ES.1802, October 21, 2026

Overview

The lesson builds from polar coordinates to the general Jacobian change-of-variables rule for double integrals. It derives the area element r dr dθ, models a circle of radius a shifted by a as r = 2a cos θ, calculates the mass of a quarter-annulus with density x² + y², and explains how the absolute Jacobian determinant scales area between coordinate systems.

Key takeaways

For polar coordinates, the area element is dA = r dr dθ, not simply dr dθ; the factor r accounts for the widening of angular wedges as their distance from the origin increases.
A circle of radius a centered at (a, 0) has polar boundary r = 2a cos θ, demonstrating that a circle not centered at the origin generally does not have a constant polar radius.
For the first-quadrant annulus with radii 1 and 2 and density σ = x² + y², the mass integral is ∫₀^{π/2}∫₁² r³ dr dθ = 15π/8.
A smooth coordinate transformation scales local area by the magnitude of its Jacobian determinant, because the transformed coordinate directions span a parallelogram.
When changing an integral from Cartesian coordinates to new coordinates, the area factor must be solved in the direction needed; the inverse mapping contributes the reciprocal determinant when converting dx dy into du dv.

Chapters

0:00 Polar Coordinates and the Area Element

A point in Cartesian coordinates is represented in polar form by x = r cos θ and y = r sin θ.
The class introduces the polar area element, which includes an extra factor of r rather than simply using dr dθ.
Double integration in polar coordinates is presented as a way to simplify regions with circular boundaries.

2:00 Finding a Circle’s Area with Polar and Cartesian Bounds

For a circle of radius a centered at the origin, polar bounds are 0 ≤ r ≤ a and 0 ≤ θ ≤ 2π.
Integrating r dr dθ gives πa², matching the familiar area formula.
A Cartesian setup requires bounds involving y = ±√(a² − x²), illustrating why polar coordinates are more convenient for circles.

7:00 Deriving the Polar Equation of a Shifted Circle

A circle of radius a centered at (a, 0) is substituted into x = r cos θ and y = r sin θ.
Simplifying the resulting equation gives r² = 2ar cos θ, and division by r yields r = 2a cos θ for the circle’s boundary.
The class notes that division by r omits the origin as a solution, so it checks the boundary behavior at relevant angles.

13:50 Setting Polar Bounds for the Shifted Circle

The shifted circle’s interior is described by 0 ≤ r ≤ 2a cos θ over the angles that sweep the region.
A constant radial bound such as r = 2a would describe a different shape, showing why the boundary must depend on θ.
The circle’s center is offset from the origin, so the polar coordinate r measures a varying distance from the origin, not the circle’s fixed radius.

18:10 Modeling Mass in a Quarter-Annulus

A planar region in the first quadrant is bounded by circular arcs with radii 1 and 2.
The mass density is σ(x, y) = x² + y², so points farther from the origin have greater mass per unit area.
The mass is modeled as a double integral of density over the region, with polar coordinates matching both the circular boundaries and the density.

26:10 Evaluating the Quarter-Annulus Mass Integral

In polar form, the density becomes r² and the area element is r dr dθ, making the integrand r³.
The bounds are 1 ≤ r ≤ 2 and 0 ≤ θ ≤ π/2.
The radial integral is ∫₁² r³ dr = 15/4, so the total mass is (15/4)(π/2) = 15π/8.

30:00 Visualizing Area Scaling Under Coordinate Changes

A coordinate-mapping applet illustrates how a Cartesian image changes when x and y are replaced by polar coordinates.
A small Cartesian square maps to a wedge in polar coordinates, and its area scaling produces the r factor in r dr dθ.
The class also tests mappings such as u = x³ and v = x + y to show that general coordinate changes distort local shapes and areas.

36:30 Using the Jacobian to Transform Local Directions

For coordinates u(x, y) and v(x, y), the Jacobian matrix is [[uₓ, uᵧ], [vₓ, vᵧ]].
Applying the Jacobian to the Cartesian direction vectors (dx, 0) and (0, dy) gives two transformed vectors in uv-space.
Those transformed vectors span a small parallelogram, whose area measures how the coordinate mapping scales a local area element.

43:00 The Determinant and the General Change-of-Variables Rule

The area of the transformed parallelogram is the absolute value of the Jacobian determinant multiplied by dx dy.
For the mapping from (x, y) to (u, v), the relation is du dv = |∂(u, v)/∂(x, y)| dx dy.
When rewriting an integral in uv-coordinates, the inverse relation introduces the reciprocal Jacobian factor, extending the derivative adjustment used in single-variable u-substitution.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, ChristianAltoSax.