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)

Quick facts

  • Formula: z = (1 - x^2 - y^2) * exp(-(x^2 + y^2) / 2)
  • Category: surface
  • Tags: surface, calculus, z=f(x,y)

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.

How to use this graph

  1. Rotate the surface by dragging with your mouse or finger
  2. Use the Embed button to copy an <iframe> for your LMS or lesson page
  3. Click PNG to export a classroom-ready image
  4. Click Formula or LaTeX to copy the equation for your notes

Embed this graph

Use the Embed button in the calculator to copy a ready iframe for blogs, LMS pages, and lesson notes.

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Teacher prompt

Question: Why is there a ring-shaped valley?

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

Related graphs

Explore more surface examples and compare shapes, 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.

Monkey Saddle

z = x^3 - 3*x*y^2

A three-way saddle with three valleys and three ridges.

Sine Wave Surface

z = sin(x) + sin(y)

Two perpendicular sine waves combined into one surface.

Explore more tools

Use these free browser tools to extend your graphing workflow.

3D Surface Plotter

Plot z=f(x,y), rotate, and export

Embed Builder

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Worksheet Generator

Print graph-based classroom worksheets

How to Graph z=f(x,y)

Step-by-step guide for beginners