Level Sets & Gradient

A level curve is f(x,y)=c. At a regular point, ∇f is perpendicular to its tangent and points toward increasing f. Gold field arrows can show only direction or use one common proportional scale; the selected gradient is emphasized. The green tangent is obtained by rotating the normalized gradient90°. At ∇f=0 no gradient normal is drawn.

The bowl has ∇f=(2x,2y) and minimum at0. The saddle has ∇f=(2x,−2y), an isolated critical saddle and zero-level crossing y=±x. The ring minimum has ∇f=4(x²+y²−1)(x,y), a critical minimum circle r=1 and a local maximum at the origin. For0<c<1 its level set has two circles; c=1 also includes an isolated point. Nonlinear spacing keeps these inner levels visible.

Preset contours use analytic circle/hyperbola coordinates, displayed as finite polylines; the background is sampled on a100×100 grid. Axes have equal scale. Critical sets/zero levels are shown separately. Click/drag picks coordinates rounded to0.001; numeric fields keep entered precision. Invalid probes hide their overlay/readings, input changes discard stale plots, resize uses cached data. Arrows and curves are clipped to the shown domain [−2,2]².

OpenStax: directional derivatives, gradient and level-curve normals.

Choosing a function or arrow display updates the diagram immediately. Valid contour counts update after a short pause or Enter. The compact legend names the colors; its tooltips explain directions and the shared arrow scale. A completed plot is reconstructed from its exact preset parameters using the bounded analytic contour calculation. New or unfinished sessions do not automatically calculate. Invalid imports keep your current work.

Only input values are remembered in this browser. Calculated results and experiment histories are not saved.