CalculusLesson 1 of 1 in the course Vector calculus

Divergence: Flux, Coordinate Formulas, and Differential Identities

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Divergence describes how a vector field behaves locally: does it spread outward, converge inward, or have balanced flow? This idea is useful for understanding fluids, gases, and other fields that transport something through space.

A vector field assigns a vector to every point in a region. The vectors may represent fluid velocity, for example. Divergence is a scalar quantity that measures the field's local tendency to alter volume around a point. In two dimensions, the corresponding measure concerns area.

All flashcards

Divergence of a vector field
The limiting net outward flux per unit volume as a region around a point shrinks to that point.
Source-like point
A point with positive divergence, where more field flows outward than inward.
Sink-like point
A point with negative divergence, where more field flows inward than outward.
Zero divergence
Balanced local inflow and outflow; the field itself need not be zero.
Divergence in spherical coordinates
A coordinate formula using radial and angular derivatives with factors accounting for spherical volume geometry.

Quiz questions

  1. For a vector field, what type of quantity is its divergence?Answer: A scalar-valued field
  2. For a continuously differentiable field F=(Fx,Fy,Fz)\mathbf{F}=(F_x,F_y,F_z) in three-dimensional Cartesian coordinates, which expression gives its divergence?Answer: ∂Fx∂x+∂Fy∂y+∂Fz∂z\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}
  3. How is the divergence of a vector field related to its Jacobian matrix?Answer: It is the trace of the Jacobian matrix.
  4. What defines a solenoidal vector field?Answer: Its divergence is zero everywhere.
  5. For a sufficiently differentiable scalar field φ\varphi, what is div⁡(∇φ)\operatorname{div}(\nabla\varphi)?Answer: The scalar Laplacian Δφ\Delta\varphi

Based on the Wikipedia article Divergence, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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