Definition:Curl Operator/Determinant Form
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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
- 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: $(4.9 \ bis)$