Chapter 1 is a minimal course on assimilating data into models using the calculus of variations. The theory is introduced with a “toy” model in the form of a single linear partial differential equation of first order. The independent variables are a spatial coordinate, and time. The well-posedness of the mixed initial-boundary value problem or “forward model” is established, and the solution is expressed explicitly with the Green's function. The introduction of additional data renders the problem ill-posed. This difficulty is resolved by seeking a weighted least-squares best fit to all the information. The fitting criterion is a penalty functional that is quadratic in all the misfits to the various pieces of information, integrated over space and time as appropriate. The best-fit or “generalized inverse” is expressed explicitly with the representers for the penalty functional, and with the Green's function for the forward model. The behavior of the generalized inverse is examined for various limiting choices of weights. The smoothness of the inverse is seen to depend upon the nature of the weights, which will be subsequently identified as kernel inverses of error covariances. After reading Chapter 1, it is possible to carry out the first four computing exercises in Appendix A.