A Quote by Sophus Lie
Among all of the mathematical disciplines the theory of differential equations is the most important... It furnishes the explanation of all those elementary manifestations of nature which involve time.
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The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics.
Hydrodynamics procreated complex analysis, partial differential equations, Lie groups and algebra theory, cohomology theory and scientific computing.
Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry.
If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).
Like music or art, mathematical equations can have a natural progression and logic that can evoke rare passions in a scientist. Although the lay public considers mathematical equations to be rather opaque, to a scientist an equation is very much like a movement in a larger symphony. Simplicity. Elegance. These are the qualities that have inspired some of the greatest artists to create their masterpieces, and they are precisely the same qualities that motivate scientists to search for the laws of nature. LIke a work of art or a haunting poem, equations have a beauty and rhythm all their own.
Yes, we now have to divide up our time like that, between politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever.
The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.
What appear to be the most valuable aspects of the theoretical physics we have are the mathematical descriptions which enable us to predict events. These equations are, we would argue, the only realities we can be certain of in physics; any other ways we have of thinking about the situation are visual aids or mnemonics which make it easier for beings with our sort of macroscopic experience to use and remember the equations.
The most important single thing about string theory is that it's a highly mathematical theory, and the mathematics holds together in a very tight and consistent way. It contains in its basic structure both quantum mechanics and the theory of gravity. That's big news.
The standard high school curriculum traditionally has been focused towards physics and engineering. So calculus, differential equations, and linear algebra have always been the most emphasized, and for good reason - these are very important.
Knowing what is big and what is small is more important than being able to solve partial differential equations.
The difficulty involved in the proper and adequate means of describing changes in continuous deformable bodies is the method of differential equations. ... They express mathematically the physical concept of contiguous action.
Einstein's Theory of Relativity
Even in relativity theory even though you can analyze space - time in terms of this four-dimensional geometrical structure one of the dimensions is different. And this shows up in the equations. It has a different sign - rather than plus it shows up as a negative minus. So even in relativity theory time is distinct from space in terms of the way in which these dimensions manifest themselves in the equations.
We talk of communing with Nature, but 'tis with ourselves we commune... Nature furnishes the conditions - the solitude - and the soul furnishes the entertainment.
What is the origin of the urge, the fascincation that drives physicists, mathematicians, and presumably other scientists as well? Psychoanalysis suggests that it is sexual curiosity. You start by asking where little babies come from, one thing leads to another, and you find yourself preparing nitroglycerine or solving differential equations. This explanation is somewhat irritating, and therefore probably basically correct.
If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.
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