Jacobian determinant for change of variables
The Jacobian determinant measures the local area or volume scaling of a differentiable coordinate transformation.
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Formula
J has units (unit x · unit y)/(unit u · unit v); it is dimensionless when all coordinates are dimensionless.
Variables & units
| Symbol | Meaning | Unit |
|---|---|---|
| Determinant of the derivative matrix for (u,v) ↦ (x,y) | unit ratio (x·y)/(u·v) | |
| Coordinates in the target region | respective coordinate unit | |
| Coordinates in the source region | respective coordinate unit | |
| Area element in the target coordinates | area unit |
LaTeX
J=\det\!\left(\frac{\partial(x,y)}{\partial(u,v)}\right), \qquad dA=|J|\,du\,dv
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Derivation & background
For a continuously differentiable, locally invertible map T(u,v)=(x(u,v),y(u,v)), the area element transforms as dA=|J(u,v)|du dv. In an integral, this gives ∬D f(x,y)dA= ∬G f(T(u,v))|J(u,v)|du dv. The absolute value is needed because a negative determinant reverses orientation but does not create negative area. For a global change of variables, also check the transformed region, possible multiple coverage, and points where J=0.
Exam blueprint
Validity range
For a continuously differentiable, locally invertible transformation; points with J=0 and multiple coverage of the target region need separate treatment.
Derivation steps
The derivative maps a small coordinate rectangle locally to a parallelogram whose area scale is |det J|.
- 1Form the matrix of partial derivatives of x,y with respect to u,v.
- 2Compute its determinant and take the absolute value for area integrals.
- 3Replace dA by |J|du dv and transform both integrand and region.
Rearrangements
Transform an area element
Only where J≠0 on a locally invertible region.
Task variant
Find the area scale for x=2u, y=3v.
J=det[[2,0],[0,3]]=6.
The unit square maps to a rectangle of area 6.
Common mistakes
Interpreting a negative J as negative area.
Use |J| for area; the sign describes orientation only.
Exam context
- Coordinate changes in double and triple integrals, such as Cartesian to polar.
Formula cluster
Multivariable calculus
The determinant connects derivatives to local area and volume scaling.
Worked example
For x=2u and y=3v, the Jacobian is det([[2,0],[0,3]])=6. The unit square in the uv-plane maps to a rectangle with side lengths 2 and 3, so its area is 6. Check: ∬D 1 dA=∫01∫016 du dv=6.
Applications
Simplify double and triple integrals by switching to polar, cylindrical, or other coordinates, while tracking how area or volume changes.
How do you calculate with Jacobian determinant for change of variables?
Here is how to work through a typical Jacobian determinant for change of variables (J = ∂(x,y)/∂(u,v); dA = |J| du dv) task step by step:
- 1
Task
Find the area scale for x=2u, y=3v.
Solution path
J=det[[2,0],[0,3]]=6.
The unit square maps to a rectangle of area 6.
Frequently asked questions about Jacobian determinant for change of variables
What is the geometric meaning of a Jacobian determinant?+
The Jacobian determinant gives the local factor by which a coordinate transformation enlarges or shrinks small areas or volumes. For x=2u and y=3v, J=6, so a small unit-area rectangle in uv coordinates becomes an area-six rectangle in xy coordinates. In multiple integrals this factor corrects for the changed integration coordinates. For a nonlinear transformation it generally depends on location. J is therefore a local scale factor, not a global area measurement for an arbitrarily large region. For a nonlinear map, J is recalculated at every integration point and therefore belongs in the integrand; for a linear map, the area factor is constant.
Why does the integral use |J|?+
The determinant can be negative when a transformation reverses orientation, for example by reflecting an axis. A geometric area is still positive, so the area element transforms as dA=|J|du dv. The determinant’s sign records orientation, not negative area. The same absolute-value rule applies to volume scaling. Orientation can matter in vector integrals, however; there the absolute value must not be inserted automatically, and the relevant transformation law must be used instead. A reflection reverses orientation while geometric area remains positive. For oriented curve or flux integrals, orientation must instead be handled separately.
What does it mean when J=0?+
If J vanishes at a point, the linearized transformation loses at least one direction there: a small area element is locally compressed to a line or a lower-dimensional set. The map is not locally invertible at that point. A change of variables may therefore be singular there; check whether the point is merely a measure-zero boundary or whether the transformation covers the region more than once. One such point does not make every integral impossible, but the standard formula must be applied only on a valid domain.
What is the Jacobian factor in polar coordinates?+
With x=r cos θ and y=r sin θ, the derivative matrix has determinant r, so the area element is dA=r dr dθ. For a disk of radius R, integrate r from 0 to R and θ from 0 to 2π: ∫02π∫0ᴿ r dr dθ=πR2. The factor r matters: without it, every thin circular ring would have the same area even though circumference grows with radius. The polar map is singular at the origin, but that single point has zero area. For sectors or annuli, adjust the integration limits.
When must a transformed region be split?+
The standard substitution formula assumes that each point in the target region is covered appropriately, without unaccounted multiple counting. A map can be locally invertible yet send several source points to the same target point globally. Integrating over the whole source region would then count part of the target more than once. Split the domain into regions where the map is one-to-one and transform each part separately. You may instead correct by the coverage number when it is constant and known. Also inspect points where J=0 and the boundary curves. The absolute value |J| handles orientation, not multiple coverage.
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