Line integrals in polar coordinates

Given the thorem:
If

Unparseable latex formula:

$$ z = x+iy, f(z)=u(x,y)+iv(x,y) $$

Then

Unparseable latex formula:

$$ \int_\gamma f(z) dz = \int_\gamma {(u dx - v dy)} + i \int_\gamma {(u dy + v dx) } $$

How do modify this theorem for

z = re^{i\theta}

?

is it simply:

f(z)=u(r,\theta)+iv(r,\theta)

Unparseable latex formula:

$$ \int_\gamma f(z) dz = \int_\gamma {(u dr - vr d\theta)} + i \int_\gamma {(ur d\theta + v dr) } $$

?

Thanks,
jb444

Th theorem works because

dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y}dy = dx + idy

by the chain rule.

The expression for dz in terms of r and theta is

dz = \frac{\partial z}{\partial r} dr + \frac{\partial z}{\partial \theta}d\theta

.

I'll leave you to work out how this affects the integrands.

LaTex

What is an LLM and how could it benefit your career?

Powerful questions business students forget to ask their career advisers

Why I chose a law degree and what studying law was like