这是一份anu澳大利亚国立大学MATH2305/MATH6405的成功案例

$$
\nabla \wedge \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} .
$$
This is Faraday’s law of induction which says that time-varying magnetic fields generate electric fields.
$$
\frac{1}{\mu_{0}} \nabla \wedge \mathbf{B}=\varepsilon_{0} \frac{\partial \mathbf{E}}{\partial t}+\mathbf{j}
$$
When there are currents present they appear as a source term $\mathbf{j}$, the current density, on the right-hand side of this equation, which is revealed as the model for generation of magnetic fields by currents. The term $\varepsilon_{0} \partial \mathbf{E} / \partial t$ is Maxwell’s inspiration, the displacement current.
$$
\nabla \cdot \mathbf{B}=0, \quad \nabla \cdot \mathbf{E}=0 .
$$
The first of these says that there are no ‘magnetic monopoles’ (magnetic fields are only generated by currents, and magnetic lines of force have no ends), and the second is a special case of $\nabla \cdot \mathbf{E}=\rho / \varepsilon_{0}$, showing the generation of electric fields by charges.

MATH2305/MATH6405 COURSE NOTES ：

$$
\frac{\rho c U a}{k}\left(\frac{\partial T^{\prime}}{\partial t^{\prime}}+\mathbf{u}^{\prime} \cdot \nabla^{\prime} T^{\prime}\right)=\nabla^{\prime 2} T^{\prime}
$$
We see that there is just one dimensionless combination in this problem,
$$
\mathrm{Pe}=\frac{\rho c U a}{k}=\frac{U a}{\kappa}
$$
known as the Peclet number. There is no dimensionless parameter in the boundary conditions because they scale linearly, to become
$$
T^{\prime} \rightarrow 0 \quad \text { as } \quad r^{\prime} \rightarrow \infty, \quad T^{\prime}=1 \quad \text { on } \quad r^{\prime}=1
$$