A change of variables rewrites a problem using new coordinates that better match its shape or symmetry. In one variable, substitution uses dx = (dx/du) du to adjust for stretching along a line. In two or three variables, the same idea works, but the stretching can happen in several directions at once.
The Jacobian measures how much a tiny area or volume changes during the transformation.
For a transformation x = x(u, v) and y = y(u, v), a tiny rectangle in the u-v plane becomes a small parallelogram in the x-y plane. The area scale factor is the absolute value of the determinant J = det([[∂x/∂u, ∂x/∂v], [∂y/∂u, ∂y/∂v]]). This factor must be included when changing variables in a double or triple integral.
It is especially useful for polar, cylindrical, spherical, and custom coordinate systems that turn complicated regions into simpler ones.
Understanding Calculus: The Jacobian and Change of Variables
The determinant has a geometric meaning that is more useful than memorising a formula. Hold one new coordinate fixed and change the other by a tiny amount. Each change produces a small direction arrow in the original plane.
Together, those arrows form the sides of a parallelogram. Its area tells you the local scale change. A determinant near three means that a tiny coordinate patch covers about three times as much physical area.
A determinant near one half means the patch is compressed. The sign records whether the mapping preserves the usual turning direction or flips it. Integrals measure positive area, so the absolute value is used.
Polar coordinates show why a scale factor can depend on position. A small change in angle does not have one fixed length. Near the origin, rotating through a small angle moves only a short distance.
Farther out, the same angular change traces a longer arc. For a narrow polar patch, one side has length close to a small radial change. The other has length close to radius times a small angular change.
Its area is therefore radius times the two small coordinate changes. Forgetting the radius factor gives too much weight to patches near the centre and too little weight to patches farther away. This is a common error in area, mass, and probability calculations.
A reliable method starts by choosing coordinates that make the boundary simple. Then rewrite the function using the new variables. Next, find the scale factor from the determinant.
Finally, describe the new region carefully with limits that cover every point once. The limits are often the hardest part. A circle may become a rectangle in radius and angle, while a slanted region may need split limits.
Sketching both coordinate pictures helps. Check a sample point on each boundary to make sure the new limits match the old region. If a transformation covers part of the region twice, the integral will count that area twice unless the domain is restricted.
A zero determinant is a warning sign. At such a point, two independent coordinate directions have become aligned, or one direction has been flattened. The mapping loses local information there, so nearby original points may share the same new coordinates.
Polar coordinates have this issue at the origin because every angle describes the same point when the radius is zero. This does not always prevent an integral from being evaluated, since one point has zero area, but it matters when claiming that a transformation has an inverse.
Students meet these ideas in physics when calculating mass from a density over a disk, electric charge over a curved region, fluid flow through a volume, and probability over regions with circular symmetry. The Jacobian is what keeps those measurements physically correct after coordinates are changed.
Key Facts
- For x = x(u, v), y = y(u, v), the Jacobian determinant is J = ∂(x, y)/∂(u, v) = (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u).
- Area scaling in 2D is dA_xy = |J| dA_uv = |J| du dv.
- The change-of-variables formula is ∬_R f(x, y) dA = ∬_S f(x(u, v), y(u, v)) |J| du dv.
- For polar coordinates, x = r cos θ, y = r sin θ, and |J| = r, so dA = r dr dθ.
- For 3D transformations, dV_xyz = |∂(x, y, z)/∂(u, v, w)| du dv dw.
- If J = 0 at a point, the transformation locally collapses area or volume there and may not be locally invertible.
Vocabulary
- Jacobian determinant
- The determinant of the matrix of first partial derivatives that gives the local area or volume scale factor of a transformation.
- Change of variables
- A method for rewriting an integral in new coordinates by transforming both the integrand and the differential area or volume element.
- Transformation
- A rule that maps points from one coordinate system to another, such as (u, v) to (x, y).
- Local scale factor
- The factor by which a transformation stretches or shrinks a tiny length, area, or volume near a point.
- Invertible mapping
- A transformation that can be reversed so each output point in the region corresponds to exactly one input point.
Common Mistakes to Avoid
- Forgetting the absolute value of the Jacobian: the determinant can be negative because of orientation, but area and volume scale factors must be nonnegative.
- Using dx dy = du dv without a scale factor: changing coordinates changes the size of tiny pieces, so the factor |J| is required.
- Computing the wrong Jacobian direction: if you express x and y in terms of u and v, use ∂(x, y)/∂(u, v), not its reciprocal.
- Changing the integrand but not the region: the old region R in x-y space must be converted into the new region S in u-v space.
Practice Questions
- 1 Let x = 3u and y = 2v. Compute the Jacobian determinant ∂(x, y)/∂(u, v), then find the area in the x-y plane of the rectangle 0 ≤ u ≤ 4, 0 ≤ v ≤ 5.
- 2 Use polar coordinates to evaluate ∬_R (x^2 + y^2) dA, where R is the disk x^2 + y^2 ≤ 9.
- 3 A transformation maps a square grid in the u-v plane into a curved grid in the x-y plane. Explain why a tiny rectangle is approximated by a parallelogram and why the determinant of the derivative matrix gives its area scale factor.