Partial Differentiation

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Partial Differentiation

Differentiation is the mathematical act of taking the derivative of a function. Partial differentiation is the mathematical act of taking the derivative of a function that depends on more than one variable with respect to one or more of those variables. If a differential equation contains partial derivatives with respect to more than one independent variable then it is called a partial differential equation (PDE). Partial differentiation is a pivotal technique used in chemistry, physics and engineering. Partial differential equations are in general very difficult to solve but their importance in applications warrants their use. Many partial differential equations describe situations in which a property is dependent upon not only time but also on position. Such a situation would be heat flow through a thin wire which involves position in the wire as well as time.

Partial differentiation involves a process by which the derivatives of a function containing multiple independent variables are found by considering all but the variable of interest as fixed during differentiation. Partial differentiation corresponds to the same thing as ordinary differentiation in that it represents an infinitesimal change in the function with respect to a given parameter. The difference is that partial differentiation is performed on an equation with more than one independent variable whereas ordinary differentiation is performed on an equation with only one independent variable. The partial derivative is usually denoted as: ∂f/∂x or f_x. These denote the partial derivative of the function f, that is comprised of more independent variables than just x, with respect to x. If a partial derivative is of second order or greater with respect to two or more different variables then it is called a mixed partial derivative. For example if f is a function of x, y, z (f(x, y, z)) and the partial derivative with respect to x is taken followed by the partial derivative with respect to y then it is called a mixed partial derivative and is denoted as: f_xy or ∂²f/∂x∂y. For such functions whose partial derivatives exist and are continuous, for nice functions, the mixed partial derivatives are equal no matter which differentiation is performed first: f_xy = f_yx.

In general partial differential equations are more difficult to solve using analytical methods than are ordinary differential equations. This is because often times they are more complex because of the multiple variables involved. There are several methods that have been developed over the years specifically designed to solve partial differential equations. Some of these methods include the Bäcklund transformation, characteristic partial differential equation, Green's function, Lagrange multiplier method, integral transform, Lax pair, and separation of variables. The Lagrange multiplier method was probably the first of these methods formally devised to solve partial differential equations. Joseph Louis Lagrange, an Italian mathematician, published this method involving multipliers to investigate the motion of a particle in space that is constrained to move on a surface defined by an equation involving three, independent variables. This method was published in Lagrange's book, Mecanique analytique, in 1778 and is currently used to maximize or minimize a function that is subject to a constraint. It can be employed in a variety of situations such as minimizing the fuel required for a spacecraft to reach its desired trajectory as well as maximizing the productivity of a commercial enterprise limited by the availability of financial, natural, and personnel resources. As well as these methods numerical methods can be applied to solve partial differential equations.

Although partial differential equations are in general difficult to solve, second-order partial differential equations are often easily solved via analytical solutions. They can be classified as elliptic, hyperbolic, or parabolic on the basis of a particular matrix or the discriminate of that matrix. Each class has a solution that is quite different from the other classes. Second-order partial differential equations that fall into the elliptic class produce stationary and energy-minimizing solutions such as Laplace's equation and Poisson's equation. Those that are classified as hyperbolic yield a propagating disturbance such as the wave equation. The last of the classes, the parabolic equations, produce a smooth-spreading flow of an initial disturbance such as the heat conduction equation and other diffusion equations.