integrate over a cube given some differential form

Question

What is process of integrating a differential form given some cube (hyperdimensional obejcts)? I read a lot qualitative problems on this, but seem to find rare examples on how to compute such integrations step by step. For example, if I have a 2-cube$$[0,1]^2->R^3$$ with c defined to be $$c(t_1,t_2)=(t_1^2,t_1t_2,t_2^2)$$ and I want to integrate dα on this, where α is:$α=x_1dx_2+x_1dx_3+x_2dx_3$, how should I approach this generally? In a neater way, what is $\int_{c}dα$?

Answer 1

Pull back the form to $[0,1]^2$ and integrate it there. The computation looks like this: we have $(x_1,x_2,x_3) = (t_1^2, t_1 t_2, t_2^2)$ and thus $(dx_1,dx_2,dx_3) = (2t_1 dt_1, t_1 dt_2 + t_2 dt_1, 2t_2 dt_2)$. Differentiating $\alpha$ we have $$d\alpha = dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_2 \wedge dx_3.$$ Substituting the expressions for $dx_i$ in we get $$c^* d\alpha = 2t_1^2 dt_1 \wedge dt_2 + 4t_1 t_2 dt_1 \wedge dt_2 +2t_2^2 dt_1 \wedge dt_2.$$ Thus $$\int_c d\alpha = \int_{[0,1]^2} c^* d\alpha = \int_{[0,1]^2}2(t_1 + t_2)^2dt_1 \wedge dt_2 = \int_0^1 \int_0^1 2(t_1 + t_2)^2 dt_1 dt_2.$$