Jacobian Calculator
Work out jacobian instantly with clear inputs, formula shown and shareable results.
About Jacobian Calculator
The Jacobian determinant of a two-variable transformation is (∂u/∂x)(∂v/∂y) − (∂u/∂y)(∂v/∂x). It measures how much the map stretches area locally, and it is the factor that appears when changing variables in a double integral.
Formulas
Jacobian
J = ∂u/∂x · ∂v/∂y − ∂u/∂y · ∂v/∂x
Change of variables
dA = |J| du dv
Worked example
With the default values below, Jacobian Calculator returns Jacobian determinant of 5.00000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- ∂u/∂x
- 2
- ∂u/∂y
- 1
- ∂v/∂x
- 3
- ∂v/∂y
- 4
Results
- Jacobian determinant
- 5.00000000
- Local area scaling factor
- 5.00000000
- Local invertibility
- Locally invertible
- Orientation
- Orientation preserved
Step by step
- 1Values used
- 2Jacobian
- 3Change of variables
- 4Jacobian determinant= 5.00000000
- 5Local area scaling factor= 5.00000000
- 6Local invertibility= Locally invertible
- 7Orientation= Orientation preserved
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
For the partials 2, 1, 3, 4?
J = 8 − 3 = 5, so areas are stretched fivefold and orientation is preserved.
What is the polar Jacobian?
r, which is why dA = r dr dθ in polar coordinates.