Conjugate points of extermals of functional refer to pairs of points on a curve or surface where the tangent lines are parallel to each other. They play an important role in the theory of extremal problems.
Conjugate points of extermals of functional are closely related to extremal problems because they provide information about the behavior of a functional at critical points. They can help identify optimal solutions for these problems.
Conjugate points of extermals of functional are significant in mathematical optimization because they allow for the determination of optimal solutions and the characterization of the behavior of a functional at these points. This can aid in the development of efficient algorithms for solving optimization problems.
Conjugate points of extermals of functional are calculated using the concept of the second variation of a functional. This involves taking the second derivative of the functional and setting it equal to zero, and then solving for the critical points.
Yes, conjugate points of extermals of functional can exist for higher dimensional problems. In this case, they refer to points where the tangent hyperplanes are parallel to each other. The concept of conjugate points can be extended to higher dimensions, making it applicable to a wide range of optimization problems.