Implicit Functions:

Derivatives, Graphing, and Orthogonal Trajectories

Functions are often defined implicitly, G(x,y,c) = 0, where c is a parameter.� In the case that c can be solved for explicitly we have F(x,y) = c, which is the form we will be dealing with in this module.� For instance

can be rewritten as�� , an example we discuss later.

Implicit differentiation can be accomplished by differentiating the equation while treating y as a differentiable function of x� and applying the chain rule,.� If

�giving

, which can be readily solved for y�(x):

.� Our first example will allow you to enter an implicit function, view its derivative and also specify the number of contours to be graphed.

Orthogonal trajectories to a given family of curves is based on the simple idea that perpendicular lines have slopes that are negative reciprocals of each other.� Starting with a family of curves one can implicitly differentiate, obtain the derivative, take its negative reciprocal, and integrate, if possible, to obtain the orthogonal family. One can easily verify this procedure by starting with �leading to the orthogonal trajectory equation .�

There is also a multivariable approach using the gradient concept. The surface z = F(x,y) contains the level curves F(x,y) = Const and the gradient vector, �is perpendicular to them.� Since (dx, dy) is tangent to these level curves we obtain the differential equation of the level curves from the scalar product� ��with , which can be viewed as an alternative formula for finding implicit derivatives.� The differential equation of the orthogonal trajectories is obtained by taking the negative reciprocal of the right hand side, namely� .� Written as �this will be readily integrable if it passes the exactness test which occurs if and only if F(x,y) is a harmonic function; i.e., it satisfies LaPlace�s equation.

.� We also note that the Cauchy-Riemann condition for analytic functions implies that the real and imaginary part of an analytic function gives rise to orthogonal curves.� A particular application relates to two-dimensional flow of an irrotational and incompressible fluid. A specific example is the flow around an inside corner. If �, the streamlines of the flow arise from the Im(f(z)) and the velocity potential from Re(f(z)). In this case the application relates to the curves in the first quadrant.

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