({\boldsymbol {G}}\cdot {\boldsymbol {F}}^{\textsf {T}}\right)\,d\Gamma -\int _{\Omega }({\boldsymbol {\nabla }}{\boldsymbol {F}}):{\boldsymbol {G}}^{\textsf {T}}\,d\Omega \,.$
En notación de índice,
    $\int _{\Omega }F_{ij}\,G_{pj,p}\,d\Omega =\int _{\Gamma }n_{p}\,F_{ij}\,G_{pj}\,d\Gamma -\int _{\Omega }G_{pj}\,F_{ij,p}\,d\Omega \,.$