Mathematical physics:

 Mathematical physics is the scientific discipline concerned with interface of mathematics and physics. It can be defined as the application of the mathematics to problems in physics and development of mathematical methods appropriate for such applications and for the formulation of physical theories. The term 'mathematical' physics is sometimes used in a special sense, to denote research aimed at studying and solving problems inspired by physics within the mathematically rigorous framework.

Mathematical physics in this sense covers a broad range of topics with the common feature that they blend pure mathematics and physics. Although related to theoretical physics, 'mathematical' physics in this sense emphasizes mathematical rigor of same type as found in the mathematics. On the other hand, theoretical physics emphasizes the links to observations and experimental physics which often requires theoretical to use, intuitive, and approximate arguments. Such types of arguments are not considered rigorous by mathematicians. Arguably, rigorous mathematical physics is closer to mathematics, and hypothetical physics is closer to physics.

The usage of the term 'Mathematical physics' is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not considered parts of mathematical physics, while other related fields are. For example, ordinary differential equations and simplistic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to physics. We know that physics is a systematic attempt to explore and understand natural phenomenon in a much detail and depth as possible.

There are distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The theory of partial differential equations is perhaps closely associated with the mathematical physics. These were developed intensively from the 2nd half of the 18th century. Theories in physics frequently make use of mathematical concepts. Mathematics has served as a powerful tool in development of the modern theoretical physics.