Level Surface

If a function w = f (x,y,z) is assigned to a value c, i.e. f (x,y,z) = c, then the surface f (x,y,z) = c is named as the level surface of f. 

There are two functions given in the box below: f (x,y,z) and g(x,y,z). By default, f (x,y,z) = 1 is shown. Tick on other choices of c from the same function to observe the changes on the level suface. You may click here if you want to rotate the graph.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.

Level Curve

If a function z = f (x,y) is assigned to a value c, i.e. f (x,y) = c, then the curve f (x,y) = c is named as the level curve of f. 

If you take a look at the graph below, there is a paraboloid z = 10 - x² - y² at the right. By default, the choice of 'Show level curve z = 6' is ticked. The surface is 'cut' through by the plane z = 6 and the intersection yields the circle x² + y² = 4.

The circle is then drawn on the xy-plane at both columns. The circle on the xy-plane is the level curve.

Choose different level of c and observe the changes on level curves. Click here if you want to rotate the 3D graph.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.