L7.1 Electrostatics: Poisson's and Laplace equations

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Lecture Notes
electrostatics, Poisson's equation, Laplace equation, classical electrodynamics, JD Jackson
Classical Electrodynamics, Third Edition, by John David Jackson, John Wiley and Sons, (1998).
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Lecture Transcript

From Wikipedia, the free encyclopedia
In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson.
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

is the gradient operator (also symbolized "grad"), and {\displaystyle f(x,y,z)}{\displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.

If the right-hand side is specified as a given function, {\displaystyle h(x,y,z)}{\displaystyle h(x,y,z)}, we have

{\displaystyle \Delta f=h.}{\displaystyle \Delta f=h.}
This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations.

Laplace’s equation is also a special case of the Helmholtz equation.

The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions,[1] which are important in branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation.

[sayanjitb]

Dear sir, when you take a gradient of \phi, which is essentially 1/|x-x'|, w.r.t, which coordinate are you taking derivatives, x or x'? Hope I cleared my doubt!
Thanks