Multivariate functions: Basic notions
Visualizing bivariate functions
Bivariate functions can be visualized in different ways; the most notable uses the concept of a graph. We recall that the graph of a function of a single variable is the set of points with and that the graph of a function of two variables is the set of points with . Usually, the graph is a surface in 3-dimensional space. Here are a few notions that help to gain visual insight into the graph.
Level curve, contour graph, and coordinate curve
Let be a bivariate function.
- The level curve of at level is the set of points with .
- The projection on the -plane of a level curve at level is called a contour graph for level . The contour graph at level is the set of all in the domain of such that for a fixed value .
- A curve of the form for a fixed value or of the form for a fixed value is called a coordinate curve of .
If is a point of the domain of , then the contour graph going through that point has equation . Note that this only makes sense if belongs to the domain of .
Level curves are curves on the graph of with constant function values, in other words, points on the surface where the function has the same value. On such a curve the value of is constant. In other words, the level curve at level is the intersection of the graph of with the plane with equation .
The difference between level curves and contour graphs is small: contour graphs are copies of level curves elevated at height , where is an arbitrary point of the contour graph.
In a drawing of a graph of a bivariate function, coordinate curves are among the means used to increase spatial suggestion. In mathematical software hues and shades are also used to suggest depth in the surface.
Some of these methods are illustrated in the examples below.

The contour graphs for are the sets of points given by the equation .
The first of these is and the last is . Here, every color represents a different value of .
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