Calculus
Lab: functions of two variables and surfaces
Connect domain, surface, level sets and traces. Rotate the 3D graph and choose a point in the plane: check domain membership before computing its height.
Read the results
Surface z=f(x,y)
Drag the surface or use the sliders. Arrow keys also work when the graph is focused. Play rotates 15° per second.
Domain and level set in the xy-plane
Tap the plane to choose (x₀,y₀), or enter coordinates above. Green: domain; grey: excluded; orange: f(x,y)=k; blue: chosen point. Solid boundary included, dashed excluded.
Trace: fix one variable
Value table
The window [−R,R]² is not the domain. The surface uses an approximate mesh: faces with excluded points or |z|>H are omitted, not flattened; valid zones near boundaries may be missing. The vertical scale adapts to H. Level sets use analytical formulas but are drawn as segments. A plot does not prove limits or continuity.
Calculations use floating point. Very close to a boundary, check the algebraic condition: color and numerical rounding are not exact proofs.