CAL-26

Polar Jacobian

Area element r dr dθ.

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CalculusJacobian

Governing equation

|\\partial(x,y)/\\partial(r,\\theta)|=r

$r \in \left[0.1, 10\right]$

where

r: r (—)
J: J (—)

The figure, part by part

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①

Governing law — red

|\\partial(x,y)/\\partial(r,\\theta)|=r

$r \in \left[0.1, 10\right]$

$r = 0.1$ $J = 0.1$

Along the curve

②

Domain of the variable — blue

r_{\min} = 0.1,\quad r_{\max} = 10

$r \in \left[0.1, 10\right]$

$r = 0.1$ $r = 0.1$

Along the curve

③

Response curve — green

|\\partial(x,y)/\\partial(r,\\theta)|=r

$r \in \left[0.1, 10\right]$

$r = 0.1$ $J = 0.1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}J}{\mathrm{d}r}

$r \in \left[0.1, 10\right]$

$r = 0.162$ $J^{\prime} = 1$

Along the curve

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-26 — Polar Jacobian) is the form associated with Jacobian. Working symbols: $r$ $\rightarrow$ $J$ . Area element r dr dθ.

Purpose

Purpose: compute

J

from

r

in Calculus via

|\\partial(x,y)/\\partial(r,\\theta)|=r

Area element r dr dθ. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

r = 2.000\,\mathrm{—}

, the governing relation

|\\partial(x,y)/\\partial(r,\\theta)|=r

yields

J = 2.000\,\mathrm{—}

. Area element r dr dθ. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

r r2 —

Outputs

J J2.000 —

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Narration of this film

Area element r dr dθ.

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