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L73 Electrostatics Poissons and Laplace equations
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#electrostatics #classicalelectrodynamics #jdjackson • Lecture Notes • https://drive.google.com/file/d/1Xxxg... • electrostatics, Poisson's equation, Laplace equation, classical electrodynamics, JD Jackson • Classical Electrodynamics, Third Edition, by John David Jackson, John Wiley and Sons, (1998). • Lecture board snap • https://drive.google.com/open?id=1Nh9... • https://drive.google.com/open?id=12ee... • Lecture Transcript • https://drive.google.com/open?id=10jR... • https://drive.google.com/open?id=1n8f... • https://drive.google.com/open?id=1WYx... • 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.
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