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

where

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

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