Laplace's Equation

In Chapter 4 we have discussed the PDEs that control the heat flow in two and three dimensional spaces given by 1 https://www.w3.org/1998/Math/MathML" display="block"> u t = k ¯ ( u x x + u y y ) u t = k ¯ ( u x x + u y y + u z z ) respectively, where https://www.w3.org/1998/Math/MathML" display="inline"> k ¯ is the thermal diffusivity. If the temperature u reaches a steady state, that is, when u does not depend on time t and depends only on the space variables, then the time derivative u t vanishes as t → ∞. In view of this, we substitute u t = 0 into (1), hence we obtain the Laplace’s equations in two and three dimensions given by 2 https://www.w3.org/1998/Math/MathML" display="block"> u x x + u y y = 0 , u x x + u y y + u z z = 0.