Foundations — Double integrals in polar coordinates — Jacobian r

This page assumes you know nothing about the notation on the parent note. We will build every symbol — $x$, $y$, $r$, $\theta$, $\int$, $\iint$, $dA$, $\det$, $\partial$ — from the ground up, each anchored to a picture, each used only after it is earned.

1. The plane and its two numbers $x$ and $y$

Draw two number lines that cross at a point called the origin. The horizontal one is the $x$-axis, the vertical one is the $y$-axis.

Why the topic needs this: the whole problem is "measure area in the plane." You cannot measure area until you can name locations, and $(x,y)$ is the first naming scheme.

2. The circle equation $x^2+y^2=a^2$

Look at the red circle in the figure above. Every point on it is the same distance $a$ from the origin. By the Pythagorean theorem — for a right triangle, the two short sides squared add up to the long side squared — the distance from origin to $(x,y)$ is $\sqrt{x^2+y^2}$.

3. Angle $\theta$ and the radian

Instead of "right and up," describe a point by turning and walking out. Stand at the origin, face along the positive $x$-axis, rotate by an angle, then walk forward.

But angles need a unit. We measure them in radians, not degrees — and this choice is the secret hero of the whole topic.

See more: Arc Length and Radian Measure.

4. Polar coordinates $(r,\theta )$

Now combine "how far out" with "which direction."

Here $\cos \theta$ ("cosine") is adjacent-over-hypotenuse of that triangle and $\sin \theta$ ("sine") is opposite-over-hypotenuse. With hypotenuse $r$, the adjacent side is $r\cos \theta$ and the opposite side is $r\sin \theta$ — exactly the formulas above.

5. The integral sign $\int$ and the double integral $\iint$

See more: Area of Regions Bounded by Polar Curves.

6. The differentials $dr$ and $d\theta$, and the polar tile

Slide $r$ by a whisker $dr$ and $\theta$ by a whisker $d\theta$. The four grid lines (two circles, two rays) trap a tiny curved box.

7. Partial derivatives $\partial$ and the determinant $\det$

The parent gives a second, machine-like proof of the same $r$ using the Jacobian. Here are its two ingredients.

See more: Jacobian Determinant and Change of Variables Theorem.

8. How it all fits together

Read it top to bottom: coordinates and radians feed the tile; the tile gives $dA=rdrd\theta$; the determinant confirms it; and the sum-idea of the integral wraps it into the master formula.

Equipment checklist

Test yourself — reveal only after you have answered aloud.

Connections

• Double integrals in polar coordinates — Jacobian r — the parent this page prepares you for.
• Arc Length and Radian Measure — source of the arc side $rd\theta$.
• Jacobian Determinant — the determinant tool that reconfirms $r$.
• Change of Variables Theorem — the general law polar is a special case of.
• Area of Regions Bounded by Polar Curves — where $\iint 1dA$ becomes area.
• Gaussian Integral — the killer payoff once the foundations are in place.
• Triple Integrals in Cylindrical and Spherical Coordinates — the same idea, one dimension up.