Coarea Formula, part I: the Jacobian

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, (x,y) and change to a set of coordinates (u,v), according to a map \phi: \mathbb{R}^2 \to \mathbb{R}^2 which, since the domain is two dimensional, we may write as \phi(x,y) = (u(x,y),v(x,y)).  In this case, we have for an open set

\int_{\mathbb{R}^2}f(x,y)J\phi(x,y)~dx~dy = \int_{\mathbb{R}^2} f(u,v) ~du~dv

Let me finally define precisely what the Jacobian from calculus is- for a general map \phi: \mathbb{R}^n \to \mathbb{R}^n, we define

J\phi(x_1,\ldots, x_n) = \det \left(\phi^j_{x_k}\right)_{j,k = 1}^n,

where we are writing \phi = (\phi^1,\ldots, \phi^n), and \phi^j_{x_k} :=\frac{\partial \phi^j}{\partial x_j} .  As a quick example, one might recall changing coordinates from Euclidean (rectangular) to polar.  Typically it went x = r \cos{\theta} and y = r \sin{\theta} (that is to say, our change of coordinates is \phi(r, \theta) = (r\cos{theta},r\sin{theta})).  Then, using the “absolute value” notation for determinant, we have

J \phi(r,\theta) = \left| \begin{array}{cc} \cos{\theta} & \sin{\theta} \\ -r \sin{\theta} & r \cos{\theta} \end{array} \right| = r \cos^2 \theta + r \sin^2 \theta = r,

which returns us to the (somewhat) familiar formula,

\int f(x,y)~dx~dy = \int f(r,\theta) r~dr~d \theta.

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.


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