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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

∇⋅F=∂xPx+∂yPy+∂zPz\nabla\cdot\mathbf F=\partial_x P_x+\partial_y P_y+\partial_z P_z

∂xPx∈[−10,10]\partial_x P_x \in \left[-10, 10\right]

where

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

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

∇⋅F=∂xPx+∂yPy+∂zPz\nabla\cdot\mathbf F=\partial_x P_x+\partial_y P_y+\partial_z P_z

∂xPx∈[−10,10]\partial_x P_x \in \left[-10, 10\right]

∂xPx=−10\partial_x P_x = -10∇⋅F=−10.2\nabla\cdot F = -10.2

②

Domain of the variable — blue

\partial_x P_x_{\min} = -10,\quad \partial_x P_x_{\max} = 10

∂xPx∈[−10,10]\partial_x P_x \in \left[-10, 10\right]

∂xPx=−10\partial_x P_x = -10∂xPx=−10\partial_x P_x = -10

③

Response curve — green

∇⋅F=∂xPx+∂yPy+∂zPz\nabla\cdot\mathbf F=\partial_x P_x+\partial_y P_y+\partial_z P_z

∂xPx∈[−10,10]\partial_x P_x \in \left[-10, 10\right]

∂xPx=−10\partial_x P_x = -10∇⋅F=−10.2\nabla\cdot F = -10.2

④

Rate of change — amber

d∇⋅Fd∂xPx\frac{\mathrm{d}\nabla\cdot F}{\mathrm{d}\partial_x P_x}

∂xPx∈[−10,10]\partial_x P_x \in \left[-10, 10\right]

∂xPx=−9.874\partial_x P_x = -9.874∇⋅F′=1\nabla\cdot F^{\prime} = 1

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: ∂xPx\partial_x P_x, ∂yPy\partial_y P_y, ∂zPz\partial_z P_z →\rightarrow ∇⋅F\nabla\cdot F. The divergence theorem: ∭ div F = ∯ F · n. Positive div is a source.

Purpose

Purpose: compute ∇⋅F\nabla\cdot F from ∂xPx\partial_x P_x, ∂yPy\partial_y P_y, ∂zPz\partial_z P_z in Calculus via ∇⋅F=∂xPx+∂yPy+∂zPz\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 ∂xPx=1.000 —\partial_x P_x = 1.000\,\mathrm{—}, ∂yPy=−0.400 —\partial_y P_y = -0.400\,\mathrm{—}, ∂zPz=0.200 —\partial_z P_z = 0.200\,\mathrm{—}, the governing relation ∇⋅F=∂xPx+∂yPy+∂zPz\nabla\cdot\mathbf F=\partial_x P_x+\partial_y P_y+\partial_z P_z yields ∇⋅F=0.8000 —\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.

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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.

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