There’s a formula called the coarea formula which I have been researching for the past year or so. There are two good ways to think about it. One is to look at the so-called “Jacobian” and seek to interpret the integral of that number. The second is to look at it as a natural dual (in a colloquial, rather than mathematical sense) to its more-famous-brother, the area formula. We deal with the Jacobian today.
The Jacobian is typically introduced in calculus courses, and associated with a change of variables. In a typical case, you would like to take and integral in one set of coordinates, and change to a set of coordinates , according to a map which, since the domain is two dimensional, we may write as . In this case, we have for an open set
Let me finally define precisely what the Jacobian from calculus is- for a general map , we define
where we are writing , and . As a quick example, one might recall changing coordinates from Euclidean (rectangular) to polar. Typically it went and (that is to say, our change of coordinates is ). Then, using the “absolute value” notation for determinant, we have
which returns us to the (somewhat) familiar formula,
That seems like well over enough for a first post. Next up: an intuition for what the Jacobian measures, as well as a definition of Jacobian for maps between spaces of different dimensions.
Also! I should mention that the words I use are almost surely wrong- typically what I call the “Jacobian” is called the “Jacobian determinant”, while the actual “Jacobian” is the matrix of derivatives. It just seems like a mouthful.