Multivariable Differential Operators
Multivariable Differential Operators
These are some differential operators that apply to multivariable and/or vector valued functions.
Gradient
Given a scalar function $f : \mathbb{R}^n \to \mathbb{R}$, the gradient of $f$, denoted as $\nabla f$, is defined as the vector of its partial derivatives. Specifically, for a function $f(x_1, x_2, \cdots, x_n)$, the gradient is given by
\[\nabla f = \begin{bmatrix} \frac{\partial f}{\partial x_1} \\ \frac{\partial f}{\partial x_2} \\\ \vdots \\\ \frac{\partial f}{\partial x_n} \end{bmatrix}\]The gradient points in the direction of steepest ascent of the function $f$ at any given point and its magnitude gives the rate of ascent.
Curl
For a vector field $\vec{F} = (F_1, F_2, F_3)$ defined in three-dimensional space $\mathbb{R}^3$, with each copmonent function $F_i$ depending on the variables $x$, $y$, and $z$, the curl of $\vec{F}$ is defined as
\[\text{curl} \vec{F} = \nabla \times \vec{F} = \left ( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right ) \mathbf{\hat{i}} + \left ( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right ) \mathbf{\hat{j}} + \left ( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right ) \mathbf{\hat{k}}\]The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. Circulation is the line integral of a vector field around a closed curve.
More intuitively, curl measures the rotation of the vector field at a given point.
Curl can also be expressed as the determinant of a 3x3 matrix invovling the unit vectors, partial derivatives, and the components of the vector field:
\[\nabla \times \vec{F} = \begin{vmatrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\\ F_1 & F_2 & F_3 \end{vmatrix}\]Divergence
The divergence of a vector field quantifies the extent to which the vector field behaves as a source or a sink at a given point. For a vector field $\vec{F} = (F_1, F_2, F_3)$ defined in three-dimension space $\mathbb{R}^3$, the divergence is defined as
\[\text{div} \vec{F} = \nabla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}.\]Divergence gives a scalar value for each point, i.e., it is a scalar field. A positive value indicates a net flow away from the point, while a negative value indicates a net flow towards a point.
An important theorem related to divergence is the Divergence Theorem (also known as Gauss’s theorem) which connects the flux of a vector field through a closed surface to the divergence of the field inside the volume bounded by the surface:
\[\iiint_V (\nabla \cdot \vec{F}) dV = \oint_s \vec{F} \cdot d\vec{S}\]