Yanming Wang, Saad Bhamla and Wei Cai
May, 2012
V ( r i ) = V ( R i ) + 1 2 ∑ i , j U i T ⋅ [ ∂ 2 V ∂ r i ∂ r j ] ⋅ U j {\displaystyle V({r_{i}})=V({R_{i}})+{\frac {1}{2}}\sum _{i,j}U_{i}^{T}\cdot \left[{\frac {\partial ^{2}V}{\partial r_{i}\partial r_{j}}}\right]\cdot U_{j}}
[ ∂ 2 V ∂ x i ∂ x j ∂ 2 V ∂ x i ∂ y j ∂ 2 V ∂ x i ∂ z j ∂ 2 V ∂ y i ∂ x j ∂ 2 V ∂ y i ∂ y j ∂ 2 V ∂ y i ∂ z j ∂ 2 V ∂ z i ∂ x j ∂ 2 V ∂ z i ∂ y j ∂ 2 V ∂ z i ∂ z j ] {\displaystyle \left[{\begin{array}{ccc}{\frac {\partial ^{2}V}{\partial x_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial x_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial x_{i}\partial z_{j}}}\\{\frac {\partial ^{2}V}{\partial y_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial y_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial y_{i}\partial z_{j}}}\\{\frac {\partial ^{2}V}{\partial z_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial z_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial z_{i}\partial z_{j}}}\\\end{array}}\right]}
![{\displaystyle V({r_{i}})=V({R_{i}})+{\frac {1}{2}}\sum _{i,j}U_{i}^{T}\cdot \left[{\frac {\partial ^{2}V}{\partial r_{i}\partial r_{j}}}\right]\cdot U_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d92cb16bec16dedbaa6b3191f642325e26f9ff3e)
![{\displaystyle \left[{\begin{array}{ccc}{\frac {\partial ^{2}V}{\partial x_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial x_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial x_{i}\partial z_{j}}}\\{\frac {\partial ^{2}V}{\partial y_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial y_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial y_{i}\partial z_{j}}}\\{\frac {\partial ^{2}V}{\partial z_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial z_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial z_{i}\partial z_{j}}}\\\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1e8d00b8cf996877d65657e66d81c236067446)