Critical point (mathematics)

Critical point is a wide term used in many branches of mathematics.

The abscissae ("x-coordinates") of the red circles are stationary points; the blue squares are inflection points.

When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero.[1] When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero.[2][3] Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero.[4]

The value of the function at a critical point is a critical value.

This sort of definition extends to differentiable maps between Rm and Rn, a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called bifurcation points.

In particular, if C is a plane curve, defined by an implicit equation f(x,y) = 0, the critical points of the projection onto the x-axis, parallel to the y-axis are the points where the tangent to C are parallel to the y-axis, that is the points where . In other words, the critical points are those where the implicit function theorem does not apply.

The notion of a critical point allows the mathematical description of an astronomical phenomenon that was unexplained before the time of Copernicus. A stationary point in the orbit of a planet is a point of the trajectory of the planet on the celestial sphere, where the motion of the planet seems to stop before restarting in the other direction. This occurs because of a critical point of the projection of the orbit into the ecliptic circle.