The derivatives of scalars , vectors , and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics . These derivatives are used in the theories of nonlinear elasticity and plasticity , particularly in the design of algorithms for numerical simulations .[ 1]
The directional derivative provides a systematic way of finding these derivatives.[ 2]
Derivatives with respect to vectors and second-order tensors
The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.
Derivatives of scalar valued functions of vectors
Let f (v ) be a real valued function of the vector v . Then the derivative of f (v ) with respect to v (or at v ) in the direction u is the vector defined as
∂
f
∂
v
⋅
u
=
D
f
(
v
)
[
u
]
=
[
d
d
α
f
(
v
+
α
u
)
]
α
=
0
{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =Df(\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~f(\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}}
for all vectors u .
Properties:
1) If
f
(
v
)
=
f
1
(
v
)
+
f
2
(
v
)
{\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )+f_{2}(\mathbf {v} )}
∂
f
∂
v
⋅
u
=
(
∂
f
1
∂
v
+
∂
f
2
∂
v
)
⋅
u
{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}+{\frac {\partial f_{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} }
2) If
f
(
v
)
=
f
1
(
v
)
f
2
(
v
)
{\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )~f_{2}(\mathbf {v} )}
∂
f
∂
v
⋅
u
=
(
∂
f
1
∂
v
⋅
u
)
f
2
(
v
)
+
f
1
(
v
)
(
∂
f
2
∂
v
⋅
u
)
{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)~f_{2}(\mathbf {v} )+f_{1}(\mathbf {v} )~\left({\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}
3) If
f
(
v
)
=
f
1
(
f
2
(
v
)
)
{\displaystyle f(\mathbf {v} )=f_{1}(f_{2}(\mathbf {v} ))}
∂
f
∂
v
⋅
u
=
∂
f
1
∂
f
2
∂
f
2
∂
v
⋅
u
{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial f_{1}}{\partial f_{2}}}~{\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} }
Derivatives of vector valued functions of vectors
Let f (v ) be a vector valued function of the vector v . Then the derivative of f (v ) with respect to v (or at v ) in the direction u is the second order tensor defined as
∂
f
∂
v
⋅
u
=
D
f
(
v
)
[
u
]
=
[
d
d
α
f
(
v
+
α
u
)
]
α
=
0
{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =D\mathbf {f} (\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~\mathbf {f} (\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}}
for all vectors u .
Properties: 1) If
f
(
v
)
=
f
1
(
v
)
+
f
2
(
v
)
{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )+\mathbf {f} _{2}(\mathbf {v} )}
∂
f
∂
v
⋅
u
=
(
∂
f
1
∂
v
+
∂
f
2
∂
v
)
⋅
u
{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}+{\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} }
2) If
f
(
v
)
=
f
1
(
v
)
×
f
2
(
v
)
{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )\times \mathbf {f} _{2}(\mathbf {v} )}
∂
f
∂
v
⋅
u
=
(
∂
f
1
∂
v
⋅
u
)
×
f
2
(
v
)
+
f
1
(
v
)
×
(
∂
f
2
∂
v
⋅
u
)
{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)\times \mathbf {f} _{2}(\mathbf {v} )+\mathbf {f} _{1}(\mathbf {v} )\times \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}
3) If
f
(
v
)
=
f
1
(
f
2
(
v
)
)
{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {f} _{2}(\mathbf {v} ))}
∂
f
∂
v
⋅
u
=
∂
f
1
∂
f
2
⋅
(
∂
f
2
∂
v
⋅
u
)
{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {f} _{2}}}\cdot \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)}
Derivatives of scalar valued functions of second-order tensors
Let
f
(
S
)
{\displaystyle f({\boldsymbol {S}})}
S
{\displaystyle {\boldsymbol {S}}}
f
(
S
)
{\displaystyle f({\boldsymbol {S}})}
S
{\displaystyle {\boldsymbol {S}}}
S
{\displaystyle {\boldsymbol {S}}}
T
{\displaystyle {\boldsymbol {T}}}
second order tensor defined as
∂
f
∂
S
:
T
=
D
f
(
S
)
[
T
]
=
[
d
d
α
f
(
S
+
α
T
)
]
α
=
0
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=Df({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~f({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}}
for all second order tensors
T
{\displaystyle {\boldsymbol {T}}}
Properties: 1) If
f
(
S
)
=
f
1
(
S
)
+
f
2
(
S
)
{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})+f_{2}({\boldsymbol {S}})}
∂
f
∂
S
:
T
=
(
∂
f
1
∂
S
+
∂
f
2
∂
S
)
:
T
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}}
2) If
f
(
S
)
=
f
1
(
S
)
f
2
(
S
)
{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})~f_{2}({\boldsymbol {S}})}
∂
f
∂
S
:
T
=
(
∂
f
1
∂
S
:
T
)
f
2
(
S
)
+
f
1
(
S
)
(
∂
f
2
∂
S
:
T
)
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)~f_{2}({\boldsymbol {S}})+f_{1}({\boldsymbol {S}})~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
3) If
f
(
S
)
=
f
1
(
f
2
(
S
)
)
{\displaystyle f({\boldsymbol {S}})=f_{1}(f_{2}({\boldsymbol {S}}))}
∂
f
∂
S
:
T
=
∂
f
1
∂
f
2
(
∂
f
2
∂
S
:
T
)
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial f_{2}}}~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
Derivatives of tensor valued functions of second-order tensors
Let
F
(
S
)
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}
S
{\displaystyle {\boldsymbol {S}}}
F
(
S
)
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}
S
{\displaystyle {\boldsymbol {S}}}
S
{\displaystyle {\boldsymbol {S}}}
T
{\displaystyle {\boldsymbol {T}}}
fourth order tensor defined as
∂
F
∂
S
:
T
=
D
F
(
S
)
[
T
]
=
[
d
d
α
F
(
S
+
α
T
)
]
α
=
0
{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=D{\boldsymbol {F}}({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~{\boldsymbol {F}}({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}}
for all second order tensors
T
{\displaystyle {\boldsymbol {T}}}
Properties: 1) If
F
(
S
)
=
F
1
(
S
)
+
F
2
(
S
)
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})+{\boldsymbol {F}}_{2}({\boldsymbol {S}})}
∂
F
∂
S
:
T
=
(
∂
F
1
∂
S
+
∂
F
2
∂
S
)
:
T
{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}}
2) If
F
(
S
)
=
F
1
(
S
)
⋅
F
2
(
S
)
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})}
∂
F
∂
S
:
T
=
(
∂
F
1
∂
S
:
T
)
⋅
F
2
(
S
)
+
F
1
(
S
)
⋅
(
∂
F
2
∂
S
:
T
)
{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})+{\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot \left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
3) If
F
(
S
)
=
F
1
(
F
2
(
S
)
)
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))}
∂
F
∂
S
:
T
=
∂
F
1
∂
F
2
:
(
∂
F
2
∂
S
:
T
)
{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
4) If
f
(
S
)
=
f
1
(
F
2
(
S
)
)
{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))}
∂
f
∂
S
:
T
=
∂
f
1
∂
F
2
:
(
∂
F
2
∂
S
:
T
)
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}
Gradient of a tensor field
The gradient ,
∇
T
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}}
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
c is defined as:
∇
T
⋅
c
=
lim
α
→
0
d
d
α
T
(
x
+
α
c
)
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}\cdot \mathbf {c} =\lim _{\alpha \rightarrow 0}\quad {\cfrac {d}{d\alpha }}~{\boldsymbol {T}}(\mathbf {x} +\alpha \mathbf {c} )}
The gradient of a tensor field of order n is a tensor field of order n +1.
Cartesian coordinates
If
e
1
,
e
2
,
e
3
{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}
Cartesian coordinate system, with coordinates of points denoted by (
x
1
,
x
2
,
x
3
{\displaystyle x_{1},x_{2},x_{3}}
T
{\displaystyle {\boldsymbol {T}}}
∇
T
=
e
i
⊗
∂
T
∂
x
i
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}=\mathbf {e} _{i}\otimes {\cfrac {\partial {\boldsymbol {T}}}{\partial x_{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
ϕ
{\displaystyle \phi }
v , and a second-order tensor field
S
{\displaystyle {\boldsymbol {S}}}
∇
ϕ
=
∂
ϕ
∂
x
i
e
i
∇
v
=
∂
(
v
j
e
j
)
∂
x
i
⊗
e
i
=
∂
v
j
∂
x
i
e
j
⊗
e
i
∇
S
=
∂
(
S
j
k
e
j
⊗
e
k
)
∂
x
i
⊗
e
i
=
∂
S
j
k
∂
x
i
e
j
⊗
e
k
⊗
e
i
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial x_{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}\\{\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}\end{aligned}}}
Curvilinear coordinates
If
g
1
,
g
2
,
g
3
{\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}}
contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (
ξ
1
,
ξ
2
,
ξ
3
{\displaystyle \xi ^{1},\xi ^{2},\xi ^{3}}
T
{\displaystyle {\boldsymbol {T}}}
[ 3]
∇
T
=
∂
T
∂
ξ
i
⊗
g
i
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}={\cfrac {\partial {\boldsymbol {T}}}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}}
From this definition we have the following relations for the gradients of a scalar field
ϕ
{\displaystyle \phi }
v , and a second-order tensor field
S
{\displaystyle {\boldsymbol {S}}}
∇
ϕ
=
∂
ϕ
∂
ξ
i
g
i
∇
v
=
∂
(
v
j
g
j
)
∂
ξ
i
⊗
g
i
=
(
∂
v
j
∂
ξ
i
+
v
k
Γ
i
k
j
)
g
j
⊗
g
i
=
(
∂
v
j
∂
ξ
i
−
v
k
Γ
i
j
k
)
g
j
⊗
g
i
∇
S
=
∂
(
S
j
k
g
j
⊗
g
k
)
∂
ξ
i
⊗
g
i
=
(
∂
S
j
k
∂
ξ
i
−
S
l
k
Γ
i
j
l
−
S
j
l
Γ
i
k
l
)
g
j
⊗
g
k
⊗
g
i
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial \xi ^{i}}}~\mathbf {g} ^{i}\\{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial (v^{j}\mathbf {g} _{j})}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\cfrac {\partial v^{j}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{j}\right)~\mathbf {g} _{j}\otimes \mathbf {g} ^{i}=\left({\cfrac {\partial v_{j}}{\partial \xi ^{i}}}-v_{k}~\Gamma _{ij}^{k}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{i}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\cfrac {\partial (S_{jk}~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k})}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\cfrac {\partial S_{jk}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ij}^{l}-S_{jl}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k}\otimes \mathbf {g} ^{i}\end{aligned}}}
where the Christoffel symbol
Γ
i
j
k
{\displaystyle \Gamma _{ij}^{k}}
Γ
i
j
k
g
k
=
∂
g
i
∂
ξ
j
⟹
Γ
i
j
k
=
∂
g
i
∂
ξ
j
⋅
g
k
=
−
g
i
⋅
∂
g
k
∂
ξ
j
{\displaystyle \Gamma _{ij}^{k}~\mathbf {g} _{k}={\cfrac {\partial \mathbf {g} _{i}}{\partial \xi ^{j}}}\quad \implies \quad \Gamma _{ij}^{k}={\cfrac {\partial \mathbf {g} _{i}}{\partial \xi ^{j}}}\cdot \mathbf {g} _{k}=-\mathbf {g} _{i}\cdot {\cfrac {\partial \mathbf {g} ^{k}}{\partial \xi ^{j}}}}
Cylindrical polar coordinates
In cylindrical coordinates , the gradient is given by
∇
ϕ
=
∂
ϕ
∂
r
e
r
+
1
r
∂
ϕ
∂
θ
e
θ
+
∂
ϕ
∂
z
e
z
∇
v
=
∂
v
r
∂
r
e
r
⊗
e
r
+
1
r
(
∂
v
r
∂
θ
−
v
θ
)
e
r
⊗
e
θ
+
∂
v
r
∂
z
e
r
⊗
e
z
+
∂
v
θ
∂
r
e
θ
⊗
e
r
+
1
r
(
∂
v
θ
∂
θ
+
v
r
)
e
θ
⊗
e
θ
+
∂
v
θ
∂
z
e
θ
⊗
e
z
+
∂
v
z
∂
r
e
z
⊗
e
r
+
1
r
∂
v
z
∂
θ
e
z
⊗
e
θ
+
∂
v
z
∂
z
e
z
⊗
e
z
∇
S
=
∂
S
r
r
∂
r
e
r
⊗
e
r
⊗
e
r
+
1
r
[
∂
S
r
r
∂
θ
−
(
S
θ
r
+
S
r
θ
)
]
e
r
⊗
e
r
⊗
e
θ
+
∂
S
r
r
∂
z
e
r
⊗
e
r
⊗
e
z
+
∂
S
r
θ
∂
r
e
r
⊗
e
θ
⊗
e
r
+
1
r
[
∂
S
r
θ
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
r
⊗
e
θ
⊗
e
θ
+
∂
S
r
θ
∂
z
e
r
⊗
e
θ
⊗
e
z
+
∂
S
r
z
∂
r
e
r
⊗
e
z
⊗
e
r
+
1
r
[
∂
S
r
z
∂
θ
−
S
θ
z
]
e
r
⊗
e
z
⊗
e
θ
+
∂
S
r
z
∂
z
e
r
⊗
e
z
⊗
e
z
+
∂
S
θ
r
∂
r
e
θ
⊗
e
r
⊗
e
r
+
1
r
[
∂
S
θ
r
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
θ
⊗
e
r
⊗
e
θ
+
∂
S
θ
r
∂
z
e
θ
⊗
e
r
⊗
e
z
+
∂
S
θ
θ
∂
r
e
θ
⊗
e
θ
⊗
e
r
+
1
r
[
∂
S
θ
θ
∂
θ
+
(
S
r
θ
+
S
θ
r
)
]
e
θ
⊗
e
θ
⊗
e
θ
+
∂
S
θ
θ
∂
z
e
θ
⊗
e
θ
⊗
e
z
+
∂
S
θ
z
∂
r
e
θ
⊗
e
z
⊗
e
r
+
1
r
[
∂
S
θ
z
∂
θ
+
S
r
z
]
e
θ
⊗
e
z
⊗
e
θ
+
∂
S
θ
z
∂
z
e
θ
⊗
e
z
⊗
e
z
+
∂
S
z
r
∂
r
e
z
⊗
e
r
⊗
e
r
+
1
r
[
∂
S
z
r
∂
θ
−
S
z
θ
]
e
z
⊗
e
r
⊗
e
θ
+
∂
S
z
r
∂
z
e
z
⊗
e
r
⊗
e
z
+
∂
S
z
θ
∂
r
e
z
⊗
e
θ
⊗
e
r
+
1
r
[
∂
S
z
θ
∂
θ
+
S
z
r
]
e
z
⊗
e
θ
⊗
e
θ
+
∂
S
z
θ
∂
z
e
z
⊗
e
θ
⊗
e
z
+
∂
S
z
z
∂
r
e
z
⊗
e
z
⊗
e
r
+
1
r
∂
S
z
z
∂
θ
e
z
⊗
e
z
⊗
e
θ
+
∂
S
z
z
∂
z
e
z
⊗
e
z
⊗
e
z
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial r}}~\mathbf {e} _{r}+{\cfrac {1}{r}}~{\cfrac {\partial \phi }{\partial \theta }}~\mathbf {e} _{\theta }+{\cfrac {\partial \phi }{\partial z}}~\mathbf {e} _{z}\\{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{r}}{\partial \theta }}-v_{\theta }\right)~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{r}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\cfrac {\partial v_{\theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\&\quad +{\cfrac {\partial v_{\theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\cfrac {\partial v_{z}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}{\cfrac {\partial v_{z}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{z}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rr}}{\partial \theta }}-(S_{\theta r}+S_{r\theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rr}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\\&\quad +{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{r\theta }}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rz}}{\partial \theta }}-S_{\theta z}\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\\&\quad +{\frac {\partial S_{rz}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {\partial S_{\theta r}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta r}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\&\quad +{\frac {\partial S_{\theta \theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta \theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {\partial S_{\theta z}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}\\&\quad +{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta z}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\&\quad +{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {\partial S_{z\theta }}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{z\theta }}{\partial \theta }}+S_{zr}\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\&\quad +{\frac {\partial S_{zz}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}~{\frac {\partial S_{zz}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}}
Divergence of a tensor field
The divergence of a tensor field
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
(
∇
⋅
T
)
⋅
c
=
∇
⋅
(
c
⋅
T
T
)
;
∇
⋅
v
=
tr
(
∇
v
)
{\displaystyle ({\boldsymbol {\nabla }}\cdot {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot (\mathbf {c} \cdot {\boldsymbol {T}}^{T})~;\qquad {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\text{tr}}({\boldsymbol {\nabla }}\mathbf {v} )}
where c is an arbitrary constant vector and v is a vector field. If
T
{\displaystyle {\boldsymbol {T}}}
n > 1 then the divergence of the field is a tensor of order n −1.
Cartesian coordinates
In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field
S
{\displaystyle {\boldsymbol {S}}}
∇
⋅
v
=
∂
v
i
∂
x
i
∇
⋅
S
=
∂
S
k
i
∂
x
k
e
i
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\cfrac {\partial v_{i}}{\partial x_{i}}}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\cfrac {\partial S_{ki}}{\partial x_{k}}}~\mathbf {e} _{i}\end{aligned}}}
Note that last relation can be found in reference [ 4]
∇
⋅
S
≠
div
S
=
∇
⋅
S
T
.
{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}\neq \operatorname {div} {\boldsymbol {S}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}^{\mathrm {T} }.}
Curvilinear coordinates
In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field
S
{\displaystyle {\boldsymbol {S}}}
∇
⋅
v
=
(
∂
v
i
∂
ξ
i
+
v
k
Γ
i
k
i
)
∇
⋅
S
=
(
∂
S
i
k
∂
ξ
i
−
S
l
k
Γ
i
i
l
−
S
i
l
Γ
i
k
l
)
g
k
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &=\left({\cfrac {\partial v^{i}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{i}\right)\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left({\cfrac {\partial S_{ik}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ii}^{l}-S_{il}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{k}\end{aligned}}}
Cylindrical polar coordinates
In cylindrical polar coordinates
∇
⋅
v
=
∂
v
r
∂
r
+
1
r
(
∂
v
θ
∂
θ
+
v
r
)
+
∂
v
z
∂
z
∇
⋅
S
=
∂
S
r
r
∂
r
e
r
+
∂
S
r
θ
∂
r
e
θ
+
∂
S
r
z
∂
r
e
z
+
1
r
[
∂
S
θ
r
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
r
+
1
r
[
∂
S
θ
θ
∂
θ
+
(
S
r
θ
+
S
θ
r
)
]
e
θ
+
1
r
[
∂
S
θ
z
∂
θ
+
S
r
z
]
e
z
+
∂
S
z
r
∂
z
e
r
+
∂
S
z
θ
∂
z
e
θ
+
∂
S
z
z
∂
z
e
z
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)+{\cfrac {\partial v_{z}}{\partial z}}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{z}\\&+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{z}\\&+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\end{aligned}}}
Curl of a tensor field
The curl of an order-n > 1 tensor field
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
(
∇
×
T
)
⋅
c
=
∇
×
(
c
⋅
T
)
;
(
∇
×
v
)
⋅
c
=
∇
⋅
(
v
×
c
)
{\displaystyle ({\boldsymbol {\nabla }}\times {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {T}})~;\qquad ({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )}
where c is an arbitrary constant vector and v is a vector field.
Consider a vector field v and an arbitrary constant vector c . In index notation, the cross product is given by
v
×
c
=
e
i
j
k
v
j
c
k
e
i
{\displaystyle \mathbf {v} \times \mathbf {c} =e_{ijk}~v_{j}~c_{k}~\mathbf {e} _{i}}
where
e
i
j
k
{\displaystyle e_{ijk}}
permutation symbol . Then,
∇
⋅
(
v
×
c
)
=
e
i
j
k
v
j
,
i
c
k
=
(
e
i
j
k
v
j
,
i
e
k
)
⋅
c
=
(
∇
×
v
)
⋅
c
{\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )=e_{ijk}~v_{j,i}~c_{k}=(e_{ijk}~v_{j,i}~\mathbf {e} _{k})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c} }
Therefore,
∇
×
v
=
e
i
j
k
v
j
,
i
e
k
{\displaystyle {\boldsymbol {\nabla }}\times \mathbf {v} =e_{ijk}~v_{j,i}~\mathbf {e} _{k}}
Curl of a second-order tensor field
For a second-order tensor
S
{\displaystyle {\boldsymbol {S}}}
c
⋅
S
=
c
m
S
m
j
e
j
{\displaystyle \mathbf {c} \cdot {\boldsymbol {S}}=c_{m}~S_{mj}~\mathbf {e} _{j}}
Hence, using the definition of the curl of a first-order tensor field,
∇
×
(
c
⋅
S
)
=
e
i
j
k
c
m
S
m
j
,
i
e
k
=
(
e
i
j
k
S
m
j
,
i
e
k
⊗
e
m
)
⋅
c
=
(
∇
×
S
)
⋅
c
{\displaystyle {\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {S}})=e_{ijk}~c_{m}~S_{mj,i}~\mathbf {e} _{k}=(e_{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times {\boldsymbol {S}})\cdot \mathbf {c} }
Therefore, we have
∇
×
S
=
e
i
j
k
S
m
j
,
i
e
k
⊗
e
m
{\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {S}}=e_{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m}}
Identities involving the curl of a tensor field
The most commonly used identity involving the curl of a tensor field,
T
{\displaystyle {\boldsymbol {T}}}
∇
×
(
∇
T
)
=
0
{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}{\boldsymbol {T}})={\boldsymbol {0}}}
This identity hold for tensor fields of all orders. For the important case of a second-order tensor,
S
{\displaystyle {\boldsymbol {S}}}
∇
×
S
=
0
⟹
S
m
i
,
j
−
S
m
j
,
i
=
0
{\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {S}}={\boldsymbol {0}}\quad \implies \quad S_{mi,j}-S_{mj,i}=0}
Derivative of the determinant of a second-order tensor
The derivative of the determinant of a second order tensor
A
{\displaystyle {\boldsymbol {A}}}
∂
∂
A
det
(
A
)
=
det
(
A
)
[
A
−
1
]
T
.
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det({\boldsymbol {A}})=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}~.}
In an orthonormal basis, the components of
A
{\displaystyle {\boldsymbol {A}}}
A . In that case, the right hand side corresponds the cofactors of the matrix.
Derivatives of the invariants of a second-order tensor
The principal invariants of a second order tensor are
I
1
(
A
)
=
tr
A
I
2
(
A
)
=
1
2
[
(
tr
A
)
2
−
tr
A
2
]
I
3
(
A
)
=
det
(
A
)
{\displaystyle {\begin{aligned}I_{1}({\boldsymbol {A}})&={\text{tr}}{\boldsymbol {A}}\\I_{2}({\boldsymbol {A}})&={\frac {1}{2}}\left[({\text{tr}}{\boldsymbol {A}})^{2}-{\text{tr}}{{\boldsymbol {A}}^{2}}\right]\\I_{3}({\boldsymbol {A}})&=\det({\boldsymbol {A}})\end{aligned}}}
The derivatives of these three invariants with respect to
A
{\displaystyle {\boldsymbol {A}}}
∂
I
1
∂
A
=
1
∂
I
2
∂
A
=
I
1
1
−
A
T
∂
I
3
∂
A
=
det
(
A
)
[
A
−
1
]
T
=
I
2
1
−
A
T
(
I
1
1
−
A
T
)
=
(
A
2
−
I
1
A
+
I
2
1
)
T
{\displaystyle {\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T}\\{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T}~(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T})=({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}})^{T}\end{aligned}}}
Proof
From the derivative of the determinant we know that
∂
I
3
∂
A
=
det
(
A
)
[
A
−
1
]
T
.
{\displaystyle {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}~.}
For the derivatives of the other two invariants, let us go back to the characteristic equation
det
(
λ
1
+
A
)
=
λ
3
+
I
1
(
A
)
λ
2
+
I
2
(
A
)
λ
+
I
3
(
A
)
.
{\displaystyle \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
∂
∂
A
det
(
λ
1
+
A
)
=
det
(
λ
1
+
A
)
[
(
λ
1
+
A
)
−
1
]
T
.
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}]^{T}~.}
Now the left hand side can be expanded as
∂
∂
A
det
(
λ
1
+
A
)
=
∂
∂
A
[
λ
3
+
I
1
(
A
)
λ
2
+
I
2
(
A
)
λ
+
I
3
(
A
)
]
=
∂
I
1
∂
A
λ
2
+
∂
I
2
∂
A
λ
+
∂
I
3
∂
A
.
{\displaystyle {\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}}}
Hence
∂
I
1
∂
A
λ
2
+
∂
I
2
∂
A
λ
+
∂
I
3
∂
A
=
det
(
λ
1
+
A
)
[
(
λ
1
+
A
)
−
1
]
T
{\displaystyle {\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}})~[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}]^{T}}
or,
(
λ
1
+
A
)
T
⋅
[
∂
I
1
∂
A
λ
2
+
∂
I
2
∂
A
λ
+
∂
I
3
∂
A
]
=
det
(
λ
1
+
A
)
1
.
{\displaystyle (\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{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
(
λ
1
+
A
T
)
⋅
[
∂
I
1
∂
A
λ
2
+
∂
I
2
∂
A
λ
+
∂
I
3
∂
A
]
=
[
λ
3
+
I
1
λ
2
+
I
2
λ
+
I
3
]
1
{\displaystyle (\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{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]=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}}
or,
[
∂
I
1
∂
A
λ
3
+
∂
I
2
∂
A
λ
2
+
∂
I
3
∂
A
λ
]
1
+
A
T
⋅
∂
I
1
∂
A
λ
2
+
A
T
⋅
∂
I
2
∂
A
λ
+
A
T
⋅
∂
I
3
∂
A
=
[
λ
3
+
I
1
λ
2
+
I
2
λ
+
I
3
]
1
.
{\displaystyle {\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}}^{T}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{T}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{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}}}
If we define
I
0
:=
1
{\displaystyle I_{0}:=1}
I
4
:=
0
{\displaystyle I_{4}:=0}
[
∂
I
1
∂
A
λ
3
+
∂
I
2
∂
A
λ
2
+
∂
I
3
∂
A
λ
+
∂
I
4
∂
A
]
1
+
A
T
⋅
∂
I
0
∂
A
λ
3
+
A
T
⋅
∂
I
1
∂
A
λ
2
+
A
T
⋅
∂
I
2
∂
A
λ
+
A
T
⋅
∂
I
3
∂
A
=
[
I
0
λ
3
+
I
1
λ
2
+
I
2
λ
+
I
3
]
1
.
{\displaystyle {\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}}^{T}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}+{\boldsymbol {A}}^{T}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{T}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{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}}}
Collecting terms containing various powers of λ, we get
λ
3
(
I
0
1
−
∂
I
1
∂
A
1
−
A
T
⋅
∂
I
0
∂
A
)
+
λ
2
(
I
1
1
−
∂
I
2
∂
A
1
−
A
T
⋅
∂
I
1
∂
A
)
+
λ
(
I
2
1
−
∂
I
3
∂
A
1
−
A
T
⋅
∂
I
2
∂
A
)
+
(
I
3
1
−
∂
I
4
∂
A
1
−
A
T
⋅
∂
I
3
∂
A
)
=
0
.
{\displaystyle {\begin{aligned}\lambda ^{3}&\left(I_{0}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{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}}^{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}}^{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}}^{T}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right)=0~.\end{aligned}}}
Then, invoking the arbitrariness of λ, we have
I
0
1
−
∂
I
1
∂
A
1
−
A
T
⋅
∂
I
0
∂
A
=
0
I
1
1
−
∂
I
2
∂
A
1
−
I
2
1
−
∂
I
3
∂
A
1
−
A
T
⋅
∂
I
2
∂
A
=
0
I
3
1
−
∂
I
4
∂
A
1
−
A
T
⋅
∂
I
3
∂
A
=
0
.
{\displaystyle {\begin{aligned}I_{0}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{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}}^{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}}^{T}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=0~.\end{aligned}}}
This implies that
∂
I
1
∂
A
=
1
∂
I
2
∂
A
=
I
1
1
−
A
T
∂
I
3
∂
A
=
I
2
1
−
A
T
(
I
1
1
−
A
T
)
=
(
A
2
−
I
1
A
+
I
2
1
)
T
{\displaystyle {\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T}\\{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T}~(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T})=({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}})^{T}\end{aligned}}}
Derivative of the second-order identity tensor
Let
1
{\displaystyle {\boldsymbol {\mathit {1}}}}
A
{\displaystyle {\boldsymbol {A}}}
∂
1
∂
A
:
T
=
0
:
T
=
0
{\displaystyle {\frac {\partial {\boldsymbol {\mathit {1}}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathsf {0}}}:{\boldsymbol {T}}={\boldsymbol {\mathit {0}}}}
This is because
1
{\displaystyle {\boldsymbol {\mathit {1}}}}
A
{\displaystyle {\boldsymbol {A}}}
Derivative of a second-order tensor with respect to itself
Let
A
{\displaystyle {\boldsymbol {A}}}
∂
A
∂
A
:
T
=
[
∂
∂
α
(
A
+
α
T
)
]
α
=
0
=
T
=
I
:
T
{\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\left[{\frac {\partial }{\partial \alpha }}({\boldsymbol {A}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}={\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}}}
Therefore,
∂
A
∂
A
=
I
{\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}={\boldsymbol {\mathsf {I}}}}
Here
I
{\displaystyle {\boldsymbol {\mathsf {I}}}}
I
=
δ
i
k
δ
j
l
e
i
⊗
e
j
⊗
e
k
⊗
e
l
{\displaystyle {\boldsymbol {\mathsf {I}}}=\delta _{ik}~\delta _{jl}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}
This result implies that
∂
A
T
∂
A
:
T
=
I
T
:
T
=
T
T
{\displaystyle {\frac {\partial {\boldsymbol {A}}^{T}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}^{T}:{\boldsymbol {T}}={\boldsymbol {T}}^{T}}
where
I
T
=
δ
j
k
δ
i
l
e
i
⊗
e
j
⊗
e
k
⊗
e
l
{\displaystyle {\boldsymbol {\mathsf {I}}}^{T}=\delta _{jk}~\delta _{il}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}
Therefore, if the tensor
A
{\displaystyle {\boldsymbol {A}}}
∂
A
∂
A
=
I
(
s
)
=
1
2
(
I
+
I
T
)
{\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}={\boldsymbol {\mathsf {I}}}^{(s)}={\frac {1}{2}}~({\boldsymbol {\mathsf {I}}}+{\boldsymbol {\mathsf {I}}}^{T})}
where the symmetric fourth order identity tensor is
I
(
s
)
=
1
2
(
δ
i
k
δ
j
l
+
δ
i
l
δ
j
k
)
e
i
⊗
e
j
⊗
e
k
⊗
e
l
{\displaystyle {\boldsymbol {\mathsf {I}}}^{(s)}={\frac {1}{2}}~(\delta _{ik}~\delta _{jl}+\delta _{il}~\delta _{jk})~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}
Derivative of the inverse of a second-order tensor
Let
A
{\displaystyle {\boldsymbol {A}}}
T
{\displaystyle {\boldsymbol {T}}}
∂
∂
A
(
A
−
1
)
:
T
=
−
A
−
1
⋅
T
⋅
A
−
1
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-1}\right):{\boldsymbol {T}}=-{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\cdot {\boldsymbol {A}}^{-1}}
In index notation with respect to an orthonormal basis
∂
A
i
j
−
1
∂
A
k
l
T
k
l
=
−
A
i
k
−
1
T
k
l
A
l
j
−
1
⟹
∂
A
i
j
−
1
∂
A
k
l
=
−
A
i
k
−
1
A
l
j
−
1
{\displaystyle {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}~T_{kl}=-A_{ik}^{-1}~T_{kl}~A_{lj}^{-1}\implies {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}=-A_{ik}^{-1}~A_{lj}^{-1}}
We also have
∂
∂
A
(
A
−
T
)
:
T
=
−
A
−
T
⋅
T
T
⋅
A
−
T
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-T}\right):{\boldsymbol {T}}=-{\boldsymbol {A}}^{-T}\cdot {\boldsymbol {T}}^{T}\cdot {\boldsymbol {A}}^{-T}}
In index notation
∂
A
j
i
−
1
∂
A
k
l
T
k
l
=
−
A
j
k
−
1
T
l
k
A
l
i
−
1
⟹
∂
A
j
i
−
1
∂
A
k
l
=
−
A
l
i
−
1
A
j
k
−
1
{\displaystyle {\frac {\partial A_{ji}^{-1}}{\partial A_{kl}}}~T_{kl}=-A_{jk}^{-1}~T_{lk}~A_{li}^{-1}\implies {\frac {\partial A_{ji}^{-1}}{\partial A_{kl}}}=-A_{li}^{-1}~A_{jk}^{-1}}
If the tensor
A
{\displaystyle {\boldsymbol {A}}}
∂
A
i
j
−
1
∂
A
k
l
=
−
1
2
(
A
i
k
−
1
A
j
l
−
1
+
A
i
l
−
1
A
j
k
−
1
)
{\displaystyle {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}=-{\cfrac {1}{2}}\left(A_{ik}^{-1}~A_{jl}^{-1}+A_{il}^{-1}~A_{jk}^{-1}\right)}
Proof
Recall that
∂
1
∂
A
:
T
=
0
{\displaystyle {\frac {\partial {\boldsymbol {\mathit {1}}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathit {0}}}}
Since
A
−
1
⋅
A
=
1
{\displaystyle {\boldsymbol {A}}^{-1}\cdot {\boldsymbol {A}}={\boldsymbol {\mathit {1}}}}
∂
∂
A
(
A
−
1
⋅
A
)
:
T
=
0
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {A}}):{\boldsymbol {T}}={\boldsymbol {\mathit {0}}}}
Using the product rule for second order tensors
∂
∂
S
[
F
1
(
S
)
⋅
F
2
(
S
)
]
:
T
=
(
∂
F
1
∂
S
:
T
)
⋅
F
2
+
F
1
⋅
(
∂
F
2
∂
S
:
T
)
{\displaystyle {\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
∂
∂
A
(
A
−
1
⋅
A
)
:
T
=
(
∂
A
−
1
∂
A
:
T
)
⋅
A
+
A
−
1
⋅
(
∂
A
∂
A
:
T
)
=
0
{\displaystyle {\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,
(
∂
A
−
1
∂
A
:
T
)
⋅
A
=
−
A
−
1
⋅
T
{\displaystyle \left({\frac {\partial {\boldsymbol {A}}^{-1}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {A}}=-{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}}
Therefore,
∂
∂
A
(
A
−
1
)
:
T
=
−
A
−
1
⋅
T
⋅
A
−
1
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\left({\boldsymbol {A}}^{-1}\right):{\boldsymbol {T}}=-{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\cdot {\boldsymbol {A}}^{-1}}
Integration by parts
Domain
Ω
{\displaystyle \Omega }
Γ
{\displaystyle \Gamma }
n
{\displaystyle \mathbf {n} }
Another important operation related to tensor derivatives in continuum mechanics is integration by parts. The formula for integration by parts can be written as
∫
Ω
F
⊗
∇
G
d
Ω
=
∫
Γ
n
⊗
(
F
⊗
G
)
d
Γ
−
∫
Ω
G
⊗
∇
F
d
Ω
{\displaystyle \int _{\Omega }{\boldsymbol {F}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {G}}\,{\rm {d}}\Omega =\int _{\Gamma }\mathbf {n} \otimes ({\boldsymbol {F}}\otimes {\boldsymbol {G}})\,{\rm {d}}\Gamma -\int _{\Omega }{\boldsymbol {G}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {F}}\,{\rm {d}}\Omega }
where
F
{\displaystyle {\boldsymbol {F}}}
G
{\displaystyle {\boldsymbol {G}}}
n
{\displaystyle \mathbf {n} }
⊗
{\displaystyle \otimes }
∇
{\displaystyle {\boldsymbol {\nabla }}}
F
{\displaystyle {\boldsymbol {F}}}
divergence theorem
∫
Ω
∇
G
d
Ω
=
∫
Γ
n
⊗
G
d
Γ
.
{\displaystyle \int _{\Omega }{\boldsymbol {\nabla }}{\boldsymbol {G}}\,{\rm {d}}\Omega =\int _{\Gamma }\mathbf {n} \otimes {\boldsymbol {G}}\,{\rm {d}}\Gamma \,.}
We can express the formula for integration by parts in Cartesian index notation as
∫
Ω
F
i
j
k
.
.
.
.
G
l
m
n
.
.
.
,
p
d
Ω
=
∫
Γ
n
p
F
i
j
k
.
.
.
G
l
m
n
.
.
.
d
Γ
−
∫
Ω
G
l
m
n
.
.
.
F
i
j
k
.
.
.
,
p
d
Ω
.
{\displaystyle \int _{\Omega }F_{ijk....}\,G_{lmn...,p}\,{\rm {d}}\Omega =\int _{\Gamma }n_{p}\,F_{ijk...}\,G_{lmn...}\,{\rm {d}}\Gamma -\int _{\Omega }G_{lmn...}\,F_{ijk...,p}\,{\rm {d}}\Omega \,.}
For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both
F
{\displaystyle {\boldsymbol {F}}}
G
{\displaystyle {\boldsymbol {G}}}
∫
Ω
F
⋅
(
∇
⋅
G
)
d
Ω
=
∫
Γ
n
⋅
(
G
⋅
F
T
)
d
Γ
−
∫
Ω
(
∇
F
)
:
G
T
d
Ω
.
{\displaystyle \int _{\Omega }{\boldsymbol {F}}\cdot ({\boldsymbol {\nabla }}\cdot {\boldsymbol {G}})\,{\rm {d}}\Omega =\int _{\Gamma }\mathbf {n} \cdot ({\boldsymbol {G}}\cdot {\boldsymbol {F}}^{T})\,{\rm {d}}\Gamma -\int _{\Omega }({\boldsymbol {\nabla }}{\boldsymbol {F}}):{\boldsymbol {G}}^{T}\,{\rm {d}}\Omega \,.}
In index notation,
∫
Ω
F
i
j
G
p
j
,
p
d
Ω
=
∫
Γ
n
p
F
i
j
G
p
j
d
Γ
−
∫
Ω
G
p
j
F
i
j
,
p
d
Ω
.
{\displaystyle \int _{\Omega }F_{ij}\,G_{pj,p}\,{\rm {d}}\Omega =\int _{\Gamma }n_{p}\,F_{ij}\,G_{pj}\,{\rm {d}}\Gamma -\int _{\Omega }G_{pj}\,F_{ij,p}\,{\rm {d}}\Omega \,.}
References
^ J. C. Simo and T. J. R. Hughes, 1998, Computational Inelasticity , Springer
^ J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity , Dover.
^ Ogden, R. W., 2000, Nonlinear Elastic Deformations , Dover.
^ http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf
See also
![{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =Df(\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~f(\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/308eadd1b18b96a60ef33e8df365bccc97f0faea)
![{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =D\mathbf {f} (\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~\mathbf {f} (\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8835d592f0ae810b9b57fa81f345a4850f1f1ce)
![{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=Df({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~f({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3916392ebeebca9ebddb31e76edb1c9f8c08b586)
![{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=D{\boldsymbol {F}}({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~{\boldsymbol {F}}({\boldsymbol {S}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2072ee23ca460f6f32851f397b5879a47033c3b)
![{\displaystyle {\begin{aligned}{\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[\mathbf {e} _{i}\otimes {\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}\right]\cdot \mathbf {c} \qquad \square \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40929ade3d51526fd6b2c4c7c9dea392dc884210)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial r}}~\mathbf {e} _{r}+{\cfrac {1}{r}}~{\cfrac {\partial \phi }{\partial \theta }}~\mathbf {e} _{\theta }+{\cfrac {\partial \phi }{\partial z}}~\mathbf {e} _{z}\\{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{r}}{\partial \theta }}-v_{\theta }\right)~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{r}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\cfrac {\partial v_{\theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\&\quad +{\cfrac {\partial v_{\theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\cfrac {\partial v_{z}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}{\cfrac {\partial v_{z}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{z}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rr}}{\partial \theta }}-(S_{\theta r}+S_{r\theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rr}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\\&\quad +{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{r\theta }}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rz}}{\partial \theta }}-S_{\theta z}\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\\&\quad +{\frac {\partial S_{rz}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {\partial S_{\theta r}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta r}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\&\quad +{\frac {\partial S_{\theta \theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta \theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {\partial S_{\theta z}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}\\&\quad +{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta z}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\&\quad +{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {\partial S_{z\theta }}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{z\theta }}{\partial \theta }}+S_{zr}\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\&\quad +{\frac {\partial S_{zz}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}~{\frac {\partial S_{zz}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/757e0b753b3c6d9e2b58108a21b9da4fe14df3f4)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)+{\cfrac {\partial v_{z}}{\partial z}}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{z}\\&+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{z}\\&+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5562be336f10361b0049f7f3608f04208a4c03f2)
![{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det({\boldsymbol {A}})=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3bee9c5d16eb2b2f5d0827d4e39e8f83c65201)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a280dfed26c159c4b698ac9db1c9e7b4ff9ce7c)
![{\displaystyle {\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}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})~{\cfrac {1}{\alpha ^{2}}}+I_{2}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})~{\cfrac {1}{\alpha }}+I_{3}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})\right)\right]\right|_{\alpha =0}\\&=\left.\det({\boldsymbol {A}})~{\cfrac {d}{d\alpha }}\left[1+I_{1}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})~\alpha +I_{2}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})~\alpha ^{2}+I_{3}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})~\alpha ^{3}\right]\right|_{\alpha =0}\\&=\left.\det({\boldsymbol {A}})~\left[I_{1}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})+2~I_{2}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})~\alpha +3~I_{3}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})~\alpha ^{2}\right]\right|_{\alpha =0}\\&=\det({\boldsymbol {A}})~I_{1}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16b8207ed9f2cc9582d921a734df86274e421fad)
![{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\det({\boldsymbol {A}})~{\text{tr}}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}:{\boldsymbol {T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80b13de7c7abc8d0c4e6853c644bfa9d82e6d0c3)
![{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {A}}}}=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/080361970fc0603caedbc79b2b1c305f37897621)
![{\displaystyle {\begin{aligned}I_{1}({\boldsymbol {A}})&={\text{tr}}{\boldsymbol {A}}\\I_{2}({\boldsymbol {A}})&={\frac {1}{2}}\left[({\text{tr}}{\boldsymbol {A}})^{2}-{\text{tr}}{{\boldsymbol {A}}^{2}}\right]\\I_{3}({\boldsymbol {A}})&=\det({\boldsymbol {A}})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5f440de0bb33a949001c6bef13f9f829fb1a42)
![{\displaystyle {\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T}\\{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T}~(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{T})=({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}})^{T}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/958082555bbf6e268f358d1286fab2eef6d5d5d0)
![{\displaystyle {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}=\det({\boldsymbol {A}})~[{\boldsymbol {A}}^{-1}]^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f59a269c296f625706d56b5bf8e45763228b86d)
![{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}]^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf5247a085970a226c820d13945cd1a7220cd49)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4ad226ea72ae236eaddfe419007ff6de53d55d)
![{\displaystyle {\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}})~[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d73a983e1a52cffc7b3a033b31ea4e166f2ae2bb)
![{\displaystyle (\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{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}}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd98450e2e00629178bedeee4200baed0a59616d)
![{\displaystyle (\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{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]=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4716bd540dce8c574d01b9b956e33ac50244ae99)
![{\displaystyle {\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}}^{T}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{T}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a103ff061c93c7199e75f5a2e355b32fd81e0fee)
![{\displaystyle {\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}}^{T}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}+{\boldsymbol {A}}^{T}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{T}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68503b4edd5ff8b08d4e88769764c4861dc553e3)
![{\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\left[{\frac {\partial }{\partial \alpha }}({\boldsymbol {A}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}={\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cf9341eabbe69c48f4ff85db571b84c8b2c318)
![{\displaystyle {\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a25e5e0ee3f8a2f287da104d5f72d8342899b9)
