How this calculation works
Build the symbolic partial-derivative matrix with output rows and input columns. Its signed determinant describes the first-order orientation and area change of the forward map. Draw the input grid and its nonlinear image in separate panels, then compare the sampled image of a finite square with its local linearized parallelogram.
det J = (∂u/∂x)(∂v/∂y)−(∂u/∂y)(∂v/∂x); forward local area scale is |det J|A worked example
For u=x²−y² and v=2xy at (1,0.5), J=[[2,−1],[1,2]] and det J=5. A unit input square therefore has a local first-order signed area prediction of five. Its finite nonlinear mapped boundary has a different oriented area, illustrating why a pointwise Jacobian is not an exact finite-region area formula.
Good to know
Two-input, two-output real transformations only. Bounded symbolic derivatives are evaluated numerically; nonsmooth points or unresolved removable singularities can fail. The finite mapped boundary is sampled, and its signed polygon area is not a certified geometric union area if the image folds. Equal coordinate scales apply within each panel, with different zoom allowed between panels. No global injectivity or change-of-variables validity proof.
Reference: OpenStax Calculus: change of variables in multiple integrals (opens in a new tab) ↗
State the direction of the transformation first
This page treats (x,y) as input coordinates and (u,v) as outputs of the map T(x,y)=(u(x,y),v(x,y)). The Jacobian matrix has one row for each output and one column for each input. Its first row contains uₓ and uᵧ; its second row contains vₓ and vᵧ. Keeping that order explicit prevents a determinant from being attached to the wrong transformation.
For this forward map, the local area-element relation is du dv=|det D(u,v)/D(x,y)| dx dy under the appropriate hypotheses. If a calculation instead needs the inverse map, its factor is the reciprocal where the inverse exists. A formula with unspecified input/output direction can accidentally apply the correct determinant in the wrong place.
Work through the original nonlinear example
Take u=x²−y² and v=2xy. Differentiation gives uₓ=2x, uᵧ=−2y, vₓ=2y and vᵧ=2x. At (1,0.5), the matrix is [[2,−1],[1,2]]. Its determinant is 2×2−(−1)×1=5, and its absolute local area scale is also five.
A small input displacement is transformed approximately by multiplying it by this matrix. The point itself maps to (0.75,1). The grid view shows the complete sampled nonlinear transformation around that point, while the comparison boundary distinguishes the actual sampled image from the first-order prediction. Neither the matrix nor the point image alone describes every point in a finite region.
Separate orientation from an unsigned area scale
A negative determinant indicates reversal of local orientation. Its absolute value gives the local area scaling used in the usual change-of-variables formula. For the reflection u=−x, v=y, the signed determinant is −1 while the unsigned scale is one. Losing the sign would hide the reflection; treating a negative determinant as a negative geometric area would also be misleading.
A determinant that is zero or near zero at the floating-point diagnostic scale needs further analysis. It can indicate local collapse or numerical cancellation, but the screen does not certify exact rank or global behavior. A nonzero determinant at one point likewise does not prove that a map is one-to-one over an entire region.
Compare a finite square with the infinitesimal prediction
The comparison square is centered at the base point, with area equal to its side length squared. Multiplying that area by the pointwise signed determinant gives a first-order prediction. For a genuinely linear affine map this agrees with the signed area of the mapped parallelogram, apart from numerical roundoff.
For a nonlinear map the derivative can vary across the square. The red boundary follows sampled transformed points; the green dashed boundary follows the linearized map at the center. Their difference is part of the explanation, rather than a reason to rename the pointwise determinant as an exact finite-area answer. Shrinking the square can illustrate local agreement, but no error bound is certified by that visual experiment.
The output area metric is the oriented area of the sampled boundary polygon. If the map folds or makes the boundary overlap itself, that algebraic area can differ from the geometric area of the union of the image. Sampling also approximates curved boundary segments. Keep these distinctions when interpreting a number as an area, orientation or integration factor.
Read the two grid panels with their scales
The input panel shows straight coordinate grid lines around the selected point. The output panel shows their mapped samples, which can curve, expand, compress, overlap or develop gaps where the formula is undefined. Both panels use equal coordinate scales internally, but each chooses its own overall zoom so that the relevant geometry fits.
Because the two panels can have different zoom, comparing their apparent square sizes in pixels is not a numerical area measurement. Use the determinant and area metrics for that comparison. The point markers identify the base input and its image, and the table supplies each symbolic partial derivative with its value at the base.
Check the domain before relying on a derivative matrix
Bounded expressions such as exp(x), sin(y) and sqrt(x) are supported in addition to polynomials. The map and the required partial-derivative expressions must be defined at the base point. A sampled square boundary must also stay within the supported real domain. Undefined grid samples are omitted and split the drawing instead of being joined across a known gap.
Symbolic differentiation here applies rules to a limited grammar rather than performing every possible algebraic simplification. A derivative with a removable singularity can still fail evaluation, and a nonsmooth expression such as absolute value at its corner needs separate mathematics. The calculator does not infer a domain proof from a finite drawing.
The polynomial Jacobian calculator offers a separate local-displacement comparison for supported polynomial maps. The interactive 2D and 3D vector field plotter displays sampled field arrows rather than a coordinate transformation of a square. Those related views should not be mistaken for interchangeable meanings of the same graph.
A few questions, answered
Why are both signed and absolute determinants shown?
The signed value carries local orientation, while its absolute value gives an unsigned local area scale.
Is det J times a unit square’s area always the exact mapped area?
No. For a nonlinear map it is a pointwise first-order prediction. The finite-region derivative can vary across the square.
Does the mapped polygon area always equal geometric area?
No. It is a sampled oriented boundary area and can differ from geometric union area when the image folds or overlaps.
Do both panels have the same zoom?
Not necessarily. Each uses equal x/y scales internally, but their overall zoom can differ.
Does a nonzero determinant prove global invertibility?
No. Global one-to-one behavior and valid integration substitutions need additional hypotheses.