Problems

01

Shoelace Formula

Let $\mathbf{F}=(-\frac{1}{2}y,\frac{1}{2}x)$ give the components of a vector field in the plane. Observe that ${\iint }_{\mathcal{D}}\nabla {\times }_z\mathbf{F}dA=\text{Area}(\mathcal{D})$.

• (a) Let $C_1$ be a line segment from $(x_1,y_1)$ to $(x_2,y_2)$. Show that: $${\oint }_{C_1}\mathbf{F}\cdot d\mathbf{s}=\frac{1}{2}(x_1{y}_2-x_2{y}_1).$$
• (b) Let $\mathcal{C}$ be the path traversing a polygon with vertices at $(x_1,y_1),\dots ,(x_n,y_n)$, with $C_i$, $i=1,\dots ,n-1$ the edge segment connecting $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$. Show that the area of this polygon is given by: $$\text{Area}=\frac{1}{2}\sum _{i=1}^n{x}_i{y}_{i+1}-x_{i+1}{y}_i.$$
• (c) Use the previous formula to find the area of this polygon:
  

02

Green’s Theorem Puzzle

Consider the following region and its boundary with circles of radius 1, 1, and 5:

Suppose we know ${\oint }_{{\mathcal{C}}_2}\mathbf{F}\cdot d\mathbf{s}=3\pi$ and ${\oint }_{{\mathcal{C}}_3}\mathbf{F}\cdot d\mathbf{s}=4\pi$. Suppose we know that $\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}=9$ on the tan region. Calculate ${\oint }_{{\mathcal{C}}_1}\mathbf{F}\cdot d\mathbf{s}$.