Vector Curl
The curl of a vector field is a vector operator that describes the infinitesimal rotation or circulation of the field at each point in space, with its direction given by the axis of rotation (right-hand rule) and its magnitude equal to the rate of rotation. A zero curl means the field is irrotational (conservative), while non-zero curl indicates rotational behaviour. Curl appears in Faraday's law of electromagnetic induction, in the Navier-Stokes equations for fluid vorticity, and in defining conservative force fields in classical mechanics.
Key Formula
curl F = ∇ × F = (∂F_z/∂y − ∂F_y/∂z) î + (∂F_x/∂z − ∂F_z/∂x) ĵ + (∂F_y/∂x − ∂F_x/∂y) k̂
LaTeX: \nabla \times \mathbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \partial_x & \partial_y & \partial_z \\ F_x & F_y & F_z \end{vmatrix}
| Symbol | Meaning | Unit |
|---|---|---|
| ∇×F | Curl of vector field F (vector result) | per unit length |
| F_x, F_y, F_z | Components of the vector field F | depends on field |
| î, ĵ, k̂ | Unit vectors along x, y, z axes | dimensionless |
Worked Example
Problem
Find the curl of F(x, y, z) = (y², −xz, x²y).
Solution
Step 1: î-component: ∂(x²y)/∂y − ∂(−xz)/∂z = x² − (−x) = x² + x. Step 2: ĵ-component: ∂(y²)/∂z − ∂(x²y)/∂x = 0 − 2xy = −2xy. Step 3: k̂-component: ∂(−xz)/∂x − ∂(y²)/∂y = −z − 2y. Step 4: Assemble: curl F = (x² + x) î + (−2xy) ĵ + (−z − 2y) k̂.
Answer
curl F = (x² + x) î − 2xy ĵ − (z + 2y) k̂
Curl in Physics and Engineering Applications
| Application | Vector Field | Curl Expression | Physical Meaning |
|---|---|---|---|
| Faraday's Law | Electric field E | ∇×E = −∂B/∂t | Changing B induces E |
| Ampere-Maxwell Law | Magnetic field H | ∇×H = J + ∂D/∂t | Current creates B |
| Fluid vorticity | Velocity field v | ω = ½∇×v | Local rotation of fluid |
| Conservative force | Force field F | ∇×F = 0 | Path-independent work |
| Stokes' Theorem | Any vector F | ∮F·dr = ∬(∇×F)·dS | Circulation = surface curl |
Interactive Tools
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Related Terms
Vector Divergence
Divergence is a scalar operator applied to a vector field that measures the net rate at which the vector field "spreads out" or "converges" at each point in space, representing the volumetric density of the outward flux of the field. A positive divergence indicates a source (field lines emanating outward), while negative divergence indicates a sink (field lines converging inward), and zero divergence means the field is incompressible. Divergence appears in Maxwell's equations for electromagnetism, the continuity equation in fluid mechanics, and Gauss's law.
Gradient Vector
The gradient of a scalar function is a vector field whose components are the partial derivatives of the function with respect to each independent variable, pointing in the direction of the steepest rate of increase of the function. Denoted ∇f (nabla f), the gradient generalizes the ordinary derivative to multivariable functions. The gradient is fundamental in optimization (gradient descent), physics (conservative force fields), and machine learning for training neural networks.
Partial Derivative
A partial derivative is the derivative of a multivariable function with respect to one variable while all other variables are held constant, measuring the function's instantaneous rate of change in a single coordinate direction. Denoted by the symbol ∂ (a stylized "d"), partial derivatives generalize single-variable differentiation to functions of two or more variables. They are essential in multivariable calculus, optimization, thermodynamics, fluid dynamics, and the formulation of partial differential equations.
From Old English and Old Norse roots meaning to curve or wind in spirals. The notation ∇× and the specific term "curl" were introduced by James Clerk Maxwell in his 1873 treatise on electromagnetism. Oliver Heaviside and Josiah Willard Gibbs later standardized vector calculus notation including the curl operator.