I'm asked to do this computation from a differential manifolds text: $${d\alpha}=\left(\frac{\partial \:R}{\partial \:y}-\frac{\partial \:Q}{\partial \:z}\right)\left(dy\wedge \:dz\right)+\left(\frac{\partial \:P}{\partial \:z}-\frac{\partial \:R}{\partial \:z}\right)\left(dz\wedge \:dx\right)+\left(\frac{\partial \:Q}{\partial \:x}-\frac{\partial \:P}{\partial \:y}\right)\left(dx\wedge \:dy\right)$$ I've waded through a Calc 1-2 answers-provided book, and started a differential equations book, but I'm really not good at all at calculations of calculus. Any help?
It comes from the chapter on "Tensor Fields: local theory" under the section "exterior derivative" in a subsection concerning grad, curl, div. Also, the differential equations textbook it too advanced for me to understand anything except the first chapter. I'm sort of looking at solving math backwards from harder problems to put in place the basics. It just how my mind works, and it keeps me interested, and interesting.
The one-form $\alpha = Pdx + Qdy + Rdz$ which is analogous to vector field $F = Pi + Qj + Rk$ where $i, j, k$ are $\frac\partial{\partial \:x}, \frac\partial{\partial \:y}, \frac\partial{\partial \:z}$. I hope this makes it more clear. Thanks.
Also, $$curl F= \left(\frac{\partial \:R}{\partial \:y}-\frac{\partial \:Q}{\partial \:z}\right)i+\left(\frac{\partial \:P}{\partial \:z}-\frac{\partial \:R}{\partial \:z}\right)j+\left(\frac{\partial \:Q}{\partial \:x}-\frac{\partial \:P}{\partial \:y}\right)k$$ I'm starting to see that the computation would entail the relation between the two large equations here.
Rather, thanks Duncan Rampage for pointing me to the beginning of the subsection on Exterior Derivative. I feel the very definition of it gives the basis for the computing required in the question. I had read the chapter a week ago without much connection to it. Feel stupid now for asking, but thank you.