surface example

3D Mexican Hat Surface Graph

A central peak surrounded by a circular valley.

z = (1 - x^2 - y^2) * exp(-(x^2 + y^2) / 2)

Teacher prompt

Why is there a ring-shaped valley?

The factor 1 minus radius squared becomes negative outside the center.

min z 0.00max z 0.0056 samples

What this graph represents

The polynomial term changes sign while the exponential term damps the edge.

Where it appears in calculus

Useful for radial symmetry and wavelet intuition.

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Related graphs

Open another surface page and compare shape, slices, and contour behavior.

Saddle Surface

z = x^2 - y^2

A saddle surface curves up in one direction and down in the perpendicular direction.

Gaussian Surface

z = exp(-(x^2 + y^2))

A smooth bell-shaped surface centered at the origin.

Elliptic Paraboloid

z = x^2 + y^2

A bowl-shaped surface that opens upward.

Inverted Paraboloid

z = 12 - x^2 - y^2

A dome-shaped surface with a highest point at the center.