I am taking a class called calculus on manifolds, we are learning about tensors, alternating tensors and k forms in hope to arrive at stokes theorem in order to integrate k-forms over manifolds. We are using Munkres book on Analysis on manifolds.
I am currently preparing for my final exam and having trouble with some past exam questions. I only have my notes and the book to go by and ive spent most of my time working through the theorems to try work it out. The question is as follows
Let $dx, dy, dz$ be a basis of elementary 1-forms on $R^3$. Consider the 1 form $\omega(x,y,z)$ and 2-form $\eta(x,y,z)$ on $R^3$ defined as follows;
$$\omega(x,y,z)=3xydx+2zdy+5dz$$ $$\eta(x,y,z)=4xydx\wedge dy +2(x^2+1)dy\wedge dz$$
(a) Detemrine $\omega\wedge\eta$ and $\omega\wedge\star\omega$, with $\star:\Omega^l(R^3)\rightarrow\Omega^{3-l}(R^3)$, the hodge star operator.
(b) Let the map $\alpha:R^2\rightarrow R^3$ be given by the equation $$\alpha(u,v)=(u+v,u-v,u^2+v^2)$$
Determine $d\omega, \alpha_*\omega, \alpha_*(d\omega)$ and $d(\alpha_*\omega)$. Which general relation does this illustrate.
End of question:
So first I will explain my understanding and attempts (a) I think I have this one, using theorems of the properties of the wedge product I found
$$\omega\wedge\eta=xydx\wedge dy\wedge dz[6x^2-17]$$
I am having trouble with putting my understanding of the hodge operator into action on an explicit function. I have tried using the fact $\star dx=dy\wedge dz $ but I don't know if what I am doing is right. I came to the following answer;
$$3xydy\wedge dz+2zdx\wedge dz +5dx\wedge dy$$ For the final part of that question I got $$(3xy)^2(dx\wedge dy\wedge dz)-4z^2dx\wedge dy\wedge dz-25dx\wedge dy\wedge dz$$ I am not sure if this is what the corrext answer is supposed to look like. Again, in the context I am not quite sure what this means.
(b) is where I am having most of my problems, I found the first part to it to be $$-xdx\wedge dy-2dy\wedge dz $$ Again, Im nots sure if this is what the answer is supposed to look like, very few examples were done in class It was mainly proving all of the theorems we were using so I am not certain if these are what my answers are supposed to look like.
The final three parts, I do not know where to begin. The $\alpha_*\omega$ was taught to us as a push forward maps as a linear map on tangent spaces where $(x;v)$ are elements of $T_x(A)\rightarrow T_{\alpha(x)}(B)$ defined as \ Let $ A\in\mathbb{R}^k $ open, and $ B\in\mathbb{R}^n $ open. $ \alpha:A\rightarrow \mathbb{R}^n $ such that $ \alpha(A)\subset B$, Then $ \alpha^{\star}\Omega^l(B)\rightarrow \Omega^l(A) $ \begin{equation} (\alpha^{\star}w)(x)((x;v_1),\ldots(x;v_l))=w(\alpha(x))(\alpha_{\star}(x;v_1),\ldots,(\alpha_{\star}(x;v_l)) \end{equation}
If anyone could help me in applying this definition to get an answer Id appreciate it. Finding it quite difficult even just to formulate a question to ask. I have tried going to my lecturer to confirm my solutions and ask for guidance but he has not gotten back to me since.
Thank you in advance.
I am not familiar with the Hodge star, but the other wedge product computation seems to be in order. As far as part $(b)$ you have a map $\alpha(u,v) $ and so $\alpha_* = \textbf{J}(d_{(u,v)} \alpha)$ which is just the differential of $\alpha$ which you know how to compute.
$$D_{(u,v)} \alpha = \begin{pmatrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v} \\\frac{\partial y}{\partial u} &\frac{\partial y}{\partial v} \\\frac{\partial z}{\partial u} &\frac{\partial z}{\partial v}\end{pmatrix}_{(u,v)}$$
where $\alpha(u,v) = (x,y,z)$. You shouldn't be writing $\alpha_*$ if you wish to do something with $\omega$ since $\alpha_*$ is a linear map on tangent vectors. Are you sure you shouldn't have the pullback instead i.e $\alpha^*$, then this will make sense.