Operator Notation, Divergence and Curl
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Operator notation
In the context of calculus, an operator is a function that takes functions as inputs and produce either a function or a number as an output.
Some examples:
- The identity operator \(I\): it receives a function and produces the exact same function as output.
- First derivative of a real function of one variable: takes a differentiable function and produces a function.
- Integral over a given interval: takes an integrable function and produces a number.
- An indefinite integral with a fixed lower limit: takes an integrable function and produces a function.
- Partial derivative of a function of two variables with respect to \(x\): takes a function of two variables and produces a function of two variables.
- Gradient of a function of two variables: takes a function of two variables and produces a two-dimensional vector field.
- Curvature: takes a vector function of single variable and produces a real function of single variable.
We usually denote operators using uppercase letters, Greek letters, or special symbols. Applying an operator to a function is usually denoted without parentheses, unless order of operations must be specified, and is considered a form of multiplication.
For example, a simple first derivative operator would be defined as
\[D f = f'\]
Then \(D \sin\) would be \(\cos\) and so on.
Operators can generally be added and multiplied by a scalar. Given two operators, \(A\) and \(B\), the operator \(2A + 3B\) would be defined the following way:
\[(2A + 3B)f = 2(Af) + 3(Bf)\]
where the addition and scalar multiplication on the right side of the equality is a regular addition and scalar multiplication of functions.
In contrast to that, multiplication or operators usually means composition, so \((AB)f = A(Bf)\). For example, taking the first derivative operator \(D\) defined above and multiplying it by itself would give us a derivative of derivative, which is the second derivative. Note that while composition in general is not commutative, when composing an operator with itself, the order clearly does not matter, so the multiplication of \(D\) by \(D\) can be abbreviated as \(D^2\).
For example, the operator \(D^2 + 2D + 5I\) would take a second derivative of a function, add to it 2 times the first derivative of the same function, and add 5 times the same function to that. In other words, \[(D^2 + 2D + 5I)f = f'' + 2f' + 5f\]
Partial derivative operators
When working with functions from \(\mathbb{R}^2\) to \(\mathbb{R}\), we have two partial derivative operators:
- Partial derivative with respect to \(x\), denoted either as \(\frac{\partial}{\partial x}\) or \(D_x\).
- Partial derivative with respect to \(y\), denoted either as \(\frac{\partial}{\partial y}\) or \(D_y\).
In \(\mathbb{R}^3\), we also have the partial derivative with respect to \(z\), denoted either as \(\frac{\partial}{\partial z}\) or \(D_z\).
This can of course be extended to any number of variables.
Vectors of operators
We can form a symbolic vector whose components are operators. Applying such operator to a function would work similarly to scalar multiplication, except that contrary to the usual convention, the “scalar” would be on the right side of the vector instead of the left side. The result of such operator would then be a vector function. For example, an operator \(\mathbf{V} = \left\langle D, D^2, D^3\right\rangle\), when applied to a function \(f\), would produce a vector function whose first component would be \(f'\), second component \(f''\) and third component \(f'''\).
Gradient operator
An important example of an operator vector is the gradient operator. In \(\mathbb{R}^2\), it is defined as \[\mathbf{\nabla} = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right\rangle\] In \(\mathbb{R}^3\), it is \[\mathbf{\nabla} = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle\]
Example: the Laplace operator
A very important operator in mathematical physics is the Laplace operator \(\Delta\), also known as Laplacian. It can be defined in a number of different ways, one of them being the dot product of the gradient operator with itself!
\[\Delta = \mathbf{\nabla}\cdot\mathbf{\nabla}\]
In fact, the Laplace operator is sometimes also denoted as \(\nabla^2\).
How does it work? Since \(\mathbf{\nabla} = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right\rangle\), and since dot product works by multiplying the corresponding components and adding all the products together,
\[\begin{align} \Delta &= \mathbf{\nabla}\cdot\mathbf{\nabla}\\ &= \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right\rangle\cdot\left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right\rangle\\ &= \frac{\partial}{\partial x}\frac{\partial}{\partial x} + \frac{\partial}{\partial y}\frac{\partial}{\partial y}\\ &= \frac{\partial^2}{(\partial x)^2} + \frac{\partial^2}{(\partial y)^2} \end{align}\]
So the Laplace operator takes the second pure partial derivatives and adds them together:
\[\Delta f(x,y) = f_{xx}(x,y) + f_{yy}(x,y)\]
The same can be done in \(\mathbb{R}^3\) (or in any number of dimensions): \[\Delta f(x,y,z) = f_{xx}(x,y,z) + f_{yy}(x,y,z) + f_{zz}(x,y,z)\]