Divergence

Divergence is a scalar differential operator that measures a vector field’s net outward flow per unit volume near a point. Positive values indicate local sources, negative values sinks, and zero values no net local expansion.

Del operator: A vector differential operator whose components are partial derivatives with respect to spatial coordinates. Taking its dot product with a vector field produces divergence.
Vector field: A function that assigns a vector to each point in a space. Divergence is defined by how a vector field varies around each point.
Continuity equation: A local conservation law relating the change in density to the divergence of its flow. It uses divergence to express how density changes when material flows.
Gradient: A differential operator that maps a scalar field to a vector pointing in its direction of steepest increase. Unlike divergence, it acts on a scalar field and returns a vector.
Divergence theorem: A theorem equating the integral of divergence over a region with flux through its boundary. It is the global integral counterpart of the local divergence operator.
Partial derivative: A derivative that measures change in a multivariable function with respect to one variable while holding others fixed. Each coordinate term in divergence is a partial derivative of one field component.
Scalar field: A function assigning a scalar value to each point in a space. Divergence converts a vector field into a scalar field.
Maxwell's equations: Four equations describing classical electric and magnetic fields and their sources. Their divergence equations relate fields to electric charge and magnetic monopole density.
Curl: A differential operator that measures the local rotation of a three-dimensional vector field. Curl detects rotational behavior, whereas divergence detects net outward flow.
Differential forms: Mathematical objects that generalize functions, differential expressions, and oriented integration. They express divergence through exterior differentiation and volume forms.

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