Case 2: $B^{(0)}(\mathbf{x}) = (1,0,0)$

$$\nabla F(\mathbf{x}' ; \mathbf{x}) = \nabla \frac{-1\,}{4\pi} \mathrm{sinh}^{-1}\left(\frac{y'}{\sqrt{{x'}^2 + {z'}^2}}\right)$$ (41)

$$= \frac{1}{4\pi} \left[ \frac{x'y'}{r'(x'^2 + z'^2)} \; , \; \frac{-1\,}{r'} \; , \; \frac{y'z'}{r'(x'^2 + z'^2)} \right]$$ (42)

Similar calculations can be performed when the applied field includes a linear gradient by taking the kernels calculated in Appendix B.