  We develop theory and software for rotation equivariant operators on scalar
and vector fields, with diverse applications in simulation, optimization and
machine learning. Rotation equivariance (covariance) means all fields in the
system rotate together, implying spatially invariant dynamics that preserve
symmetry. Extending the convolution theorems of linear time invariant systems,
we theorize that linear equivariant operators are characterized by tensor field
convolutions using an appropriate product between the input field and a
radially symmetric kernel field. Most Green's functions and differential
operators are in fact equivariant operators, which can also fit unknown
symmetry preserving dynamics by parameterizing the radial function. We
implement the Julia package EquivariantOperators.jl for fully differentiable
finite difference equivariant operators on scalar, vector and higher order
tensor fields in 2d/3d. It can run forwards for simulation or image processing,
or be back propagated for computer vision, inverse problems and optimal
control. Code at https://aced-differentiate.github.io/EquivariantOperators.jl/
