Welcome back Mechanical EI. Did you know that the descriptor harmonic in the name harmonic function originates from a point on a taught string which is undergoing harmonic motion like the swinging of a pendulum? This makes us wonder what are harmonic functions? Before we jump in, check out the previous part of this series to learn about what millenithompson method is. Now harmonic function is a twice continuously differentiable function f which maps the set u into the set r where u is an open subset of r power n that satisfies laplace equation that is, dau squared f upon dau x1 squared plus dau squared f upon dau x2 squared and so on until the sum reaches dau squared f upon dau xn squared and is equal to 0 everywhere on u. This is usually written as delta f equals to 0. 1. Among the many applications of differential equations is to find curves that intersect a given family of curves at right angles. In other words, given a family of curves capital f, we wish to find curve or curves gamma which intersect orthogonally with any member of capital f whenever the intersect. It is important to note that we are not insisting that gamma should intersect every member of f but if they intersect the angle between their tangents at every point of intersection is 90 degrees. Such a family of curves gamma is called orthogonal trajectories of the family capital f that is at common point of intersection the tangents are orthogonal. In case the family capital f1 and capital f2 are identical we say that the family is self orthogonal. Hence we first saw what harmonic functions are and then went on to see what orthogonal trajectories are. So like, subscribe and comment with your feedback to help us make better videos thanks for watching. Also thanks a lot for those constructive comments you helped the channel grow. So here are the top mechanical ears of our last videos.