Wednesday:
 Study: Finish Section 4.2 and read the first part of
section 4.3.
 Do:
 In Section 4.2: 5, 6a and 14.
 (EP1) Recall from multivariable calculus that if
$\mathbf{F}(x,y)=(P(x,y),Q(x,y))$ is a vector field continuous on a
contour $\Gamma$ parameterized by $z(t)=(x(t),y(t))$ with $t\in
[a,b]$ (we would write $z(t) =x(t)+iy(t)$ in Math 43), then the
"line integral" is $$\int_\Gamma \mathbf{F}\cdot
d\mathbf{r}=\int_\Gamma P\, dx + Q\,dy,$$ where, for example,
$$\int_\Gamma P\,dx=\int_a^b P(x(t),y(t))x'(t)\,dt.$$ (If we think
of $\mathbf{F}$ as a force field, the line integral gives us the
work done in traversing $\Gamma$ through $\mathbf{F}$.) Now
suppose that $f(x+iy)=u(x,y)+iv(x,y)$ is continuous on
$\Gamma$. Find $P$, $Q$, $R$ and $T$ such that $$ \int_\Gamma
f(z)\,dz=\int_\Gamma P\,dx+Q\,dy +i \Bigl(\int_\Gamma R\,dx +
T\,dy\Bigr). $$
