Integrating differential forms on curves

Question

How can I integrate the differential form $$\omega=x\,dx+y\,dy+z\,dz$$ in $\mathbb R^3$ on the curve $$c:[0,2\pi]\to\mathbb R^3: t\mapsto (e^{t\sin t}, t^2-2\pi t, \cos \frac{t}{2})?$$

Some advice would be nice thank you.

Answer 1

Just use the definition (with $c = (c_x, c_y, c_z)$):

$$\begin{align} \int \limits _c x \ \Bbb d x + y \ \Bbb d y + z \ \Bbb d z \\ &= \int \limits _0 ^{2\pi} c_x (t) c'_x (t) + c_y (t) c'_y (t) + c_z (t) c'_z (t) \ \Bbb d t \\ &= \int \limits _0 ^{2\pi} \frac 1 2 \left( c_x(t)^2 + c_y(t)^2 + c_z(t)^2 \right)' \ \Bbb d t \\ &= \frac 1 2 \left( c_x(t)^2 + c_y(t)^2 + c_z(t)^2 \right) \Big| _0 ^{2\pi} \\ &= \frac 1 2 (\| c(2 \pi) \|^2 - \| c(0) \|^2) \\ &= \frac 1 2 (0^2 + 0^2 + (-1)^2 - 0^2 - 0^2 - 1^2) \\ &= 0 . \end{align}$$

Notice that you didn't really have to use the concrete expression of $c$ in performing the integration, the only thing that mattered was the fact that $c$ is a closed curve.