What is the Jacobian determinant of the polar map (r, theta) -> (r cos theta, r sin theta)?

## Computing it The polar-to-Cartesian map takes `(r, theta)` to `(u, v)` with `u = r cos(theta)`, `v = r sin(theta)`. This is a map from R^2 to R^2, so its Jacobian is 2 by 2, with rows indexing the outputs u and v and columns indexing the inputs r and theta: `J = [[du/dr, du/dtheta], [dv/dr, dv/dtheta]] = [[cos(theta), -r sin(theta)], [sin(theta), r cos(theta)]]` The determinant of a 2 by 2 matrix `[[a, b], [c, d]]` is `a d - b c`, so `det J = cos(theta) * r cos(theta) - (-r sin(theta)) * sin(theta) = r cos^2(theta) + r sin^2(theta) = r`. The Pythagorean identity collapses the whole expression to r, independent of the angle — a pleasing result that is also easy to check for slips: if your answer still contains a trigonometric function, you have likely made a sign error on the off-diagonal entry. ## What the number means A Jacobian determinant is a local scaling factor for area (in two dimensions) or volume (in general). The Jacobian is the best linear approximation of the map near a point, and the determinant of a linear map is the factor by which it multiplies areas. So a tiny rectangle in the `(r, theta)` plane with sides dr and d(theta) is carried to a patch in the plane whose area is approximately `|det J| dr d(theta) = r dr d(theta)`. This is geometrically obvious once seen. A wedge at radius r spanning a small angle d(theta) has arc length `r d(theta)`, and giving it radial thickness dr makes its area about `r dr d(theta)`. The same angular slice covers much more area far from the origin than near it, and the determinant is precisely that r-dependence. It is also why rewriting a plane integral in polar coordinates carries an extra factor of r: the substitution must correct for the fact that the coordinate grid is not area-preserving. ## The sign and orientation Here `det J = r >= 0`, so with r positive the map preserves orientation: a counterclockwise loop in the `(r, theta)` rectangle maps to a counterclockwise loop in the plane. A negative determinant would mean the map flips orientation, mirroring the local picture. Area scaling uses the absolute value; the sign carries the orientation information. ## The degenerate point At r = 0 the determinant is zero. Concretely, every angle theta maps to the same point, the origin, so the map is not one-to-one there and cannot be locally inverted. More generally, a vanishing Jacobian determinant at a point says the linear approximation squashes some direction to zero, collapsing volume; the inverse function theorem guarantees a local inverse exactly where the determinant is nonzero and the derivative is continuous. The polar map is well behaved on `r > 0` with theta restricted to an interval of length 2 pi, and misbehaves only on that single degenerate ray of the parameter domain. ## Connecting back to composition Determinants multiply: `det(A B) = det(A) det(B)`. Since the chain rule makes the Jacobian of a composition the product of the stage Jacobians, the volume-scaling factors of the stages simply multiply too. If one map doubles areas locally and the next triples them, the composition scales them by six. That is a genuinely useful consistency check on a multi-stage change of coordinates: compute the determinant of each stage separately and confirm the product matches the determinant of the composed map. It also explains why a composition is locally invertible only if every stage is: one zero factor kills the whole product. ## Common errors The frequent mistakes are dropping the r from the `du/dtheta` entry — differentiating `r cos(theta)` with respect to theta gives `-r sin(theta)`, not `-sin(theta)` — and losing the minus sign, which happens to leave the determinant looking similar in special cases but is wrong in general. Another is reporting `r^2`, usually from confusing the area factor with the substitution used in a different coordinate system, or from squaring somewhere in the identity. A quick reality check helps: the determinant must have units of length here, since it converts a length times an angle into an area, and only a factor linear in r does that. ## Answering well Write the 2 by 2 matrix, compute the determinant explicitly, state the value as r, then explain the meaning as local area scaling and mention the r = 0 degeneracy. Adding that determinants multiply along a composition shows that the result is understood as part of the chain rule picture rather than as an isolated exercise.