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Contents: The Lagrangian function and the parametric integrand Extremal curves; The Euler-Lagrange equation Lagrangian functions which are linear in x sub t The Legendre condition for a minimal curve Proof of the Legendre condition Constrained problems; The Hamilton canonical equations The reciprocity between L and H; The transversality conditions Extremal fields; The Hilbert invariant integral The Weierstrass E-function; Positively regular problems A simple example of the construction of an extremal field; Rayleigh quotients and the method of Rayleigh-Ritz The principle of maupertuis; The propagation of waves Problems whose Lagrangian functions involve derivatives of higher order than the first Multiple-Integral problems of the calculus of variations Constrained problems; Characteristic numbers Multiple-integral problems whose Lagrangian functions involve derivatives of higher order than the first The Courant maximum-minimum principle