Problems

01

Faster calculation of line integral using divergence

Let $\mathcal{D}$ be the box in the $xy$-plane defined by $x\in [0,2]$ and $y\in [0,3]$. Let $\mathbf{F}=(x^2,e^y)$. Compute the line integral ${\oint }_{\partial \mathcal{D}}\mathbf{F}\cdot {\mathbf{e}}_{\mathbf{n}}ds$ in two ways:

• (a) using separate line integrals over the four sides of the box,
• (b) using a double integral over the interior of the box.

02

Heat equation

Suppose the temperature in a region is given by the scalar field $T(x,y)$. Heat flows from hot to cold, following the negative temperature gradient $-\nabla T$.

The rate of total heat flow into a microscopic disk $D_r(a,b)$ enclosed by the circle $C_r(a,b)$ (radius $r$, centered at $(a,b)$) is the vector integral of heat flow across the loop $C_r$, namely: ${\oint }_{C_r}(-\nabla T)\cdot {\mathbf{e}}_{\mathbf{n}}ds$. Furthermore, the rate of change of $T(a,b)$ is proportional to the net heat flow into $D_r(a,b)$ (assuming $r$ is small enough).

Suppose that all temperatures are static in a region $\mathcal{D}$. (This doesn’t mean all temperatures are equal, since there could be a source or sink of heat on the boundary $\partial \mathcal{D}$ and a temperature gradient across $\mathcal{D}$.) This implies that the net heat flowing into and out of any disk $D_r(x,y)$ fully contained within $\mathcal{D}$ is zero.

Using the divergence theorem, explain why the temperature across $\mathcal{D}$ must satisfy the partial differential equation $\Delta T=0$. (Recall that the Laplacian operator $\Delta$ applied to $T$ gives $\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}$.)

03

Electric field: loop detector of contained charge

The planar electric field $\mathbf{E}$ generated by a point charge at the point $\mathbf{p}=(a,b)$ has the form:

$$\mathbf{E}=\left(\frac{x-a}{(x-a{)}^2+(y-b{)}^2},\frac{y-b}{(x-a{)}^2+(y-b{)}^2}\right).$$

Find the total electric flux exiting any given loop $\mathcal{C}$ in the plane. Your answer should be one value for any loop that encloses the point $\mathbf{p}$, and another value for loops that do not.

For any loop that does not enclose $\mathbf{p}$, consider the value of $\nabla \cdot \mathbf{E}$ in the region of the loop and apply the divergence theorem.

You cannot apply the divergence theorem to a region containing the point $\mathbf{p}$ because $\mathbf{E}$ is not defined at $\mathbf{p}$. So, for a loop that does enclose $\mathbf{p}$, you should first compute the answer directly for a small circular loop inside the given loop, and then apply the divergence theorem to the region enclosed between the larger given loop and your small circular loop.