This is the question I got on my final assignment (Calculus III):
Evaluate the surface integral
\begin{equation}
\int \int_S xy \; \; dy\wedge dz – yz \; \; dz\wedge dx + xz \; \; dx\wedge dy
\end{equation}Where $S$ is the part of the plane $x+y+z=1$ lying in the first octant. Use $x$ and $y$ as parameters.
I am quite confused. I asked around and someone told me that this is also the symbol for something called the Wedge product, which I've not heard of before and appears in neither my calculus textbooks (Stweart's and Div Grad Curl) nor any of my Linear Algebra books.
From what I saw online, I still don't understand how it would make sense in this equation.
Is this a typo? Seems like a strange typo. Should it just read:
\begin{equation}
\int \int_S xy \; \; dydz – yz \; \; dzdx + xz \; \; dxdy
\end{equation}
Best Answer
On a higher level, the preference for wedge notation makes perfect sense: an object we integrate over a surface should be a differential $2$-form. But in calculus courses it is more common to talk about integrating vector fields over a surface (the flux integral $\iint_S \vec F\cdot \vec{dS}$). The flux is easier to visualize and to relate to physical concepts such as Faraday's Law.
As long as we work in $\mathbb R^3$ with its standard coordinate system, the correspondence between vector fields and $2$-forms is as follows: $$\vec i \mapsto dy\wedge dz$$ $$\vec j \mapsto dz\wedge dx$$ $$\vec k \mapsto dx\wedge dy$$ The rule is to keep the variables in cyclic order $xyz$. This is a special case of Hodge dual.