Abstract
The laws of physics are almost exclusively written in the form of differential equations (DEs). In (point) particle mechanics there is only one independent variable, leading to ordinary differential equations (ODEs). In other areas of physics in which extended objects such as fields are studied, variations with respect to position are also important. Partial derivatives with respect to coordinate variables show up in the differential equations, which are therefore called partial differential equations (PDEs). We list the most common PDEs of mathematical physics in the following.
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Notes
- 1.
- 2.
In most cases, α is chosen to be real. In the case of the Schrödinger equation, it is more convenient to choose α to be purely imaginary so that the i in the definition of
can be compensated. In all cases, the precise nature of α is determined by boundary conditions. - 3.
These operators act on the space of functions possessing enough “nice” properties as to render the space suitable. The operator x j simply multiplies functions, while
differentiates them. - 4.
It is assumed that the reader is familiar with vector algebra using indices and such objects as δ ij and ϵ ijk . For an introductory treatment, sufficient for our present discussion, see [Hass 08]. A more advanced treatment of these objects (tensors) can be found in Part VIII of the present book.
- 5.
We could just as well have chosen
and any other component as our maximal set. However, 
and 
is the universally accepted choice.
can be compensated. In all cases, the precise nature of α is determined by boundary conditions.
differentiates them.
and any other component as our maximal set. However, 
and 
is the universally accepted choice.References
Hassani, S.: Mathematical Methods for Students of Physics and Related Fields, 2nd edn. Springer, Berlin (2008)
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Hassani, S. (2013). Separation of Variables in Spherical Coordinates. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_13
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