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Hi Max, Suppose you have a piece of some material, maybe a concrete block, a piece of wood or perhaps a stone. Suppose this material is in 3-dimensional space and I can determine the co-ordinates of any point in this material. Let be the temperature of this material in degrees C at the point (x,y,z) at time t seconds. Hence u is a function of 4 variables so I can write u as u(x,y,z,t). In the heat equation \frac{\partial u}{\partial t} = a \bigtriangledown^2 u the left side is the partial derivative of u with respect to t. This is the rate at which the temperature is changing over time at each point and is in the units of degrees Celsius per second. On the right side a is a property of the material called the thermal diffusivity, and \bigtriangledown^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}. The expression \frac{\partial u}{\partial x} is the partial derivative of u with respect to x and measures the rate at which the temperature is changing as you move in the x direction. The expression \frac{\partial^2 u}{\partial x^2} is the second partial derivative of u with respect to x and measures the rate at which the temperature change is changing as you move in the x direction. It's a kind of acceleration or curvature of the temperature function u as you move in the x direction. Taken together \bigtriangledown^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} measures the "curvature" of u. Usually the people who work with the heat equation are trying to solve for u under various conditions. I hope this helps, |
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Math Central is supported by the University of Regina and the Imperial Oil Foundation. |