Kernels for Gradient Field Calculations

To calculate the gradient of the perturbed fields at the voxel centres, the same calculation method used for the main field can be employed, but using the gradient of the kernels, $\nabla F$, rather than $F$. In particular,

$$\frac{\partial F(\mathbf{x}' ; \mathbf{x})}{\partial q} = \frac{\partial F(\mathbf{x}' ; \mathbf{x})}{\partial \mathbf{x}'}... ...{x}' ; \mathbf{x})}{\partial \mathbf{x}} \frac{\partial \mathbf{x}}{\partial q}$$ (37)

which reduces to

$$\frac{\partial F(\mathbf{x}' ; \mathbf{x})}{\partial q} = \frac{\partial F(\mathbf{x}' ; \mathbf{x})}{\partial q'}$$ (38)

when $F(\mathbf{x}';\mathbf{x}) = F(\mathbf{x}')$ and $q = x, y$ or $z$.