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CAL-26

Polar Jacobian

Area element r dr dθ.

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CalculusJacobian

Governing equation

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

where

r
r (—)
J
J (—)

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: rr →\rightarrow JJ. Area element r dr dθ.

Purpose

Purpose: compute JJ from rr in Calculus via ∣partial(x,y)/partial(r,theta)∣=r|\\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 —r = 2.000\,\mathrm{—}, the governing relation ∣partial(x,y)/partial(r,theta)∣=r|\\partial(x,y)/\\partial(r,\\theta)|=r yields J=2.000 —J = 2.000\,\mathrm{—}. Area element r dr dθ. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • J J2.000 —
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Narration of this film

Area element r dr dθ.

Area element r dr dθ.

Reading speed

Watch on YouTube