CAL-13

2D Jacobian determinant

J = ∂(u,v)/∂(x,y) = ux vy − uy vx. Area scale of a map.

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MultivariableJacobian

Governing equation

J=u_x v_y-u_y v_x

$u_x \in \left[-10, 10\right]$

where

u_x: ∂u/∂x (—)
u_y: ∂u/∂y (—)
v_x: ∂v/∂x (—)
v_y: ∂v/∂y (—)
J: Jacobian (—)

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

J=u_x v_y-u_y v_x

$u_x \in \left[-10, 10\right]$

$u_x = -10$ $J = -30$

Along the curve

②

Domain of the variable — blue

u_x_{\min} = -10,\quad u_x_{\max} = 10

$u_x \in \left[-10, 10\right]$

$u_x = -10$ $u_x = -10$

Along the curve

③

Response curve — green

J=u_x v_y-u_y v_x

$u_x \in \left[-10, 10\right]$

$u_x = -10$ $J = -30$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}J}{\mathrm{d}u_x}

$u_x \in \left[-10, 10\right]$

$u_x = -9.874$ $J^{\prime} = 3$

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-13 — 2D Jacobian determinant) is the form associated with Jacobian. Working symbols: $u_x$ , $u_y$ , $v_x$ , $v_y$ $\rightarrow$ $J$ . Change of variables: dx dy = |J| du dv. J = 0 is a critical point of the map.

Purpose

Purpose: compute

J

from

u_x

u_y

v_x

v_y

in Calculus via

J=u_x v_y-u_y v_x

J = ∂(u,v)/∂(x,y) = ux vy − uy vx. Area scale of a map. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

u_x = 2.000\,\mathrm{—}

u_y = 0.000\,\mathrm{—}

v_x = 0.000\,\mathrm{—}

v_y = 3.000\,\mathrm{—}

, the governing relation

J=u_x v_y-u_y v_x

yields

J = 6.0000\,\mathrm{—}

. Enter the four first partials of a planar map. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

∂u/∂x u_x2 —∂u/∂y u_y0 —∂v/∂x v_x0 —∂v/∂y v_y3 —

Outputs

Jacobian J6.0000 —

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Narration of this film

Enter the four first partials of a planar map.

Change of variables: dx dy = |J| du dv. J = 0 is a critical point of the map.

Reading speed

Watch on YouTube