Foundations — Triple integrals in Cartesian, cylindrical, spherical coordinates

This page assumes you know nothing about the notation. We build each symbol, anchor it to a picture, and say why the topic needs it — in an order where each idea leans on the one before.

0. What is a "region in space"? ($E$, and 3D itself)

Before any integral, we need a place to integrate over.

Look at the figure: the point sits where the three dashed guide-lines meet. Each coordinate is a shadow distance along one axis.

Why the topic needs it: you can only "add over every chunk" once you've fenced off which chunks — that fence is $E$. The whole difficulty of the parent note is turning "$E$" into concrete number ranges for the sliders.

1. The function $f(x,y,z)$ — a value living at every point

Why the topic needs it: the parent's two headline uses are $f=1$ (gives volume) and $f=\rho (x,y,z)$ = density (gives mass). The integral is the machine that sums these readings.

2. Chopping and summing — $\Delta$, $\sum$, $\lim$

The integral is built by cutting $E$ into tiny pieces and adding.

In the figure, the coarse chop (left) over-/under-shoots the blob; the fine chop (right) hugs it. The limit is the perfect fit you never quite draw but always reach.

Why the topic needs it: this is the topic. Everything else is machinery for evaluating it. See Center of mass and moments of inertia for what these sums physically compute.

3. $dV$, $dxdydz$ — the shape of one tiny chunk

Why the topic needs it: $dV$ is the "one sugar cube" of the parent's Feynman story. The entire drama of cylindrical and spherical coordinates is that their chunks are not straight boxes, so their $dV$ picks up extra stretch factors ($r$, and ${\rho }^2\sin \varphi$) — which brings us to §6.

4. Nested integrals — the $\int$ sign and iterated limits

Why the topic needs it: setting these nested limits is, in the parent's words, "the only skill that matters." You already meet the 2D version in Double integrals & polar coordinates.

5. Trig you must own — $\cos ,\sin ,\tan$, and $\pi$

Cylindrical and spherical coordinates are built from angles, so we need the two functions that turn an angle into coordinates.

• At $\theta =0$: point $(1,0)$, so $\cos 0=1,\sin 0=0$.
• At $\theta =\frac{\pi }{2}$ (quarter turn, straight up): point $(0,1)$, so $\cos \frac{\pi }{2}=0,\sin \frac{\pi }{2}=1$.
• At $\theta =\pi$ (half turn): $(-1,0)$.

Why the topic needs it: the substitutions $x=r\cos \theta$, $y=r\sin \theta$ (cylindrical) and the spherical formulas are nothing but "horizontal/vertical shadows of a rotating arm." Deeper geometry lives in Spherical coordinates geometry.

6. Partial derivatives, determinants, and the Jacobian $J$

This is the one genuinely new machine. We build it in three layers.

6a. Partial derivative $\frac{\partial x}{\partial u}$

6b. Determinant — signed volume of a box

For a $3\times 3$ array, $\det \left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)=a(ei-fh)-b(di-fg)+c(dh-eg).$

6c. The Jacobian $J$ and $|J|$

Why the topic needs it: the mysterious extra $r$ (cylindrical) and ${\rho }^2\sin \varphi$ (spherical) in the parent note are nothing but $|J|$ computed for those coordinates. Master this and those factors stop being memorization. Full treatment: Jacobian and change of variables. This same volume-of-a-box idea reappears in the Divergence theorem.

7. The coordinate players — $r,\theta ,\rho ,\varphi$

Now the letters that name the new coordinates.

The prerequisite map

Read it upward: points and values feed the chunk-and-sum idea; single integrals feed nesting; trig plus partials plus determinants build the Jacobian; together they give the volume elements the parent note uses.

Equipment checklist

Cover the right side and answer each aloud before starting the parent note.