# Weierstrass–Erdmann condition

The Weierstrass–Erdmann condition is a technical tool from the calculus of variations. This condition gives the sufficient conditions for an extremal to have a corner.[1]

## Conditions

The condition says that, along a piecewise smooth extremal x(t) (i.e. an extremal which is smooth except at a finite number of corners) for an integral ${\displaystyle J=\int f(t,x,y)\,dt}$, the partial derivative ${\displaystyle \partial f/\partial x}$ must be continuous at a corner T. That is, if one takes the limit of partials on both sides of the corner as one approaches the corner T, the result must be the same answer.

## Applications

The condition allows one to prove that a corner exists along a given extremal. As a result, there are many applications to differential geometry. In calculations of the Weierstrass E-Function, it is often helpful to find where corners exist along the curves. Similarly, the condition allows for one to find a minimizing curve for a given integral.

## References

1. ^ Gelfand, I. M.; Fomin, S. V. (1963). Calculus of Variations. Englewood Cliffs, NJ: Prentice-Hall. pp. 61–63.