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Gradient, divergence, curl
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Vector calculus operators describing fields.
∇f, ∇·F, ∇×FWith the operator ∇ = (∂/∂x, ∂/∂y, ∂/∂z): the gradient ∇f points uphill in the direction of steepest increase of f; the divergence ∇·F measures how much a field spreads out from a point; the curl ∇×F measures local rotation. Maxwell's equations are written with them.
The electric field is E = −∇V; its divergence is ρ/ε₀: charge is the source of the field.
Gradient takes a scalar to a vector; divergence takes a vector to a scalar; curl takes a vector to a vector.
In more detail
The gradient points in the direction of steepest increase of a scalar field, divergence measures how much a vector field spreads out from a point, and curl measures how much it rotates. These operators underpin electromagnetism and fluid mechanics.
Real-world uses
- The electric field is the negative gradient of the electric potential.
- Divergence of the velocity field tells fluid dynamicists whether a flow is compressible.
- Weather maps use the gradient of pressure to find the wind direction, and meteorologists use curl to find rotation in storms.
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Frequently asked questions
What does the gradient of temperature tell you?
It points towards the fastest increase in temperature, so heat flows in the opposite direction.
Is curl the same as rotation?
Curl measures the local tendency of a field to rotate, which is closely related to the idea of vorticity in fluids.
Does the gradient point up or down a slope?
It points in the direction of steepest increase, so the force on a body is in the direction of the negative gradient of potential energy.
Related topics in Measurement & Math Tools
Calculus: derivatives · Calculus: integrals · Partial derivatives · Differential equations in physics · Taylor series and small angles · Complex numbers and phasors