Definition:Curl Operator/Determinant Form

Definition

Let $R$ be a region of space embedded in Cartesian $3$ space $\R^3$.

Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.

Let $\mathbf V$ be a vector field acting over $R$.

The curl of $\mathbf V$ at a point $A$ in $R$ is defined as:

$\curl \mathbf V = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ \dfrac \partial {\partial x} & \dfrac \partial {\partial y} & \dfrac \partial {\partial z} \\ V_x & V_y & V_z \end {vmatrix}$

where:

$V_x$, $V_y$ and $V_z$ denote the magnitudes of the components at $A$ of $\mathbf V$ in the directions of the coordinate axes $x$, $y$ and $z$ respectively.

Also see

• Determinant Form of Curl Operator

• Results about the curl operator can be found here.

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $4$. The Curl of a Vector Field: $(4.9 \ bis)$