Derivatives with respect to vectors and second-order tensors
edit
Gradient of a tensor field
edit
The gradient ,
∇
T
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}}
, of a tensor field
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
in the direction of an arbitrary constant vector 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
edit
If
e
1
,
e
2
,
e
3
{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}
are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by (
x
1
,
x
2
,
x
3
{\displaystyle x_{1},x_{2},x_{3}}
), then the gradient of the tensor field
T
{\displaystyle {\boldsymbol {T}}}
is given by
∇
T
=
∂
T
∂
x
i
⊗
e
i
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}={\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}\otimes \mathbf {e} _{i}}
Since the basis vectors do not vary in a Cartesian coordinate system we have the following relations for the gradients of a scalar field
ϕ
{\displaystyle \phi }
, a vector field v , and a second-order tensor field
S
{\displaystyle {\boldsymbol {S}}}
.
∇
ϕ
=
∂
ϕ
∂
x
i
e
i
=
ϕ
,
i
e
i
∇
v
=
∂
(
v
j
e
j
)
∂
x
i
⊗
e
i
=
∂
v
j
∂
x
i
e
j
⊗
e
i
=
v
j
,
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
=
S
j
k
,
i
e
j
⊗
e
k
⊗
e
i
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial x_{i}}}~\mathbf {e} _{i}=\phi _{,i}~\mathbf {e} _{i}\\{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial (v_{j}\mathbf {e} _{j})}{\partial x_{i}}}\otimes \mathbf {e} _{i}={\cfrac {\partial v_{j}}{\partial x_{i}}}~\mathbf {e} _{j}\otimes \mathbf {e} _{i}=v_{j,i}~\mathbf {e} _{j}\otimes \mathbf {e} _{i}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\cfrac {\partial (S_{jk}\mathbf {e} _{j}\otimes \mathbf {e} _{k})}{\partial x_{i}}}\otimes \mathbf {e} _{i}={\cfrac {\partial S_{jk}}{\partial x_{i}}}~\mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{i}=S_{jk,i}~\mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{i}\end{aligned}}}
Curvilinear coordinates
edit
If
g
1
,
g
2
,
g
3
{\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}}
are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (
ξ
1
,
ξ
2
,
ξ
3
{\displaystyle \xi ^{1},\xi ^{2},\xi ^{3}}
), then the gradient of the tensor field
T
{\displaystyle {\boldsymbol {T}}}
is given by (see [ 3] for a proof.)
∇
T
=
∂
T
∂
ξ
i
⊗
g
i
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}={\frac {\partial {\boldsymbol {T}}}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}}
From this definition we have the following relations for the gradients of a scalar field
ϕ
{\displaystyle \phi }
, a vector field 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 &={\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}}}
where the Christoffel symbol
Γ
i
j
k
{\displaystyle \Gamma _{ij}^{k}}
is defined using
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}={\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}}}}
Cylindrical polar coordinates
edit
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
+
∂
S
r
r
∂
z
e
r
⊗
e
r
⊗
e
z
+
1
r
[
∂
S
r
r
∂
θ
−
(
S
θ
r
+
S
r
θ
)
]
e
r
⊗
e
r
⊗
e
θ
+
∂
S
r
θ
∂
r
e
r
⊗
e
θ
⊗
e
r
+
∂
S
r
θ
∂
z
e
r
⊗
e
θ
⊗
e
z
+
1
r
[
∂
S
r
θ
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
r
⊗
e
θ
⊗
e
θ
+
∂
S
r
z
∂
r
e
r
⊗
e
z
⊗
e
r
+
∂
S
r
z
∂
z
e
r
⊗
e
z
⊗
e
z
+
1
r
[
∂
S
r
z
∂
θ
−
S
θ
z
]
e
r
⊗
e
z
⊗
e
θ
+
∂
S
θ
r
∂
r
e
θ
⊗
e
r
⊗
e
r
+
∂
S
θ
r
∂
z
e
θ
⊗
e
r
⊗
e
z
+
1
r
[
∂
S
θ
r
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
θ
⊗
e
r
⊗
e
θ
+
∂
S
θ
θ
∂
r
e
θ
⊗
e
θ
⊗
e
r
+
∂
S
θ
θ
∂
z
e
θ
⊗
e
θ
⊗
e
z
+
1
r
[
∂
S
θ
θ
∂
θ
+
(
S
r
θ
+
S
θ
r
)
]
e
θ
⊗
e
θ
⊗
e
θ
+
∂
S
θ
z
∂
r
e
θ
⊗
e
z
⊗
e
r
+
∂
S
θ
z
∂
z
e
θ
⊗
e
z
⊗
e
z
+
1
r
[
∂
S
θ
z
∂
θ
+
S
r
z
]
e
θ
⊗
e
z
⊗
e
θ
+
∂
S
z
r
∂
r
e
z
⊗
e
r
⊗
e
r
+
∂
S
z
r
∂
z
e
z
⊗
e
r
⊗
e
z
+
1
r
[
∂
S
z
r
∂
θ
−
S
z
θ
]
e
z
⊗
e
r
⊗
e
θ
+
∂
S
z
θ
∂
r
e
z
⊗
e
θ
⊗
e
r
+
∂
S
z
θ
∂
z
e
z
⊗
e
θ
⊗
e
z
+
1
r
[
∂
S
z
θ
∂
θ
+
S
z
r
]
e
z
⊗
e
θ
⊗
e
θ
+
∂
S
z
z
∂
r
e
z
⊗
e
z
⊗
e
r
+
∂
S
z
z
∂
z
e
z
⊗
e
z
⊗
e
z
+
1
r
∂
S
z
z
∂
θ
e
z
⊗
e
z
⊗
e
θ
{\displaystyle {\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}}}
Divergence of a tensor field
edit
The divergence of a tensor field
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
is defined using the recursive relation
(
∇
⋅
T
)
⋅
c
=
∇
⋅
(
c
⋅
T
T
)
;
∇
⋅
v
=
tr
(
∇
v
)
{\displaystyle ({\boldsymbol {\nabla }}\cdot {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot \left(\mathbf {c} \cdot {\boldsymbol {T}}^{\textsf {T}}\right)~;\qquad {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\text{tr}}({\boldsymbol {\nabla }}\mathbf {v} )}
where c is an arbitrary constant vector and v is a vector field. If
T
{\displaystyle {\boldsymbol {T}}}
is a tensor field of order n > 1 then the divergence of the field is a tensor of order n − 1.
Cartesian coordinates
edit
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
=
v
i
,
i
∇
⋅
S
=
∂
S
i
k
∂
x
i
e
k
=
S
i
k
,
i
e
k
{\displaystyle {\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}}}
where tensor index notation for partial derivatives is used in the rightmost expressions. Note that
∇
⋅
S
≠
∇
⋅
S
T
.
{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}\neq {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}^{\textsf {T}}.}
For a symmetric second-order tensor, the divergence is also often written as[ 4]
∇
⋅
S
=
∂
S
k
i
∂
x
i
e
k
=
S
k
i
,
i
e
k
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\cfrac {\partial S_{ki}}{\partial x_{i}}}~\mathbf {e} _{k}=S_{ki,i}~\mathbf {e} _{k}\end{aligned}}}
The above expression is sometimes used as the definition of
∇
⋅
S
{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}}
in Cartesian component form (often also written as
div
S
{\displaystyle \operatorname {div} {\boldsymbol {S}}}
). Note that such a definition is not consistent with the rest of this article (see the section on curvilinear co-ordinates).
The difference stems from whether the differentiation is performed with respect to the rows or columns of
S
{\displaystyle {\boldsymbol {S}}}
, and is conventional. This is demonstrated by an example. In a Cartesian coordinate system the second order tensor (matrix)
S
{\displaystyle \mathbf {S} }
is the gradient of a vector function
v
{\displaystyle \mathbf {v} }
.
∇
⋅
(
∇
v
)
=
∇
⋅
(
v
i
,
j
e
i
⊗
e
j
)
=
v
i
,
j
i
e
i
⋅
e
i
⊗
e
j
=
(
∇
⋅
v
)
,
j
e
j
=
∇
(
∇
⋅
v
)
∇
⋅
[
(
∇
v
)
T
]
=
∇
⋅
(
v
j
,
i
e
i
⊗
e
j
)
=
v
j
,
i
i
e
i
⋅
e
i
⊗
e
j
=
∇
2
v
j
e
j
=
∇
2
v
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \left({\boldsymbol {\nabla }}\mathbf {v} \right)&={\boldsymbol {\nabla }}\cdot \left(v_{i,j}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{i,ji}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}=\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)_{,j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)\\{\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}}}
The last equation is equivalent to the alternative definition / interpretation[ 4]
(
∇
⋅
)
alt
(
∇
v
)
=
(
∇
⋅
)
alt
(
v
i
,
j
e
i
⊗
e
j
)
=
v
i
,
j
j
e
i
⊗
e
j
⋅
e
j
=
∇
2
v
i
e
i
=
∇
2
v
{\displaystyle {\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}}}
Curvilinear coordinates
edit
In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field
S
{\displaystyle {\boldsymbol {S}}}
are
∇
⋅
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}}}
More generally,
∇
⋅
S
=
[
∂
S
i
j
∂
q
k
−
Γ
k
i
l
S
l
j
−
Γ
k
j
l
S
i
l
]
g
i
k
b
j
=
[
∂
S
i
j
∂
q
i
+
Γ
i
l
i
S
l
j
+
Γ
i
l
j
S
i
l
]
b
j
=
[
∂
S
j
i
∂
q
i
+
Γ
i
l
i
S
j
l
−
Γ
i
j
l
S
l
i
]
b
j
=
[
∂
S
i
j
∂
q
k
−
Γ
i
k
l
S
l
j
+
Γ
k
l
j
S
i
l
]
g
i
k
b
j
{\displaystyle {\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}}}
Cylindrical polar coordinates
edit
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} =\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}}}
Curl of a tensor field
edit
The curl of an order-n > 1 tensor field
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
is also defined using the recursive relation
(
∇
×
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.
Curl of a first-order tensor (vector) field
edit
Consider a vector field v and an arbitrary constant vector c . In index notation, the cross product is given by
v
×
c
=
ε
i
j
k
v
j
c
k
e
i
{\displaystyle \mathbf {v} \times \mathbf {c} =\varepsilon _{ijk}~v_{j}~c_{k}~\mathbf {e} _{i}}
where
ε
i
j
k
{\displaystyle \varepsilon _{ijk}}
is the permutation symbol , otherwise known as the Levi-Civita symbol. Then,
∇
⋅
(
v
×
c
)
=
ε
i
j
k
v
j
,
i
c
k
=
(
ε
i
j
k
v
j
,
i
e
k
)
⋅
c
=
(
∇
×
v
)
⋅
c
{\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )=\varepsilon _{ijk}~v_{j,i}~c_{k}=(\varepsilon _{ijk}~v_{j,i}~\mathbf {e} _{k})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c} }
Therefore,
∇
×
v
=
ε
i
j
k
v
j
,
i
e
k
{\displaystyle {\boldsymbol {\nabla }}\times \mathbf {v} =\varepsilon _{ijk}~v_{j,i}~\mathbf {e} _{k}}
Curl of a second-order tensor field
edit
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
)
=
ε
i
j
k
c
m
S
m
j
,
i
e
k
=
(
ε
i
j
k
S
m
j
,
i
e
k
⊗
e
m
)
⋅
c
=
(
∇
×
S
)
⋅
c
{\displaystyle {\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {S}})=\varepsilon _{ijk}~c_{m}~S_{mj,i}~\mathbf {e} _{k}=(\varepsilon _{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times {\boldsymbol {S}})\cdot \mathbf {c} }
Therefore, we have
∇
×
S
=
ε
i
j
k
S
m
j
,
i
e
k
⊗
e
m
{\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {S}}=\varepsilon _{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m}}
Identities involving the curl of a tensor field
edit
The most commonly used identity involving the curl of a tensor field,
T
{\displaystyle {\boldsymbol {T}}}
, is
∇
×
(
∇
T
)
=
0
{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}{\boldsymbol {T}})={\boldsymbol {0}}}
This identity holds for tensor fields of all orders. For the important case of a second-order tensor,
S
{\displaystyle {\boldsymbol {S}}}
, this identity implies that
∇
×
(
∇
S
)
=
0
⟹
S
m
i
,
j
−
S
m
j
,
i
=
0
{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}{\boldsymbol {S}})={\boldsymbol {0}}\quad \implies \quad S_{mi,j}-S_{mj,i}=0}
Derivative of the determinant of a second-order tensor
edit
The derivative of the determinant of a second order tensor
A
{\displaystyle {\boldsymbol {A}}}
is given by
∂
∂
A
det
(
A
)
=
det
(
A
)
[
A
−
1
]
T
.
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det({\boldsymbol {A}})=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}~.}
In an orthonormal basis, the components of
A
{\displaystyle {\boldsymbol {A}}}
can be written as a matrix A . In that case, the right hand side corresponds the cofactors of the matrix.
Derivatives of the invariants of a second-order tensor
edit
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}}}
are
∂
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}}}\\[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}}}
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}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {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}})~\left[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}\right]^{\textsf {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}})~\left[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}\right]^{\textsf {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}})^{\textsf {T}}\cdot \left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right]=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~{\boldsymbol {\mathit {1}}}~.}
Expanding the right hand side and separating terms on the left hand side gives
(
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 \left(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\right)\cdot \left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right]=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}}
or,
[
∂
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}}^{\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}}}
If we define
I
0
:=
1
{\displaystyle I_{0}:=1}
and
I
4
:=
0
{\displaystyle I_{4}:=0}
, we can write the above as
[
∂
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}}^{\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}}}
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}}^{\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}}}
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}}^{\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}}}
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}}^{\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}}}
Derivative of the second-order identity tensor
edit
Derivative of a second-order tensor with respect to itself
edit
Derivative of the inverse of a second-order tensor
edit
Integration by parts
edit
Domain
Ω
{\displaystyle \Omega }
, its boundary
Γ
{\displaystyle \Gamma }
and the outward unit normal
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}}\,d\Omega =\int _{\Gamma }\mathbf {n} \otimes ({\boldsymbol {F}}\otimes {\boldsymbol {G}})\,d\Gamma -\int _{\Omega }{\boldsymbol {G}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {F}}\,d\Omega }
where
F
{\displaystyle {\boldsymbol {F}}}
and
G
{\displaystyle {\boldsymbol {G}}}
are differentiable tensor fields of arbitrary order,
n
{\displaystyle \mathbf {n} }
is the unit outward normal to the domain over which the tensor fields are defined,
⊗
{\displaystyle \otimes }
represents a generalized tensor product operator, and
∇
{\displaystyle {\boldsymbol {\nabla }}}
is a generalized gradient operator. When
F
{\displaystyle {\boldsymbol {F}}}
is equal to the identity tensor, we get the divergence theorem
∫
Ω
∇
G
d
Ω
=
∫
Γ
n
⊗
G
d
Γ
.
{\displaystyle \int _{\Omega }{\boldsymbol {\nabla }}{\boldsymbol {G}}\,d\Omega =\int _{\Gamma }\mathbf {n} \otimes {\boldsymbol {G}}\,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}\,d\Omega =\int _{\Gamma }n_{p}\,F_{ijk...}\,G_{lmn...}\,d\Gamma -\int _{\Omega }G_{lmn...}\,F_{ijk...,p}\,d\Omega \,.}
For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both
F
{\displaystyle {\boldsymbol {F}}}
and
G
{\displaystyle {\boldsymbol {G}}}
are second order tensors, we have
∫
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}})\,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 \,.}
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}\,d\Omega =\int _{\Gamma }n_{p}\,F_{ij}\,G_{pj}\,d\Gamma -\int _{\Omega }G_{pj}\,F_{ij,p}\,d\Omega \,.}
^ 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.
^ R. W. Ogden, 2000, Nonlinear Elastic Deformations , Dover.
^ a b Hjelmstad, Keith (2004). Fundamentals of Structural Mechanics . Springer Science & Business Media. p. 45. ISBN 9780387233307 .