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Select a preset scalar field from the dropdown, or enter a custom expression in the text box.

The Point button shows a single point.

The Slice button shows a single level set.

Level sets shows contour lines (2D) or isosurfaces (3D) at evenly spaced values.

Heat map shows the field as a colour map.

In 2D mode, the Surface button plots z = φ(x,y) as a 3D surface.

Toggle ∇φ to display the gradient field.

Overview

A scalar field φ assigns a number to each point in space. It can be visualised via a heat map (colour coding the value) or via level sets (curves or surfaces where φ is constant).

In 2D we can plot the graph z = φ(x,y) and visualise the scalar field as a surface.

The gradient of φ, written ∇φ, is a vector field that points in the direction of greatest increase of φ, with magnitude equal to the rate of change:

In 2D:   ∇φ = (∂φ/∂x, ∂φ/∂y)
In 3D:   ∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z)

∇φ is always perpendicular to the level sets of φ. Where the level sets are closely spaced, |∇φ| is large; where they are far apart, |∇φ| is small.