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University:  University of Illinois at Urbana Champaign - UIUC

Greek Organization:  Theta Tau

Author:  Navier Stokes

Comment:  For a function {\displaystyle u(x,y,z,t)}{\displaystyle u(x,y,z,t)}of three spatial variables {\displaystyle (x,y,z)}(x,y,z) (see Cartesian coordinate system) and the time variable {\displaystyle t}t, the heat equation is {\displaystyle {\frac {\partial u}{\partial t}}=\alpha \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)}{\displaystyle {\frac {\partial u}{\partial t}}=\alpha \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)} where {\displaystyle \alpha }\alpha is a real coefficient called the diffusivity of the medium. Using Newton's notation for derivatives, and the notation of vector calculus, the heat equation can be written in compact form as {\displaystyle {\dot {u}}=\alpha \nabla ^{2}u}{\displaystyle {\dot {u}}=\alpha \nabla ^{2}u} Here {\displaystyle \nabla ^{2}}\nabla ^{2} denotes the Laplace operator, and {\displaystyle {\dot {u}}}{\displaystyle {\dot {u}}} is the time derivative of {\displaystyle u}u. One advantage of this formula is that the operator {\displaystyle \nabla ^{2}}\nabla ^{2} can usually be defined in purely physical terms, independently of the choice of coordinate system. This equation describes the flow of heat in a homogeneous and isotropic medium, with {\displaystyle u(x,y,z,t)}u(x,y,z,t) being the temperature at the point {\displaystyle (x,y,z)}(x,y,z) and time {\displaystyle t}t. However, it also describes many other physical phenomena as well. The value of {\displaystyle \alpha }\alpha affects the speed and spatial scale of the process; changing it has the same effect as changing the unit of measure for time (which affects the value of {\displaystyle {\dot {u}}}{\displaystyle {\dot {u}}}), and/or the unit of measure of length (that affects the value of {\displaystyle \nabla ^{2}u}{\displaystyle \nabla ^{2}u}). Therefore, in mathematical studies of this equation, one often sets {\displaystyle \alpha =1}\alpha =1. With this simplification, the heat equation is the prototypical parabolic partial differential equation.