Magnetic deformation theory of a vesicle

We have extended the Helfrich’s spontaneous curvature model [M. Iwamoto and Z. C. Ou-Yang. Chem. Phys. Lett. $\mathbf{\text{590}}$(2013)183; Y. X. Deng, et.al., EPL. $\mathbf{\text{123}}$(2018)68002] of the equilibrium vesicle shapes by adding the interaction between magnetic field and the constituent molecules to explain the phenomena of the reversibly deformation of artificial stomatocyte[P. G. van Rhee, et.al., Nat. Commun. $\mathbf{\text{Sep}}\mathbf{\text{24;5:5010}}$(2014)doi: 10.1038/ncomms6010.] and the anharmonic deformation of a self-assembled nanocapsules of bola-amphiphilic molecules and the linear birefringence[O.V. Manyuhina, et.al., Phys. Rev. Lett. $\mathbf{\text{98}}$(2007)146101.]. However, the sophistic mathematics in differential geometry is still covered. Here, we present the derivations of formulas in detailed to reveal the perturbation of deformation $\psi$ under two cases.