The constraint appears in integral form as $$\int dxdydz \psi^* \psi=\mathrm{constant},$$ and multiplying with a Lagrangian multiplier $$\lambda$$ and taking the variational minimum we obtain the final variational equation $$\delta \left\{\int dxdydz\left( \frac{1}{2}\nabla\psi^*\nabla\psi+V\psi^*\psi-\lambda\psi^*\psi\right)\right\} = 0.$$ We introduce the function $$f$$ $$f = \frac{1}{2}\nabla\psi^*\nabla\psi+V\psi^*\psi-\lambda\psi^*\psi= \frac{1}{2}(\psi^*_x\psi_x+\psi^*_y\psi_y+\psi^*_z\psi_z)+V\psi^*\psi-\lambda\psi^*\psi,$$ where we have skipped the dependence on $$x,y,z$$ and introduced the shorthand $$\psi_x$$, $$\psi_y$$ and $$\psi_z$$ for the various derivatives.