Here is a plot of a surface called a paraboloid. The surface is produced by assigning a height to every point of the plane, so it can be presented as a function of two variables, with the \( z \)-axis a function of \( x, y \):
For every \( C \in \mathbb{R} \), the equation \( f(x,y) = C \) describes a level curve of the surface: the set of points on the surface lying at height \( C \).
Each such level curve is called an isocline. The level curve
is also called a nullcline.
For example, the isoclines of the paraboloid \( f(x,y) = x^2 + y^2 \) are the circles
For \( C = 1 \) we get \( x^2 + y^2 = 1 \). On this circle we draw vectors with direction \( \arctan(1) = 45^\circ \).
For \( C = 2 \) we get \( x^2 + y^2 = 2 \). On this circle we draw vectors with direction \( \arctan(2) \approx 63^\circ \).
For \( C = 4 \) we get \( x^2 + y^2 = 4 \). On this circle we draw vectors with direction \( \arctan(4) \approx 76^\circ \).
For \( C = 8 \) we get \( x^2 + y^2 = 8 \). On this circle we draw vectors with direction \( \arctan(8) \approx 83^\circ \).
Given an ODE of the form
at every point \( (x_0, y_0) \) the value \( y' = f(x_0, y_0) \) is the slope of the tangent to the solution function \( y \) at the point \( (x_0, y_0) \).
We sketch several isoclines of \( f(x,y) \), and on each isocline of the form \( f(x,y) = C \) we draw several vectors whose slope is \( C \). Continuing in this way we obtain the dense direction field of the ODE.
The solutions of the equation must be tangent to the dense direction field. The panel shows three solutions of the equation for the initial conditions