ISEGORIA / MATH ENCYCLOPEDIA
Differential forms: circulation, curl, and pullbacks
Move between line integrals, area integrals, and coordinate changes.
Before you begin: Multivariable calculus and differential geometry
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. A line integral
Integrate a one-form ω along a circular path traced a chosen number of turns. Pick an exact form d(xy), a form with nonzero dω, or the angle form dθ, which is closed but has a hole at the origin. Drag the centre of the path and its starting point: the path is coloured by the sign of ω(γ̇), and the graph shows the integral accumulating as the path is traced.
Worked example. For ω = dθ, a circle once around the origin gives 2π; a circle that does not enclose the origin gives 0.
Watch out. dθ satisfies dω = 0 away from the origin, but it is not the differential of any function on the punctured plane; path independence needs an exact form, or a closed form on a simply connected domain.
What changes when the path winds around a hole?
A field can be locally curl-free yet have nonzero circulation around a missing point.
2. Stokes’ theorem
For the one-form ω = (−y/2 + y³/3) dx + (x/2 − x³/3 + 0.4x²) dy, the colour shows dω, positive in the middle and negative outside the dashed circle. Drag the disk and its edge: the circulation of ω around the boundary always equals the integral of dω over the disk. The graph checks this for every radius at once, and reversing the orientation flips both sides together.
Worked example. Changing orientation reverses both integrals together.
Watch out. Equality assumes a smooth form and an oriented surface with compatible boundary.
Why do both sides change sign?
Reversing orientation reverses the induced boundary orientation.
3. Pullbacks and area
The map φ(u, v) = (su + kv, v + cu²) bends the (u, v) square into the plane. Drag the small cell: its image has signed area close to det Dφ times the cell's area, and the dashed parallelogram, Dφ applied to the cell, is the linear approximation. With enough bend and shear, det Dφ changes sign along a fold line, where the image folds over and orientation reverses.
Worked example. A positive determinant preserves orientation; a negative one reverses it.
Watch out. The map is chosen for its simple Jacobian. Where it folds, the integral of the pullback counts overlapping parts of the image with opposite signs.
What does the determinant encode?
The signed local area scale of the map.