Related Concepts: 1.02 Vector Operators (Gradient, Divergence, Curl & Laplacian) | 1.03 Integral Theorems (Divergence Theorem & Stokes_s Theorem) & Identity Proofs | 2.01 Fundamental Postulates of Electrostatics & Gauss_s Law Applications

1.01 Spatial Coordinates & Differential Elements (Cartesian, Cylindrical, Spherical)

Core Idea

Every field problem in this course is solved by an integral — over a line, a surface, or a volume. You cannot set up that integral until you have written $d\vec{l}$, $d\vec{s}$ and $dv$ correctly in the coordinate system that matches the symmetry of the source. Choose the wrong system and a two-line problem becomes unsolvable; choose the right one and the integral often collapses by symmetry.

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1. Why Three Coordinate Systems?

An orthogonal coordinate system {one where the three coordinate surfaces cross each other at right angles everywhere} lets us treat the three directions independently — a dot product between different unit vectors is zero, so vector algebra stays simple.

We use three of them because charge and current distributions in this syllabus come in exactly three symmetry flavours:

graph TD
    A["Look at the source geometry"] --> B{"What symmetry?"}
    B -->|"Flat plates, boxes, straight edges"| C["Cartesian x, y, z"]
    B -->|"Long wires, coax, solenoids"| D["Cylindrical r, phi, z"]
    B -->|"Point charges, charge clouds, dipoles"| E["Spherical R, theta, phi"]
    C --> F["Write dl, ds, dv then integrate"]
    D --> F
    E --> F

1.1 Comparison of the Three Systems

Parameter	Cartesian	Cylindrical	Spherical
Variables	$(x,y,z)$	$(r,\varphi ,z)$	$(R,\theta ,\varphi )$
Bounds	$-\infty <x,y,z<\infty$	$0\le r<\infty$, $0\le \varphi <2\pi$, $-\infty <z<\infty$	$0\le R<\infty$, $0\le \theta \le \pi$, $0\le \varphi <2\pi$
Length-type variables	3 $(x,y,z)$	2 $(r,z)$	1 $(R)$
Angle-type variables	0	1 $(\varphi )$	2 $(\theta ,\varphi )$
Unit vectors	${\hat{a}}_x,{\hat{a}}_y,{\hat{a}}_z$	${\hat{a}}_r,{\hat{a}}_{\varphi },{\hat{a}}_z$	${\hat{a}}_R,{\hat{a}}_{\theta },{\hat{a}}_{\varphi }$
Best used for	Rectangular plates, boxes, planar boundaries	Coaxial cable, long straight wire, solenoid	Point charge, spherical charge cloud, dipole
Right-hand order	${\hat{a}}_x\times {\hat{a}}_y={\hat{a}}_z$	${\hat{a}}_r\times {\hat{a}}_{\varphi }={\hat{a}}_z$	${\hat{a}}_R\times {\hat{a}}_{\theta }={\hat{a}}_{\varphi }$

The one difference that trips everyone

In Cartesian coordinates the unit vectors are constant — ${\hat{a}}_x$ points the same way everywhere. In cylindrical and spherical coordinates ${\hat{a}}_r$, ${\hat{a}}_{\varphi }$, ${\hat{a}}_{\theta }$, ${\hat{a}}_R$ change direction from point to point. That is exactly why you cannot pull them outside an integral unless a symmetry argument lets you.

[FIGURE: Three side-by-side 3D sketches showing a point P located in (a) Cartesian, (b) Cylindrical, (c) Spherical coordinates with the differential volume element drawn as a shaded cube/wedge — source: David K. Cheng, Ch. 2, Figs. 2-9 to 2-12 / Senior Notes “01 Static Electric Field.pdf” pg 4–6]

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2. Metric Coefficients (Scale Factors) $h_i$ [PYQ: 2019]

Abstract

A coordinate variable $u_i$ is not always a physical distance — it can be an angle {like $\theta$ or $\varphi$, measured in radians, carrying no units of length}. The metric coefficient (or scale factor) $h_i$ is the multiplier that converts a differential change in the coordinate $d{u}_i$ into a real physical differential length $d{l}_i$: $d{l}_i=h_{i}d{u}_i$

This single idea generates every $d\vec{l}$, $d\vec{s}$ and $dv$ formula in the course, so learn it once instead of memorising nine separate expressions.

Coordinate System	Variables $(u_1,u_2,u_3)$	Metric Coefficients $(h_1,h_2,h_3)$
Cartesian	$(x,y,z)$	$(1,1,1)$
Cylindrical	$(r,\varphi ,z)$	$(1,r,1)$
Spherical	$(R,\theta ,\varphi )$	$(1,R,R\sin \theta )$

Where the scale factors physically come from

• Cylindrical $h_{\varphi }=r$: sweeping through a small angle $d\varphi$ at radius $r$ traces an arc of physical length $rd\varphi$. Further from the axis, the same angle covers more distance.
• Spherical $h_{\theta }=R$: sweeping $d\theta$ at radius $R$ traces an arc $Rd\theta$ along a great circle.
• Spherical $h_{\varphi }=R\sin \theta$: sweeping $d\varphi$ traces a horizontal circle whose radius is not $R$ but $R\sin \theta$ — the perpendicular distance from the $z$-axis. At the poles ($\theta =0$ or $\pi$) that radius shrinks to zero, which is why $h_{\varphi }\to 0$ there.

2.1 Generalised Formulas (memorise these three, derive the rest)

$d\vec{l}={\hat{a}}_1(h_{1}d{u}_1)+{\hat{a}}_2(h_{2}d{u}_2)+{\hat{a}}_3(h_{3}d{u}_3)$

$d{\vec{s}}_1={\hat{a}}_1(h_2{h}_{3}d{u}_{2}d{u}_3),d{\vec{s}}_2={\hat{a}}_2(h_1{h}_{3}d{u}_{1}d{u}_3),d{\vec{s}}_3={\hat{a}}_3(h_1{h}_{2}d{u}_{1}d{u}_2)$

$dv=h_1{h}_2{h}_{3}d{u}_{1}d{u}_{2}d{u}_3$

Read $d{\vec{s}}_1$ as: “the surface whose normal is ${\hat{a}}_1$, so it is built from the other two length elements.”

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3. Cartesian Coordinates $(x,y,z)$

Since all three scale factors are $1$, the formulas are as plain as they get.

$d\vec{l}={\hat{a}}_{x}dx+{\hat{a}}_{y}dy+{\hat{a}}_{z}dz$

$d{\vec{s}}_x={\hat{a}}_x(dydz),d{\vec{s}}_y={\hat{a}}_y(dxdz),d{\vec{s}}_z={\hat{a}}_z(dxdy)$

$dv=dxdydz$

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4. Cylindrical Coordinates $(r,\varphi ,z)$

Here $r$ is the radial distance from the $z$-axis, $\varphi$ is the azimuthal angle {how far you have rotated around the $z$-axis, measured from the $+x$ axis}, and $z$ is the axial height.

$d\vec{l}={\hat{a}}_{r}dr+{\hat{a}}_{\varphi }(rd\varphi )+{\hat{a}}_{z}dz$

Surface	Held constant	Element
Curved side of a cylinder	$r$	$d{\vec{s}}_r={\hat{a}}_r(rd\varphi dz)$
Flat radial “fin”	$\varphi$	$d{\vec{s}}_{\varphi }={\hat{a}}_{\varphi }(drdz)$
Flat top/bottom disc	$z$	$d{\vec{s}}_z={\hat{a}}_z(rdrd\varphi )$

$dv=rdrd\varphi dz$

Why this is the system for Gauss's law on a line charge

For an infinite line charge along $z$ the field is purely radial. A cylindrical Gaussian surface of radius $r$ and length $L$ makes $\vec{D}$ perfectly parallel to $d{\vec{s}}_r$ on the curved side and perpendicular to $d{\vec{s}}_z$ on the end caps — so the flux integral collapses to $D_r(2\pi rL)$ with no integration at all. See 2.01 Fundamental Postulates of Electrostatics & Gauss_s Law Applications.

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5. Spherical Coordinates $(R,\theta ,\varphi )$

Here $R$ is the radial distance from the origin, $\theta$ is the polar (zenith) angle {measured down from the $+z$ axis, ranging $0$ to $\pi$}, and $\varphi$ is the same azimuthal angle as before.

$d\vec{l}={\hat{a}}_{R}dR+{\hat{a}}_{\theta }(Rd\theta )+{\hat{a}}_{\varphi }(R\sin \theta d\varphi )$

Surface	Held constant	Element
Sphere of radius $R$	$R$	$d{\vec{s}}_R={\hat{a}}_R(R^2\sin \theta d\theta d\varphi )$
Cone of half-angle $\theta$	$\theta$	$d{\vec{s}}_{\theta }={\hat{a}}_{\theta }(R\sin \theta dRd\varphi )$
Flat half-plane	$\varphi$	$d{\vec{s}}_{\varphi }={\hat{a}}_{\varphi }(RdRd\theta )$

$dv=R^2\sin \theta dRd\theta d\varphi$

Sanity check — surface area and volume of a sphere

$S=\oint d{s}_R={\int }_0^{2\pi }{\int }_0^{\pi }{R}^2\sin \theta d\theta d\varphi =R^2(2\pi )[-\cos \theta {]}_0^{\pi }=4\pi R^2✓$ $V=\int dv={\int }_0^R{\int }_0^{\pi }{\int }_0^{2\pi }{R}^{\prime 2}\sin \theta d\varphi d\theta d{R}^{\prime }=(2\pi )(2)\frac{R^3}{3}=\frac{4}{3}\pi R^3✓$ If your differential element reproduces the formulas you learned in school, you wrote it correctly.

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6. Master Summary Table

System	$d\vec{l}$	Key $d\vec{s}$	$dv$
Cartesian	${\hat{a}}_{x}dx+{\hat{a}}_{y}dy+{\hat{a}}_{z}dz$	${\hat{a}}_{z}dxdy$	$dxdydz$
Cylindrical	${\hat{a}}_{r}dr+{\hat{a}}_{\varphi }rd\varphi +{\hat{a}}_{z}dz$	${\hat{a}}_{r}rd\varphi dz$ (side), ${\hat{a}}_{z}rdrd\varphi$ (cap)	$rdrd\varphi dz$
Spherical	${\hat{a}}_{R}dR+{\hat{a}}_{\theta }Rd\theta +{\hat{a}}_{\varphi }R\sin \theta d\varphi$	${\hat{a}}_R{R}^2\sin \theta d\theta d\varphi$	$R^2\sin \theta dRd\theta d\varphi$

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7. Coordinate Variable Conversions

Cylindrical $\leftrightarrow$ Cartesian: $x=r\cos \varphi ,y=r\sin \varphi ,z=z$ $r=\sqrt{x^2+y^2},\varphi ={\tan }^{-1}\left(\frac{y}{x}\right),z=z$

Spherical $\leftrightarrow$ Cartesian: $x=R\sin \theta \cos \varphi ,y=R\sin \theta \sin \varphi ,z=R\cos \theta$ $R=\sqrt{x^2+y^2+z^2},\theta ={\cos }^{-1}\left(\frac{z}{R}\right),\varphi ={\tan }^{-1}\left(\frac{y}{x}\right)$

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8. PYQ Coverage for This Note

Question (verbatim)	Marks	Year(s)
Show the inadequacy of circuit theory concept and necessity of electromagnetic field concept with two examples. Define metric coefficient.	08+02	2019

Honest scope note

The differential-element formulas themselves have never been asked as a standalone question in the 2015–2025 papers. They are examined indirectly — you cannot do the Gauss’s law derivations (Ch. 2), the Biot-Savart integrations (Ch. 3), or the boundary-condition pillbox arguments without them. Only “Define metric coefficient” (2019) is a direct 2-mark hit. Treat this note as a toolbox, not a memorisation target.

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9. Exam Hacks & Traps

Key Exam Checkpoints

1. Never integrate $d\theta$ or $d\varphi$ bare. They are angles, not lengths. Multiply by the scale factor ($R$, $R\sin \theta$, or $r$) every single time. This is the most common silent mark-loser in the whole course.
2. $R^2$ in the spherical volume element. Writing $dv=R\sin \theta dRd\theta d\varphi$ (missing the square) breaks every charge-cloud derivation. Check it against $\frac{4}{3}\pi R^3$.
3. Vector notation is graded. Mashuk Sir docks marks for missing arrows and hats. Write $\vec{A}$ for vectors and ${\hat{a}}_x$ for unit vectors — always.
4. Cylindrical uses $r$, spherical uses $R$. This course (Cheng’s convention) reserves lowercase $r$ for the cylindrical radial distance and uppercase $R$ for the spherical one. Mixing them makes a $1/r$ answer look like a $1/R^2$ answer.
5. Pick the system before you write anything. One line — “Because the charge distribution has cylindrical symmetry about the $z$-axis, we work in cylindrical coordinates” — earns setup marks and stops you fighting the algebra.

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10. Self-Check

1. Write $d\vec{s}$ on the curved surface of a cylinder of radius $r$. (Answer: ${\hat{a}}_{r}rd\varphi dz$)
2. What are the metric coefficients in spherical coordinates? (Answer: $1,R,R\sin \theta$)
3. Why does $h_{\varphi }$ vanish at the poles in spherical coordinates? (Answer: the circle traced by $d\varphi$ shrinks to a point on the $z$-axis)
4. A charge cloud is a sphere of radius $a$. Which system, and what is $dv$? (Answer: spherical; $R^2\sin \theta dRd\theta d\varphi$)

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Next: 1.02 Vector Operators (Gradient, Divergence, Curl & Laplacian)