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.
Key Formula
div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z
LaTeX: \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}
| Symbol | Meaning | Unit |
|---|---|---|
| ∇·F | Divergence of vector field F (scalar result) | per unit length |
| F_x, F_y, F_z | x-, y-, z-components of vector field F | depends on field |
| ∂/∂x, ∂/∂y, ∂/∂z | Partial derivative operators | dimensionless |
Worked Example
Problem
Find the divergence of F(x, y, z) = (x², 2yz, z³) at the point (1, 2, 1).
Solution
Step 1: ∂F_x/∂x = ∂(x²)/∂x = 2x. Step 2: ∂F_y/∂y = ∂(2yz)/∂y = 2z. Step 3: ∂F_z/∂z = ∂(z³)/∂z = 3z². Step 4: div F = 2x + 2z + 3z². Step 5: At (1, 2, 1): div F = 2(1) + 2(1) + 3(1)² = 2 + 2 + 3 = 7.
Answer
div F at (1, 2, 1) = 7 (positive: source region)
Physical Interpretation of Divergence in Different Fields
| Field | Vector Field | Divergence Meaning | Law / Equation |
|---|---|---|---|
| Electrostatics | Electric field E | ∇·E = ρ/ε₀ | Gauss's Law |
| Magnetostatics | Magnetic field B | ∇·B = 0 | No magnetic monopoles |
| Fluid mechanics | Velocity field v | ∇·v = 0 (incompressible) | Continuity equation |
| Heat conduction | Heat flux q | ∇·q = −∂(ρcT)/∂t | Heat equation |
| Gravity | Gravitational field g | ∇·g = −4πGρ | Gauss's law for gravity |
Interactive Tools
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Related Terms
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.
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.
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 Latin "divergere" (to move apart, spread out). The ∇· (nabla dot) notation for divergence was standardized by Oliver Heaviside and Josiah Willard Gibbs in the 1880s during the development of modern vector calculus, replacing earlier notations used by James Clerk Maxwell in his 1873 "Treatise on Electricity and Magnetism."