Integration of a 1 form Along a Curve Example
Integration of a 1 - form along a curve reduces to an ordinary integral with limits.Let
\[\omega_{\mathbf{x}} = x_1^2 dx_1 + x_2^2 dx_2 +x_3^2 dx_3\]
and \[C\]
be the curve \[(cos t, sin t, t), \: 0 \leq < 2 \pi\]
.\[x_1 =cos t \rightarrow dx_1 = - sin t dt\]
\[x_2 =sin t \rightarrow dx_2 = cos t dt\]
\[ x_3 = t \rightarrow dx_3 = dt\]
Then
\[\begin{equation} \begin{aligned} \omega_{\mathbf{x}} &= cos^2 t (-sin t)dt + sin^2 t (cos t)dt +t^2 dt \\ &= - cos^2 t sin t dt + sin^2 t cos t dt +t^2dt\end{aligned} \end{equation}\]
The integral of
\[\omega_{\mathbf{x}}\]
along \[C\]
is then\[\begin{equation} \begin{aligned} \int^{2 \pi}_0 - cos^2 t sin t dt + sin^2 t cos t dt +t^2 dt &= [\frac{cos^3 t}{3} + \frac{sin^3 t}{3} + \frac{t^3}{3}]^{2 \pi}_0 \\ &= (\frac{1}{3} + 0 + \frac{8 \pi^3}{3}) - (\frac{1}{3} +0 +0) \\ &= \frac{8 \pi^3}{3} \pi \end{aligned} \end{equation}\]