Domain Plane (x, y)
Original Space
Click/Drag crosshair to sample target point
(x: 1.50, y: 0.50)
Range Plane (u, v) = F(x, y)
Transformed Space
Mapped image point under F(x,y)
(u: 1.50, v: 3.88)
Transformation Map Definition
Invalid mathematical expression!
Jacobian Matrix J(x,y) Evaluation
∂P/∂x
1
∂P/∂y
0
∂Q/∂x
3*x^2
∂Q/∂y
1
det(J) at active point:
1.000
det(J) Expression:
1
Constant Determinant?
YES (Constant = 1)
Global Invertibility:
Globally Invertible
The Jacobian Conjecture
If F: ℂⁿ → ℂⁿ is a polynomial map with a non-zero constant Jacobian determinant, must F be globally invertible with a polynomial inverse?
• Local Area Scale: |det J| represents how unit areas dilate/contract locally.
• Non-Polynomial Counterexamples: Maps like (eˣ, y e⁻ˣ) have constant det=1 but fold locally / fail global invertibility!
Interactive Proof Bench
P(x,y), Q(x,y) ∈ ℝ[x,y]