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3D surface explorer

Shape a surface with the sliders and read the calculus off the model: height, gradient and the steepest way uphill.

Surface family
Evaluate at a point (x, y ∈ [−3, 3])

Result

z = (x² + y²)

z = 2
at (1, 1) — the orange marker on the surface
Gradient ∇f(2, 2)
Steepest slope |∇f|2.828
Steepest ascent45° from +x
z-range on domain[0, 18]

The origin is a local (and global) minimum — the bottom of the bowl. Second-derivative test: D > 0, fxx > 0.

The surface z = (x² + y²) rendered in 3D over the square domain, colored by height
How is this calculated?

Using the tool

Pick a surface family, shape it live with the sliders, and read the calculus off the model: the height z at your point, the gradient ∇f, and which way is steepest uphill — on a surface you can spin with your hand.

The math

A surface z = f(x, y) assigns a height to every point of the plane. The gradient ∇f = (∂f/∂x, ∂f/∂y)at a point tells you two things at once: it points in the direction of steepest ascent, and its length is the slope you'd climb walking that way. Where the gradient is zero the surface is momentarily flat — a critical point— and the second-derivative test (D = fxx·fyy − fxy²) sorts it into minimum (D > 0, fxx > 0), maximum (D > 0, fxx < 0), or saddle (D < 0). Each family here shows one behavior: the bowl and the bell have true extrema, the saddles do not, and the cone shows that a surface can be continuous yet have no gradient at all at a corner.

New to the math? Learn the concept: Higher dimensions, made visual

Formula sources: Standard multivariable calculus: partial derivatives, the gradient, and the second-derivative test

Gradients are exact analytic derivatives per family; the z-range is sampled on a fine grid. A reference instrument for reasoning and teaching — not a general CAS.