We now have Helmholtz's equation for the spatial variable "'r "'and a second-order ordinary differential equation in time.
12.
The interval length is not required to be the same for each spatial variable \ Delta x _ i, i = 1, \ ldots, n.
13.
Since there are two variables, we will use the Fourier transformation in both and rather than operate as Fourier did, who only transformed in the spatial variables.
14.
Kruskal's most widely known work was the discovery in the 1960s of the integrability of certain nonlinear partial differential equations involving functions of one spatial variable as well as time.
15.
If we ignore the spatial variables, the phase space of a gas with three particles is three dimensional, which permits one to sketch the n-spheres over which the integral over phase space must be taken.
16.
For example, in one dimension, the spatial variable of, say, a particle, can only be measured by the quantum mechanical " position operator " at the cost of losing information about the momentum of the particle.
17.
However, already in quantum mechanics there is one " caveat ", namely the particles acting onto each other do not only possess the spatial variable, but also a discrete intrinsic spin ", and there is the Pauli principle relating the space and the spin variables.
18.
For the problems of wave motion is free space, the basic method of separating spatial variables is the application of integral transforms, while for the problems of wave generation and propagation in the guiding systems the variables are usually separated using expansions in terms of the basic functions ( modes ) meeting the required boundary conditions at the surface of the guiding system.
19.
Although four-and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done . ( This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces . ) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.
20.
If two identical fermions ( e . g . electrons ) have a " symmetric " spin function ( e . g . parallel spins ) the spatial variables must be " antisymmetric " ( i . e . they exclude each other from their places much as if there was a repulsive force ), and vice versa, i . e . for antiparallel " spins " the " position variables " must be symmetric ( i . e . the apparent force must be attractive ).
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