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Energy


The term energy has an important physical meaning in physics and is an extremely useful concept. There are several forms energy defined in mathematics.

In measure theory, let Omega be a space with measure mu>=0 and let Phi(P,Q) be a real function on the product space Omega×Omega. When

(mu,nu)=intintPhi(P,Q)dmu(Q)dnu(P)
(1)
=intPhi(P,mu)dnu(P)
(2)

exists for measures mu,nu>=0, (mu,nu) is called the mutual energy and (mu,mu) is called the energy (Iyanaga and Kawada 1980, p. 1038)

In harmonic function theory, let h be a real-valued harmonic function on a bounded domain Omega, then the Dirichlet energy is defined as int_Omega|del h|^2dx, where del is the gradient.

In graph theory, graph energy is defined as the sum of absolute values of the graph eigenvalues (i.e., eignvalues of a graph's adjacency matrix). Other varieties of graph energy are defined analogously using different matrices associated with a graph (and in particular, a weighted adjacency matrix).


See also

Dirichlet Energy, Graph Energy, Mutual Energy

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References

Iyanaga, S. and Kawada, Y. (Eds.). "General Potential." §335.B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1038, 1980.

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Energy

Cite this as:

Weisstein, Eric W. "Energy." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Energy.html

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