Every smooth function in several variables with a Lipschitz derivative, when considered on a compact convex set, is the difference of a convex function and a convex quadratic function.
We use this result to decompose anti - derivatives of continuous Lipschitz functions and augment the fundamental theorem of calculus. The augmentation makes it possible to convexify
and monotonize ordinary differential equations and obtain possibly new results for integrals of scalar functions and for line integrals. The result is also used in linear algebra where new bounds for the determinant and the spectral radius of symmetric matrices are obtained
