 Formal transpose of a differential operator

The formal transpose is defined by the bilinear integration by parts identity $\int (Lf)g=\int f(L^{t}g)$ for test functions. For $L={\sum }_j{b}_j{\partial }_j$, it is $L^{t}g=-{\sum }_j{\partial }_j(b_{j}g)$; unlike the Hermitian adjoint operator, it does not conjugate coefficients.