CAL-15

Divergence snapshot

div F = ∂Px/∂x + ∂Py/∂y + ∂Pz/∂z. Source density of a flux.

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Vector calculusDivergence

Governing equation

\nabla\cdot\mathbf F=\partial_x P_x+\partial_y P_y+\partial_z P_z

where

\partial_x P_x: ∂Px/∂x (—)
\partial_y P_y: ∂Py/∂y (—)
\partial_z P_z: ∂Pz/∂z (—)
\nabla\cdot F: Divergence (—)

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-15 — Divergence snapshot) is the form associated with Divergence. Working symbols: $\partial_x P_x$ , $\partial_y P_y$ , $\partial_z P_z$ $\rightarrow$ $\nabla\cdot F$ . The divergence theorem: ∭ div F = ∯ F · n. Positive div is a source.

Purpose

Purpose: compute

\nabla\cdot F

from

\partial_x P_x

\partial_y P_y

\partial_z P_z

in Calculus via

\nabla\cdot\mathbf F=\partial_x P_x+\partial_y P_y+\partial_z P_z

div F = ∂Px/∂x + ∂Py/∂y + ∂Pz/∂z. Source density of a flux. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\partial_x P_x = 1.000\,\mathrm{—}

\partial_y P_y = -0.400\,\mathrm{—}

\partial_z P_z = 0.200\,\mathrm{—}

, the governing relation

\nabla\cdot\mathbf F=\partial_x P_x+\partial_y P_y+\partial_z P_z

yields

\nabla\cdot F = 0.8000\,\mathrm{—}

. Enter the three diagonal partials of a vector field. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

∂Px/∂x \partial_x P_x1 —∂Py/∂y \partial_y P_y-0.4 —∂Pz/∂z \partial_z P_z0.2 —

Outputs

Divergence \nabla\cdot F0.8000 —

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Narration of this film

Enter the three diagonal partials of a vector field.

The divergence theorem: ∭ div F = ∯ F · n. Positive div is a source.

Reading speed

Watch on YouTube