Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on $\mathbb{R}^3$ and let $\omega$ be the $1$-form on $\mathbb{R}^3$ given by $$\omega=xy^2dz$$

i. Compute $c^*\omega$

ii. Compute $c^*d\omega$

iii. Compute $\int_cd\omega$

iv. Without using stokes theorem compute $\int_{\partial c}\omega$

Answer:

i. $(c^*\omega)(s,t)=(\omega\circ c)(s,t)=\omega\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)=(\frac{1}{2}s^2)(st)(d(\frac{1}{2}t^2))=(\frac{1}{2}s^3t)(tdt)=\frac{1}{2}s^3t^2dt$

ii. $d\omega=d(xy^2dz)=y^2dx\wedge dz+2xydy\wedge dz+xy^2d(dz)=y^2dx\wedge dz+2xydy\wedge dz$

$c^*d\omega=(st)^2d(\frac{1}{2}s^2)\wedge d(\frac{1}{2}t^2)+s^3td(st)\wedge d(\frac{1}{2}t^2)$

$=(st)^3ds\wedge dt+d^3t^3ds\wedge dt=2(st)^3ds\wedge dt$

I am stuck on parts iii. and iv.

Is there a formula for these types of problems?

I have not seen an example of one of these computed before

What is the name of this type of problem so that I can look up methods for it?

Any help would be greatly appreciated

Answer 1

Remember that $$\int_c d\omega=\int_{I^2}c^*d\omega.$$

I supppose $I=[0,1]$, now its just a easy calculation.