Calculus

How a Jacobian Warps Space

How transformations locally scale area

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How a Jacobian Matrix Warps Space

Blue = originalOrange = transformed

Transformation

Parameter t0.50
Probe X1.00
Probe Y1.00

Jacobian Matrix

F(x,y) = (x·cos θ − y·sin θ, x·sin θ + y·cos θ)

0.878-0.4790.4790.878
det(J)1.0000
Area scaling1.0000×

💡 What this shows

The Jacobian measures how a transformation locally warps space. The determinant tells you how much a tiny square's area is scaled at that point. If negative, orientation flips.