Subject: Calculus

Differential Operator

In calculus, we define differential operator as an operator that indicates the operation of differentiation or simply the process of calculating a derivative. Suppose we have a variable y = f(x) which indicates that the independent variable x is and the dependent variable is y. We can therefore write the derivative of y in different notations which I will be discussing in the next part.

Notations

Generally, the notations for the derivatives of y = f(x) can be written as follows,

f'(x)=y'=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}f(x)=Df(x)=D_{x}f(x)

Please do take note that the notation dy/dx is not a ratio. This was introduced by Leibniz which is simply a synonym for f(x). Nonetheless, it is a very useful and suggestive notation, especially when used in conjunction with increment notation.

We can also rewrite the definition of a derivative in terms of Leibniz notation which takes the form of,

\frac{dy}{dx}=\lim_{\Delta x \to \infty}\frac{\Delta y}{\Delta x}

If we want to indicate a value of a derivative dy/dx in Leibniz notation at a specific number a, we therefore write it as

\frac{dy}{dx}|_{x-a}

which is just a synonym of f(a).

In higher calculus, you will frequently see a differential operator called the Laplacian operator. This is defined by

\Delta = \nabla^2 = \sum_{k=1}^{n}\frac{\partial^2}{\partial x_{k}^{2}}

The symbols Δ and ∇^2denotes the divergence of the gradient of a function on Euclidean space.

Properties of Differential Operator

Differential operators are linear. When we say linear, a differential operator follows the sum rule of differentiation and the constant factor rule in differentiation.

To expound that idea, suppose we have functions f and g and constants a and b. Now, if we have this expression

\frac{d}{dx}\left (a, f(x) + b, g(x)\right )

by sum rule of differentiation, we can express the above equation as

\frac{d}{dx}(a, f(x)) + \frac{d}{dx}(b, g(x))

Furthermore, by constant factor rule in differentiation, we could further simplify the above equation as

a\frac{d}{dx}f(x) + b\frac{d}{dx}g(x)

So we now have,

\frac{d}{dx}\left (a, f(x) + b, g(x)\right ) = a\frac{d}{dx}f(x) + b\frac{d}{dx}g(x)

In multivariable functions, differential operators may be used to only one variable of the function. This is the basic foundation of partial differential equations. In example, if we have z = f(x, y), we denote the partial differential operators as

f_{x}(x, y) = f_x = \frac{\partial f}{\partial x}= \frac{\partial }{\partial x}f(x, y)=\frac{\partial z}{\partial x}=f_1=D_1f=D_xf
f_{y}(x, y) = f_y = \frac{\partial f}{\partial y}= \frac{\partial }{\partial y}f(x, y)=\frac{\partial z}{\partial y}=f_2=D_2f=D_yf

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