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MathidiaMathematics Academy
Multivariable calculus
Lesson 31 of 75
Introduction

Surfaces and level sets

Objective. Represent multivariable functions geometrically.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A surface in multivariable calculus is the graph z = f(x, y) of a two-variable function, visualised as a curved sheet in three-dimensional space, while a level set (or level curve) is the set of points where f(x, y) = k for a fixed constant k, like a contour line on a map. Sketching level curves for several values of k and stacking them mentally builds a picture of the surface's shape without plotting it fully in 3D. For a paraboloid f(x, y) = x² + y², the level sets x² + y² = k are circles of radius √k, showing the surface is a bowl that widens as height increases. Level sets are especially useful for visualising steepness: closely spaced contours indicate a steep surface region.

Worked example

Describe the level set of the surface f(x, y) = x² + y² when f(x, y) = 9.

y) = x² + y² when f(x
Illustrated card showing y) = x² + y² when f(x
  1. 1Set the function equal to the given constant: x² + y² = 9.
  2. 2Recognise this as the equation of a circle centred at the origin.
  3. 3Identify the radius by comparing to x² + y² = r², so r² = 9.
  4. 4Take the square root to find the radius of the circle.

Answer: A circle of radius 3 centred at the origin

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Common mistake. Students confuse a level set f(x,y) = k, which is a curve in the xy-plane, with the surface z = f(x,y) itself, which lives in three dimensions; the level curve is a flat slice, not the surface.

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