 Welcome back MechanicalEI, 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 Milner-Thompson method is. Now, a 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, dou squared f upon dou x1 squared plus dou squared f upon dou x2 squared and so on until the sum reaches dou squared f upon dou xn squared and is equal to 0 everywhere on u. This is usually written as delta f equals to 0, one 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 they 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.