∬

Surface Lab

Explore multivariable surfaces geometrically

1Surface

f(x,y)

x min

x max

y min

y max

x(s,t)

y(s,t)

z(s,t)

s min

s max

t min

t max

F(x,y,z) = 0

x radius

y radius

z radius

Implicit surfaces use marching tetrahedra inside the box ± the three radii. Click the surface or drag the white point to choose a location.

Mesh resolution

Opacity

Shading

Surface color

Teaching presets

00

00

P = (0, 0, 0)

2Derivatives & gradient

Partial derivatives

fₓ slicefᵧ slice

vertical planestangent lines

For z=f(x,y), this panel shows the two coordinate-direction slice curves and their tangent lines.

Gradient & directional derivative

∇f in xy-planelocal level curve

Direction angle θ = 30°

snap direction to ∇funit direction vectorvertical planeslice curvetangent line

∇f gives steepest increase; choose any unit direction u to compare Dᵤf = ∇f · u.

Vertical-plane appearance

Plane color

Transparency 28%

strong outline

3Level curves

Level z = c, c = 0

link point to level curve horizontal slicing plane intersection curve show many level curves

Stack spacing Δz

align to multiples of Δz

Plane color

Transparency 18%

strong plane outline

Move the horizontal plane to reveal the level curve where the surface has height z=c. With Δz = 1 and alignment on, the stack uses integer heights.

4Tangent plane / normal

tangent planenormal vector

Plane size

Tangent-plane appearance

Plane color

Transparency 22%

strong outline

For implicit F=0, the tangent plane uses ∇F as its normal.

5Second derivative test

D = fₓₓfᵧᵧ − fₓᵧ²
Move to a point to inspect local curvature and critical-point classification.

The max/min/saddle conclusion is mathematically decisive only at a critical point (where ∇f≈0). Away from critical points, D still describes the local Hessian determinant.

6Laplacian & harmonic mean-value view

Circle radius r = 0.75

circle on surfacesurrounding samplesclip to this disk

Δf = fₓₓ + fᵧᵧ. For a harmonic function, the center value equals the average value around every sufficiently small circle.

7View & window

Projection

axesshow xy-planeground gridsurface grid linescurrent parameter lineswireframedouble-sided

x window

y window

z window

clip to windowshow window box

© 2026 John Chase · Made with help from ChatGPT
Back to Teaching Apps
Version 1.1 · Version history

Surface Lab
Drag to orbit · right-drag to pan · wheel to zoom
Click the surface to choose a point · drag the white point along the surface.

x =

Speed 1.0×

y =

Speed 1.0×

Orbit: left drag · Pan: right drag · Zoom: wheel
Drag white point: move along surface · Double-click background: fit surface

Share this construction

This link stores the surface equations/type, domains, point, investigation settings, view settings, theme, and camera view.

Copied