Calculus
Surfaces: area, mass and flux
Follow a cell from the parameter square to the spatial surface. Tangents determine the normal; area weight and field produce three different sums.
Standard normal: upward on plane and paraboloid; outward on sphere, hemisphere, cylinder, cone and torus. The cone bounds the solid below its mantle. Reversing the normal multiplies N by −1 without changing the surface.
With u=s² the first tangent includes 2s. This increasing change is not a double covering and preserves the chosen normal direction. Displayed formulas describe the base map before substitution.
This experiment replaces 2πu by 4πu in the cylinder formula and its derivatives.
The surface in space
Green: added cells; pale: pending; purple: current cell. Orange/blue arrows: the two tangents; red: unit normal; dark green: F. Arrows are rescaled for display: do not compare lengths across types. The current cell and arrows remain highlighted even on the back. Drawn faces approximate geometry; calculation uses derivatives.
The parameter square
Select a cell to jump to its step. Back/step buttons provide the same keyboard navigation.
D=[0,1]²; ΔuΔv=1/n²
A=∬J du dv; M=∬ρJ du dv; Φ=∬F·N du dv
J=‖r_u×r_v‖; N=±(r_u×r_v)
Parametrized area A
Mass M
Oriented flux Φ
Current cell: every step
All cells and contributions
Compare convergence
Midpoint sums over n² cells with case-specific analytic references. Each step adds a cell, not a physical time interval. Error comparisons are neither certified bounds nor proofs. No free expressions are executed.