Given that our field variable \(u\) or \(\boldsymbol{u}\) is represented as \(u = \sum_k c_k \psi_k\) or \(\boldsymbol{u} = \sum_k \boldsymbol{c_k} \psi_k\) we can express the change in the field variable in terms of nodal value \({c_j}\) as

\[\frac{\partial u}{\partial c_j} = \frac{\partial}{\partial c_j} \sum_k c_k \psi_k = \sum_k \frac{\partial c_k}{\partial c_j} \psi_k\]

Knowing that \(\frac{\partial c_k}{\partial c_j} = 1\) when \(k = j\) and \(\frac{\partial c_k}{\partial c_j} = 0\) when \(k \neq j\) simplifies to

\[\frac{\partial u}{\partial c_j} = \psi_j\]

Similar the following can be shown:

\[\frac{\partial}{\partial c_j}\nabla u = \nabla \psi_j \\\]

Vector Field

For vector fields \(\boldsymbol{u} = u_c \hat{e}_c\)

\[\frac{\partial \boldsymbol{u}}{\partial c_{c,j}} = \psi_j \hat{e}_c \\\]

For illustrative purposes, this can be show in cartesian coordinates for a single component of \(c_j\).

\[\boldsymbol{u} = u_1 \hat{e}_1 + u_2 \hat{e}_2 + u_3 \hat{e}_3 \\ u_1 = \sum_k c_{1,k} \psi_k , u_2 = \sum_k c_{2,k} \psi_k , u_3 = \sum_k c_{3,k} \psi_k \\ \therefore \boldsymbol{u} = \sum_k c_{1,k} \psi_k \hat{e}_1 + \sum_k c_{2,k} \psi_k \hat{e}_2 + \sum_k c_{3,k} \psi_k \hat{e}_3 \\ \frac{\partial \boldsymbol{u}}{\partial c_{1,j}} = \sum_k \frac{c_{1,k}}{c_{1,j}} \psi_k \hat{e}_1 + \sum_k \frac{c_{2,k}}{c_{1,j}} \psi_k \hat{e}_2 + \sum_k \frac{c_{3,k}}{c_{1,j}} \psi_k \hat{e}_3\]

when \(j = k\), \(\frac{c_{1,j}}{c_{1,j}} = 1\), \(\frac{c_{2,j}}{c_{1,j}} = 0\), \(\frac{c_{3,j}}{c_{1,j}} = 0\), resulting in

\[\frac{\partial \boldsymbol{u}}{\partial c_{1,j}} = \psi_j \hat{e}_1\]

Similarly for a gradient of a vector field

\[\frac{\partial \nabla\boldsymbol{u}}{\partial c_{c,j}} = \hat{e}_l \frac{\partial \psi_j}{\partial x_l} \hat{e}_c\]

So for \(c_{1,j}\) this would reduce to

\[\frac{\partial \nabla\boldsymbol{u}}{\partial c_{1,j}} = \hat{e}_1 \frac{\partial \psi_j}{\partial x_1} \hat{e}_1 + \hat{e}_2 \frac{\partial \psi_j}{\partial x_2} \hat{e}_1 + \hat{e}_3 \frac{\partial \psi_j}{\partial x_3} \hat{e}_1\]

For the divergence of a vector field

\[\nabla \cdot \boldsymbol{u} = \frac{\partial u_{cc}}{\partial x_{cc}} = \sum_k c_{cc,k} \frac{\partial \psi_k}{\partial x_{cc}} \\ \frac{\partial \nabla \cdot \boldsymbol{u}}{\partial c_{c,j}} = \frac{\partial \psi_j}{\partial x_c}\]