grad, div, curl defintions

[KnutAM]

In the literature, there are different definitions for divergence and curl for second order and higher tensor fields. With the introduction of 3rd order Tensors, #205, we need to define this clearly. This issue is to get an overview over different sources with the goal to make a decision for which definition should be used.

Let's denote a general second-order tensor as $\boldsymbol{S}$ for the discussion. Note that we always assume an orthonormal, right-handed Cartesian coordinate system.

Just comment below the additional definitions and references, and I'll try to keep the tables updated (ping me on Slack if I forget)

Gradient

AFAIK, this is not problematic (correct me if I'm wrong). To my knowledge, there are just different notations (which can be confusing in itself), i.e.
$$\mathrm{grad}(\boldsymbol{S}) = \nabla \boldsymbol{S} = \boldsymbol{S} \otimes \nabla = \frac{\partial S_{ij}}{x_k} \boldsymbol{e}_i\otimes\boldsymbol{e}_j\otimes\boldsymbol{e}_k$$

Divergence

Tensor form	Index form	Sources	Comment
$\nabla \cdot \boldsymbol{S} $	$d_i = \frac{\partial S_{ji}}{x_j}$	[1]
$\boldsymbol{S} \cdot \nabla $	$d_i = \frac{\partial S_{ij}}{x_j}$	[2] (2.134), [3] (2.3.11), [4] (2.112)	Common in mechanics?

Curl

Here it is important that our definition fulfills $\mathrm{grad}(\mathrm{curl}(\boldsymbol{S}))=\boldsymbol{0}$. There exist definitions in the literature that don't. As a precursor, we haven't defined the cross-product for 2nd-order tensors, so for the discussion, let's define the cross product with a vector $\boldsymbol{v}$ as

$$\begin{align*} \boldsymbol{S}\times\boldsymbol{v} &= S_{ij} v_k \boldsymbol{e}_i \otimes \boldsymbol{e}_j \times \boldsymbol{e}_k = S_{ij} v_k \boldsymbol{e}_i \otimes [\varepsilon_{jkm} \boldsymbol{e}_m] = S_{ij} v_k \varepsilon_{jkm} \boldsymbol{e}_i \otimes \boldsymbol{e}_m \\\ \boldsymbol{v}\times \boldsymbol{S} &= v_i S_{jk} \boldsymbol{e}_i \times \boldsymbol{e}_j \otimes \boldsymbol{e}_k = v_i S_{jk} [\varepsilon_{ijm} \boldsymbol{e}_m] \otimes \boldsymbol{e}_k = v_i S_{jk} \varepsilon_{ijm} \boldsymbol{e}_m \otimes \boldsymbol{e}_k \end{align*}$$

Tensor form	Index form	Sources	Comment
$- \boldsymbol{S} \times \nabla $	$d_{ij} = \varepsilon_{opj}\frac{\partial S_{ip}}{x_o}$	[3] (2.3.18)
$\nabla \times \boldsymbol{S}$	$d_{ij} = \varepsilon_{opi}\frac{\partial S_{jp}}{x_o}$	[1]	From the definition of the cross-product, this should be $\nabla \times \boldsymbol{S}^\mathrm{T}$

where $\varepsilon_{ijk}$ is the Levi-Civita symbol.

Sources

[1] https://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)
[2] Bonet and Wood (2008)
[3] Rubin (2000)
[4] Itskov (2015)

[KnutAM]

Just pointing out that we can, if we follow the definitions of the tensor products, support the following operations

julia> using Tensors

julia> import Tensors: ∇

julia> f(x::Vec{3}) = Vec{3}((norm(x), sum(x), prod(x)))
f (generic function with 1 method)

julia> v = rand(Vec{3});

julia> divergence(f, v)
2.0866914680779223

julia> (f ⋅ ∇)(v)
2.0866914680779223

julia> gradient(f, v)
3×3 Tensor{2, 3, Float64, 9}:
 0.630406  0.673204  0.386502
 1.0       1.0       1.0
 0.279749  0.261964  0.456285

julia> (f ⊗ ∇)(v)
3×3 Tensor{2, 3, Float64, 9}:
 0.630406  0.673204  0.386502
 1.0       1.0       1.0
 0.279749  0.261964  0.456285

julia> curl(f, v)
3-element Vec{3, Float64}:
 -0.738035882169441
  0.10675354891495425
  0.3267956925012888

julia> (f × ∇)(v)
3-element Vec{3, Float64}:
 -0.738035882169441
  0.10675354891495425
  0.3267956925012888

with

LinearAlgebra.dot(f::Function, ::typeof(∇)) = Base.Fix1(divergence, f)  # d_{⋯} = ∂f(x)_{⋯i}/∂xᵢ
LinearAlgebra.cross(f::Function, ::typeof(∇)) = Base.Fix1(curl, f)      # d_{⋯j} εₒₚⱼ ∂f(x)_{⋯p}/∂xₒ
otimes(f::Function, ::typeof(∇)) = Base.Fix1(gradient, f)               # d_{⋯jk} = ∂f(x)_{⋯j}/∂xₖ