Gradient and level curves

Parametric equations

The level curve of a function $f(x,y)$ of two variables is the set of points where $f(x,y)=c$. The gradient $\nabla f$ is the vector of partial derivatives. It points where $f$ increases fastest, and it is orthogonal to the level curve through that point.

This page uses three standard examples: circular level curves $f=x^2+y^2$ (the gradient points outward along the radius), the saddle $f=x^2-y^2$, and $f=xy$, whose level curves are rectangular hyperbolas. For the parallel level lines of the linear objective $z=px+qy$, see Level curves of the objective.

Interaction

• Circle x²+y² / Saddle x²−y² / Hyperbola xy: switch among the three functions and watch the level curves change from circles into two families of hyperbolas
• Level curve family: lay the neighboring level curves over the figure and compare them with the one through the test point
• Drag the test point $P$ on the figure and read $f(P)$ and $\nabla f(P)$. The gold arrow is the gradient. The faint line is the tangent to the level curve at $P$. The two stay orthogonal
• Move the test point to the origin and the gradient arrow disappears. The figure marks the critical point $\nabla f=0$

What to notice

• The gradient is orthogonal to the level curve. Along the tangent, $f$ hardly changes. Along the arrow, $f$ changes fastest.
• The larger $\Vert \nabla f\Vert$ is, the more sharply the same displacement changes the function. In the readout, $\Vert \nabla f\Vert$ gets smaller near the origin, and at a critical point the arrow disappears.
• On the circle, the gradient lies along the radius. On the saddle and on $xy$, the level curves are hyperbolas. The gradient is still orthogonal to the tangent there, but the direction is not necessarily along the radius.

Related

• Level curves of the objective
• Basic patterns of a vector field
• Tangent approximation
• A function and its derivative

Further reading

• Gradient (Wikipedia)
• Level set (Wikipedia)