Hey Mitch—just popping in here momentarily without the tooootally full context, but I think you can build some more insight on how these differential surface patches and the corresponding integrals are done in the Introduction to Multivariable Calculus course, particularly in the exercises for the lessons in Section 5 (Three Great Theorems). I worked on these solutions and the corresponding visualizations, and I think they're pretty great!

(As for Minors, if you check out the Applications section in the documentation, you can see how it comes in here—when one calculates a determinant by hand, they often will write out the minors of the matrix to make the calculation easier. You can indeed totally avoid it when just computing the differential "size" element for an integral in WL.)

Hi Gianluca;

Thank you so much for your response, because it certainly appears that your solution absolutely nailed the answer. I found it extremely interesting that you took both derivatives at once, within one derivative function, which created a single matrix containing both derivatives (x, y). Whereas, textbook examples show, taking two separate partial derivatives (one for x and another for y) and then combining them using a cross product before taking the norm. This method producing a different answer than obtained from using the SurfaceIntegrate[] function. Additionally, it seems that the method of combining two separate derivatives depends on whether the values are scalars or vectors (by adding scalars together or using the cross product to combine vectors) - is this correct?

Lastly, I noted that you used a function Minors[] which I am totally unfamiliar with. However, the function Minors[] did not seem to have any effect on the matrix, at least in my example, and therefore I was curious as to the intention of using it.

Again, thank you so much for your response, I certainly appreciate your time and insight, and I am still studying your solution to get a better understanding surface integrals.

Mitch Sandlin