## Level Curves

We need a method of visualising a function of two variables,We can find the value offor anyby substitution, but cannot plotin two dimensions. We can however equatewithand construct a plot in three dimensions, or we can put whereis a constant and plot the set of pointssatisfyingfor various values ofThe set of points satisfyingis called a level curve ofThe relationship between a 3D sketch ofand the level curves ofis shown below.

Each value ofdefines a level curve ofshown by the black contour lines above. We may reduce the sketch to two dimensions by labelling each contour with the associated(or constant).