Wednesday, October 28, 2015

the scalar differential operator , is called the Laplacian. The Laplacian is a good scalar operator (i.e., it is coordinate independent) because it is formed from a combination of divergence (another good scalar operator) and gradient (a good vector operator).

http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node163.html




Next: Curl Up: Vectors and Vector Fields Previous: Divergence



Laplacian Operator

So far we have encountered
\begin{displaymath} \nabla\phi = \left(\frac{\partial \phi}{\partial x},\, \frac... ...\phi} {\partial y},\, \frac{\partial \phi}{\partial z}\right), \end{displaymath}(1444)

which is a vector field formed from a scalar field, and
\begin{displaymath} \nabla\cdot{\bf A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z}, \end{displaymath}(1445)

which is a scalar field formed from a vector field. There are two ways in which we can combine gradient and divergence. We can either form the vector field $\nabla(\nabla\cdot{\bf A})$ or the scalar field $\nabla\cdot(\nabla\phi)$. The former is not particularly interesting, but the scalar field $\nabla\cdot(\nabla\phi)$ turns up in a great many physical problems, and is, therefore, worthy of discussion.
Let us introduce the heat flow vector ${\bf h}$, which is the rate of flow of heat energy per unit area across a surface perpendicular to the direction of ${\bf h}$. In many substances, heat flows directly down the temperature gradient, so that we can write
\begin{displaymath} {\bf h} = - \kappa \,\,\nabla T, \end{displaymath}(1446)

where $\kappa$ is the thermal conductivity. The net rate of heat flow $\oint_S {\bf h}\cdot d{\bf S}$ out of some closed surface $S$ must be equal to the rate of decrease of heat energy in the volume $V$ enclosed by $S$. Thus, we have
\begin{displaymath} \oint_S {\bf h}\cdot d{\bf S} = - \frac{\partial}{\partial t}\left( \int c\, T\,dV\right), \end{displaymath}(1447)

where $c$ is the specific heat. It follows from the divergence theorem that
\begin{displaymath} \nabla\cdot{\bf h} = -c\,\frac{\partial T}{\partial t}. \end{displaymath}(1448)

Taking the divergence of both sides of Equation (1446), and making use of Equation (1448), we obtain
\begin{displaymath} \nabla\cdot\left(\kappa \,\nabla T\right) = c\,\frac{\partial T}{\partial t}. \end{displaymath}(1449)

If $\kappa$ is constant then the above equation can be written
\begin{displaymath} \nabla\cdot (\nabla T) = \frac{c}{\kappa} \frac{ \partial T}{\partial t}. \end{displaymath}(1450)

The scalar field $\nabla\cdot (\nabla T)$ takes the form
$\displaystyle \nabla\cdot(\nabla T)$$\textstyle =$$\displaystyle \frac{\partial}{\partial x}\!\left(\frac{\partial T}{\partial x}\... ...right)+ \frac{\partial}{\partial z}\!\left(\frac{\partial T}{\partial z}\right)$ 
 $\textstyle =$$\displaystyle \frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \equiv \nabla^2 T.$(1451)



Here, the scalar differential operator
\begin{displaymath} \nabla^2 \equiv \frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}+ \frac{\partial^2}{\partial z^2} \end{displaymath}(1452)

is called the Laplacian. The Laplacian is a good scalar operator (i.e., it is coordinate independent) because it is formed from a combination of divergence (another good scalar operator) and gradient (a good vector operator).
What is the physical significance of the Laplacian? In one dimension, $\nabla^2 T$ reduces to $\partial^2 T/\partial x^2$. Now, $\partial^2 T/\partial x^2$ is positive if $T(x)$ is concave (from above) and negative if it is convex. So, if $T$ is less than the average of $T$ in its surroundings then $\nabla^2 T$ is positive, and vice versa. In two dimensions,
\begin{displaymath} \nabla^2 T = \frac{\partial^2 T}{\partial x^2}+ \frac{\partial ^2 T}{\partial y^2}. \end{displaymath}(1453)

Consider a local minimum of the temperature. At the minimum, the slope of $T$ increases in all directions, so $\nabla^2 T$ is positive. Likewise, $\nabla^2 T$ is negative at a local maximum. Consider, now, a steep-sided valley in $T$. Suppose that the bottom of the valley runs parallel to the $x$-axis. At the bottom of the valley $\partial^2 T/\partial y^2$ is large and positive, whereas $\partial^2 T/\partial x^2$ is small and may even be negative. Thus, $\nabla^2 T$ is positive, and this is associated with $T$ being less than the average local value.
Let us now return to the heat conduction problem:
\begin{displaymath} \nabla^2 T = \frac{c}{\kappa} \frac{\partial T}{\partial t}. \end{displaymath}(1454)

It is clear that if $\nabla^2 T$ is positive then $T$ is locally less than the average value, so $\partial T/\partial t>0$: i.e., the region heats up. Likewise, if $\nabla^2 T$ is negative then $T$ is locally greater than the average value, and heat flows out of the region: i.e., $\partial T/\partial t<0$. Thus, the above heat conduction equation makes physical sense.




为什么 空间二阶导（拉普拉斯算子）这么重要？
《数理方程》课上讲的三类基本方程，方程的一边都是拉普拉斯算符，另一边分别是时间二阶导、一阶导和0，为什么空间二阶导这么重要？它有什么样的数学和物理意义？
按投票排序按时间排序

12 个回答

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