Sure, it's mostly a mnemonic. But you can do it easily in your head, even in infinite dimensions.
For example, if you recall that (due to stokes theorem) the divergence is minus the transpose of the gradient, you can solve many variational problems that way.
The canonical example is: find a function u whose gradient is a given vector field F. This is an over-determined system without a solution. Applying the above trick, you compute the divergence of both sides to obtain Poisson equation Δu=div(F), that you can solve. This is equivalent to finding the least-square minimizer of the energy E(u)=∫|∇u-F|²
For example, if you recall that (due to stokes theorem) the divergence is minus the transpose of the gradient, you can solve many variational problems that way. The canonical example is: find a function u whose gradient is a given vector field F. This is an over-determined system without a solution. Applying the above trick, you compute the divergence of both sides to obtain Poisson equation Δu=div(F), that you can solve. This is equivalent to finding the least-square minimizer of the energy E(u)=∫|∇u-F|²