Hyperbolic System and Conservation Laws
There is a connection between a hyperbolic system and a conservation law. Consider a hyperbolic system of one partial differential equation for one unknown function . Then the system has the form
Now can be some quantity with a flux . To show that this quantity is conserved, integrate over a domain
If and are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and to get a conservation law for the quantity in the general form
which means that the time rate of change of in the domain is equal to the net flux of through its boundary . Since this is an equality, it can be concluded that is conserved within .
Read more about this topic: Hyperbolic Partial Differential Equations
Famous quotes containing the words system, conservation and/or laws:
“We are now going through a period of demolition. In morals, in social life, in politics, in medicine, and in religion there is a universal upturning of foundations. But the day of reconstruction seems to be looming, and now the grand question is: Are there any sure and universal principles that will evolve a harmonious system in which we shall all agree?”
—Catherine E. Beecher (1800–1878)
“The putting into force of laws which shall secure the conservation of our resources, as far as they may be within the jurisdiction of the Federal Government, including the more important work of saving and restoring our forests and the great improvement of waterways, are all proper government functions which must involve large expenditure if properly performed.”
—William Howard Taft (1857–1930)
“Language is a process of free creation; its laws and principles are fixed, but the manner in which the principles of generation are used is free and infinitely varied. Even the interpretation and use of words involves a process of free creation.”
—Noam Chomsky (b. 1928)