Laplace’s equation A classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplace’s equation is ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0. Show that the following functions are harmonic ; that is, they satisfy Laplace’s equation. 80 . u ( x , y ) = e – x sin y
Laplace’s equation A classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplace’s equation is ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0. Show that the following functions are harmonic ; that is, they satisfy Laplace’s equation. 80 . u ( x , y ) = e – x sin y
Solution Summary: The author explains that the function u(x,y)=e-xmathrmsiny is harmonic if it satisfies the Laplace
Laplace’s equationA classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplace’s equation is
∂
2
u
∂
x
2
+
∂
2
u
∂
y
2
=
0.
Show that the following functions are harmonic; that is, they satisfy Laplace’s equation.
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY