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Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus. The mass of an object of known density , the moment of inertia of objects, and the potential energies due to gravitational and electromagnetic forces can all be found by the use of calculus.
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". [ 1 ]
Marston Morse applied calculus of variations in what is now called Morse theory. [6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. [6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations. [7] [8] [9] [c]
The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the 18th century these applications were multiplied, until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of analysis.
During this period there was little distinction between physics and mathematics; [18] as an example, Newton regarded geometry as a branch of mechanics. [19] In the 19th century Auguste Comte in his hierarchy of the sciences, placed physics and astronomy as less general and more complex than mathematics, as both depend on it. [20]
The action principles have applications as broad as physics, including many problems in classical mechanics but especially in modern problems of quantum mechanics and general relativity. These applications built up over two centuries as the power of the method and its further mathematical development rose.
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.