第二章
前置知识：傅里叶定律， 热量守恒定律
热传导方程 \(dQ=-k(x,y,z)\frac{\partial{u}}{\partial{n}}dSdt\)
在曲面S包围的$\Omega$区域内部由于热传导而流失的热量为 \(Q=\int_{t_2}^{t1}\iint\limits_{S}k(x,y,z)\frac{\partial{u}}{\partial{n}}dsdt\)
由$\frac{\partial{u}}{\partial{n}}ds$在x,y,z方向的分量为 \((\frac{\partial{u}}{\partial{x}}i+\frac{\partial{u}}{\partial{y}}j+\frac{\partial{u}}{\partial{z}}k)ds=\frac{\partial{u}}{\partial{x}}dydz +\frac{\partial{u}}{\partial{y}}dxdz+\frac{\partial{u}}{\partial{z}}dxdy\) 由于高斯公式： \(\int\int\limits_{S}\nabla\cdot{F}dS=\int\int\int\limits_{V}\nabla\cdot{F}dxdydz\) 那 \(\frac{\partial{u}}{\partial{n}}ds=\nabla{u}\cdot{ds}\) \(\int_{t_2}^{t1}\iint\limits_{S}k(x,y,z)\frac{\partial{u}}{\partial{n}}dSdt=\int_{t_2}^{t1}\int\int\limits_{V}\nabla\cdot(k\nabla{u})dxdydz\)
从物体热量变换的角度看，由比热容$C$和温度$T$组成的物体，其热量变化为 \(\int\int\int Cp\int_{t_1}^{t_2}\frac{\partial{u}}{\partial{t}}dxdydz\) 根据热量守恒得到：
\(\int\int\int Cp\int_{t_1}^{t_2}\frac{\partial{u}}{\partial{t}}dxdydz=-\int_{t_2}^{t1}\int_{S}k(x,y,z)\frac{\partial{u}}{\partial{n}}dSdt\) 变量分离法
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