Related Concepts: 1.01 Spatial Coordinates & Differential Elements (Cartesian, Cylindrical, Spherical) | 1.03 Integral Theorems (Divergence Theorem & Stokes_s Theorem) & Identity Proofs | 2.01 Fundamental Postulates of Electrostatics & Gauss_s Law Applications

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

Core Idea

Maxwell’s equations are written entirely in terms of four operations built from the del operator $\nabla$: gradient (scalar $\to$ vector), divergence (vector $\to$ scalar), curl (vector $\to$ vector), and the Laplacian (scalar $\to$ scalar). Each answers a specific physical question about a field at a point, and every postulate you meet later is one of these four applied to $\vec{E}$, $\vec{D}$, $\vec{B}$ or $\vec{H}$.

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1. The Del Operator $\nabla$

In Cartesian coordinates: $\nabla ={\hat{a}}_x\frac{\partial }{\partial x}+{\hat{a}}_y\frac{\partial }{\partial y}+{\hat{a}}_z\frac{\partial }{\partial z}$

$\nabla$ is a vector differential operator {it behaves like a vector in the algebra, but each “component” is an instruction to differentiate rather than a number}. Because of that dual nature it can act on a field in three distinct ways:

graph LR
    S["Scalar field V"] -->|"grad"| V1["Vector field"]
    A["Vector field A"] -->|"div"| S1["Scalar field"]
    A -->|"curl"| V2["Vector field"]
    S -->|"Laplacian"| S2["Scalar field"]

Terminology & Concept Breakdown

• Scalar field: a quantity with magnitude only, defined at every point — e.g. electric potential $V(x,y,z)$, temperature.
• Vector field: a quantity with magnitude and direction at every point — e.g. $\vec{E}$, $\vec{B}$, wind velocity.
• Point (differential) form: a law written with $\nabla$, valid at a single point in space — as opposed to the integral form, which describes a whole region.

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2. Gradient of a Scalar Field $\nabla V$

Definition — Gradient

The gradient of a scalar field $V$ is a vector that points in the direction of the maximum rate of increase of $V$, with magnitude equal to that maximum rate of change per unit distance.

General curvilinear form (using the metric coefficients from 1.01 Spatial Coordinates & Differential Elements (Cartesian, Cylindrical, Spherical)): $\nabla V={\hat{a}}_1\frac{1}{h_1}\frac{\partial V}{\partial u_1}+{\hat{a}}_2\frac{1}{h_2}\frac{\partial V}{\partial u_2}+{\hat{a}}_3\frac{1}{h_3}\frac{\partial V}{\partial u_3}$

2.1 Explicit forms

System	$\nabla V$
Cartesian	${\hat{a}}_x\frac{\partial V}{\partial x}+{\hat{a}}_y\frac{\partial V}{\partial y}+{\hat{a}}_z\frac{\partial V}{\partial z}$
Cylindrical	${\hat{a}}_r\frac{\partial V}{\partial r}+{\hat{a}}_{\varphi }\frac{1}{r}\frac{\partial V}{\partial \varphi }+{\hat{a}}_z\frac{\partial V}{\partial z}$
Spherical	${\hat{a}}_R\frac{\partial V}{\partial R}+{\hat{a}}_{\theta }\frac{1}{R}\frac{\partial V}{\partial \theta }+{\hat{a}}_{\varphi }\frac{1}{R\sin \theta }\frac{\partial V}{\partial \varphi }$

Notice the $1/h_i$ factors — the gradient divides by the scale factor where the differential element multiplied by it.

The relation you will use a hundred times

$\boxed{\vec{E}=-\nabla V}$ The electric field points “downhill” in potential — in the direction of maximum decrease of $V$, hence the minus sign. Derived properly in 2.02 Electric Potential, Equipotential Contours & Dipole Derivations.

Why field lines are perpendicular to equipotentials

Moving along a surface where $V$ is constant means $dV=\nabla V\cdot d\vec{l}=0$. A zero dot product means $\nabla V\perp d\vec{l}$ — so $\vec{E}$ (parallel to $\nabla V$) is perpendicular to every equipotential surface. That single line answers a recurring sketch question [PYQ: 2024, 2025].

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3. Divergence of a Vector Field $\nabla \cdot \vec{A}$ [PYQ: 2018]

Abstract

The divergence of a vector field $\vec{A}$ is a scalar giving the net outward flux per unit volume as the volume shrinks to a point: $\nabla \cdot \vec{A}={\lim }_{\Delta v\to 0}\frac{{\oint }_S\vec{A}\cdot d\vec{s}}{\Delta v}$

3.1 Physical significance in electromagnetics [PYQ: 2018]

This is the exact wording the 2018 paper asked for, so answer it like this:

Sign of $\nabla \cdot \vec{A}$	Meaning at that point	EM example
$>0$	Source — flux lines originate here	Positive charge: $\nabla \cdot \vec{D}={\rho }_v>0$
$<0$	Sink — flux lines terminate here	Negative charge
$=0$	Solenoidal — no sources or sinks; lines close on themselves	Magnetic field: $\nabla \cdot \vec{B}=0$ (no monopoles)

The one-sentence answer for the 2-mark version

“Divergence measures the net outward electric (or magnetic) flux emanating per unit volume from a point. In electromagnetics, $\nabla \cdot \vec{D}={\rho }_v$ tells us that electric flux lines have their sources and sinks strictly at electric charges, while $\nabla \cdot \vec{B}=0$ tells us magnetic flux lines have no sources at all and must close upon themselves.”

3.2 Explicit forms

$\nabla \cdot \vec{A}=\frac{1}{h_1{h}_2{h}_3}\left[\frac{\partial }{\partial u_1}(h_2{h}_3{A}_1)+\frac{\partial }{\partial u_2}(h_1{h}_3{A}_2)+\frac{\partial }{\partial u_3}(h_1{h}_2{A}_3)\right]$

System	$\nabla \cdot \vec{A}$
Cartesian	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$
Cylindrical	$\frac{1}{r}\frac{\partial }{\partial r}(r{A}_r)+\frac{1}{r}\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}$
Spherical	$\frac{1}{R^2}\frac{\partial }{\partial R}(R^2{A}_R)+\frac{1}{R\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{R\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }$

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4. Curl of a Vector Field $\nabla \times \vec{A}$ [PYQ: 2022]

Abstract

The curl of $\vec{A}$ is a vector whose magnitude is the maximum circulation per unit area as the area shrinks to a point, and whose direction ${\hat{a}}_n$ is normal to the plane in which that circulation is maximum (right-hand rule): $\nabla \times \vec{A}={\lim }_{\Delta s\to 0}{\hat{a}}_n\frac{{\left[{\oint }_C\vec{A}\cdot d\vec{l}\right]}_{\max }}{\Delta s}$

4.1 Properties of the Curl Operation [PYQ: 2022]

The 2022 paper asked: “Write down the properties of curl operation. What are the consequences of a vector being curl free?” Write all of these — the mark scheme is a checklist:

1. Curl measures rotation. $\nabla \times \vec{A}$ describes the rotational (circulating, vortex-like) character of the field at a point.
2. The output is a vector, unlike divergence which is a scalar. Its direction follows the right-hand rule about the axis of maximum circulation.
3. Curl of a gradient is identically zero: $\nabla \times (\nabla V)\equiv 0$ for any scalar $V$.
4. Divergence of a curl is identically zero: $\nabla \cdot (\nabla \times \vec{A})\equiv 0$ for any vector $\vec{A}$.
5. Linearity: $\nabla \times (\vec{A}+\vec{B})=\nabla \times \vec{A}+\nabla \times \vec{B}$.
6. Product rule with a scalar: $\nabla \times (V\vec{A})=V(\nabla \times \vec{A})+(\nabla V)\times \vec{A}$.
7. Curl connects to circulation via Stokes’s theorem: ${\int }_S(\nabla \times \vec{A})\cdot d\vec{s}={\oint }_C\vec{A}\cdot d\vec{l}$.

4.2 Consequences of a Curl-Free (Irrotational) Field [PYQ: 2022]

If $\nabla \times \vec{A}=0$ everywhere, then:

1. Zero circulation on every closed path: by Stokes’s theorem ${\oint }_C\vec{A}\cdot d\vec{l}=0$.
2. The field is conservative — ${\int }_a^b\vec{A}\cdot d\vec{l}$ depends only on the endpoints, never on the path.
3. A scalar potential exists: $\vec{A}=-\nabla V$. This is precisely why electrostatics has a potential $V$ at all.
4. No net work in a closed loop — the field-theory statement of Kirchhoff’s Voltage Law.
5. Physical example: the static electric field, $\nabla \times \vec{E}=0$.

4.3 Explicit forms

Cartesian determinant: $\nabla \times \vec{A}=\left|\begin{array}{ccc}{\hat{a}}_x& {\hat{a}}_y& {\hat{a}}_z\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ A_x& A_y& A_z\end{array}\right|$

Expanded: $\nabla \times \vec{A}={\hat{a}}_x\left(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\right)+{\hat{a}}_y\left(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}\right)+{\hat{a}}_z\left(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}\right)$

General curvilinear: $\nabla \times \vec{A}=\frac{1}{h_1{h}_2{h}_3}\left|\begin{array}{ccc}{h}_1{\hat{a}}_1& h_2{\hat{a}}_2& h_3{\hat{a}}_3\\ \frac{\partial }{\partial u_1}& \frac{\partial }{\partial u_2}& \frac{\partial }{\partial u_3}\\ h_1{A}_1& h_2{A}_2& h_3{A}_3\end{array}\right|$

Cylindrical (used constantly in Ch. 3 for $\vec{B}=\nabla \times \vec{A}$): $\nabla \times \vec{A}={\hat{a}}_r\left(\frac{1}{r}\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z}\right)+{\hat{a}}_{\varphi }\left(\frac{\partial A_r}{\partial z}-\frac{\partial A_z}{\partial r}\right)+{\hat{a}}_z\frac{1}{r}\left(\frac{\partial (r{A}_{\varphi })}{\partial r}-\frac{\partial A_r}{\partial \varphi }\right)$

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5. Solenoidal vs. Irrotational Fields

Field Type	Condition	Physical Meaning	Can be written as	EM Example
Solenoidal (divergence-less)	$\nabla \cdot \vec{F}=0$	Flux lines form continuous closed loops; no sources or sinks	$\vec{F}=\nabla \times \vec{A}$	Magnetic flux density $\vec{B}$
Irrotational (curl-free / conservative)	$\nabla \times \vec{F}=0$	Zero circulation; work around any closed loop is zero	$\vec{F}=-\nabla V$	Static electric field $\vec{E}$

Memory hook

Solenoidal $\to$ Source-free $\to$ divergence zero $\to$ $\vec{B}$. Irrotational $\to$ no rotatIon $\to$ curl zero $\to$ $\vec{E}$ (static).

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6. The Laplacian ${\nabla }^{2}V$

The Laplacian of a scalar field is the divergence of its gradient: ${\nabla }^{2}V=\nabla \cdot (\nabla V)$

System	${\nabla }^{2}V$
Cartesian	$\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}$
Cylindrical	$\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}V}{\partial {\varphi }^2}+\frac{{\partial }^{2}V}{\partial z^2}$
Spherical	$\frac{1}{R^2}\frac{\partial }{\partial R}\left(R^2\frac{\partial V}{\partial R}\right)+\frac{1}{R^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial V}{\partial \theta }\right)+\frac{1}{R^2{\sin }^2\theta }\frac{{\partial }^{2}V}{\partial {\varphi }^2}$

These three expressions are the entire machinery behind Poisson’s and Laplace’s equations and every capacitance derivation in 2.04 Electrostatic Boundary Conditions, Refraction Law & Poisson-Laplace Equations: ${\nabla }^{2}V=-\frac{{\rho }_v}{ϵ}\text{(Poisson)},{\nabla }^{2}V=0\text{(Laplace)}$

Vector Laplacian (needed for the wave equation and the vector magnetic potential): ${\nabla }^2\vec{A}=\nabla (\nabla \cdot \vec{A})-\nabla \times (\nabla \times \vec{A})$

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7. Master Solved PYQ Numerical [PYQ: 2022, 2025]

PYQ — 2022 & 2025 (12 / 10 marks)

Let $\vec{A}=(3y-z){\hat{a}}_x+2xz{\hat{a}}_y$ (Wb/m) in a certain region of free space. (i) Find $\nabla \cdot \vec{A}$. (ii) Find $\vec{A}$, $\vec{B}$, $\vec{H}$ and $\vec{J}$ at $P(3,-1,2)$.

Part (i) — Divergence

Components: $A_x=3y-z$, $A_y=2xz$, $A_z=0$.

$\nabla \cdot \vec{A}=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}=\frac{\partial }{\partial x}(3y-z)+\frac{\partial }{\partial y}(2xz)+\frac{\partial }{\partial z}(0)$

$=0+0+0=\boxed{0}$

Read the physics, don't just stop at "0"

$\nabla \cdot \vec{A}=0$ means $\vec{A}$ is solenoidal. This is exactly the Coulomb-gauge condition $\nabla \cdot \vec{A}=0$ that lets $\vec{A}$ serve as a legitimate vector magnetic potential — which is why the question then asks for $\vec{B}$. State this; it is worth a mark.

Part (ii) — Fields at $P(3,-1,2)$

Step 1 — $\vec{A}$ at $P$ (substitute $x=3,y=-1,z=2$): ${\vec{A}}_P=(3(-1)-2){\hat{a}}_x+2(3)(2){\hat{a}}_y=-5{\hat{a}}_x+12{\hat{a}}_y\text{(Wb/m)}$

Step 2 — $\vec{B}=\nabla \times \vec{A}$: $\vec{B}=\left|\begin{array}{ccc}{\hat{a}}_x& {\hat{a}}_y& {\hat{a}}_z\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ 3y-z& 2xz& 0\end{array}\right|$

$\vec{B}={\hat{a}}_x\left[\frac{\partial (0)}{\partial y}-\frac{\partial (2xz)}{\partial z}\right]-{\hat{a}}_y\left[\frac{\partial (0)}{\partial x}-\frac{\partial (3y-z)}{\partial z}\right]+{\hat{a}}_z\left[\frac{\partial (2xz)}{\partial x}-\frac{\partial (3y-z)}{\partial y}\right]$

$\vec{B}={\hat{a}}_x[0-2x]-{\hat{a}}_y[0-(-1)]+{\hat{a}}_z[2z-3]$

$\vec{B}=-2x{\hat{a}}_x-{\hat{a}}_y+(2z-3){\hat{a}}_z\text{(T)}$

At $P(3,-1,2)$: ${\vec{B}}_P=-6{\hat{a}}_x-{\hat{a}}_y+{\hat{a}}_z\text{(T)}$

Step 3 — $\vec{H}$ at $P$. The region is free space, so $\mu ={\mu }_0=4\pi \times 10^{-7}$ H/m: ${\vec{H}}_P=\frac{{\vec{B}}_P}{{\mu }_0}=\frac{-6{\hat{a}}_x-{\hat{a}}_y+{\hat{a}}_z}{4\pi \times 10^{-7}}=\left(-4.77{\hat{a}}_x-0.796{\hat{a}}_y+0.796{\hat{a}}_z\right)\times 10^6\text{(A/m)}$

Step 4 — $\vec{J}=\nabla \times \vec{H}$: $\nabla \times \vec{B}=\left|\begin{array}{ccc}{\hat{a}}_x& {\hat{a}}_y& {\hat{a}}_z\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ -2x& -1& 2z-3\end{array}\right|$

$={\hat{a}}_x\left[\frac{\partial (2z-3)}{\partial y}-\frac{\partial (-1)}{\partial z}\right]-{\hat{a}}_y\left[\frac{\partial (2z-3)}{\partial x}-\frac{\partial (-2x)}{\partial z}\right]+{\hat{a}}_z\left[\frac{\partial (-1)}{\partial x}-\frac{\partial (-2x)}{\partial y}\right]=0$

$\therefore {\vec{J}}_P=\frac{1}{{\mu }_0}(\nabla \times \vec{B})=\boxed{0{\text{A/m}}^2}$

Interpretation worth a mark

$\vec{J}=0$ everywhere means $P$ sits in a current-free region — the magnetic field there is produced by currents located elsewhere. Consistency check: $\nabla \cdot \vec{B}=-2+0+2=0$, confirming $\vec{B}$ is solenoidal as any real magnetic field must be.

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8. PYQ Coverage for This Note

Question (verbatim)	Marks	Year(s)
…Explain the physical significance of divergence in terms of electromagnetic field.	02	2018
Write down the properties of curl operation. What are the consequences of a vector being curl free?	05+02	2022
Let $\vec{A}=(3y-z){\hat{a}}_x+2xz{\hat{a}}_y$ (Wb/m)… (i) Find $\nabla \cdot \vec{A}$. (ii) Find $\vec{A},\vec{B},\vec{H},\vec{J}$ at $P(3,-1,2)$.	12 / 10	2022, 2025

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9. Exam Hacks & Traps

Key Exam Checkpoints

1. Divergence gives a scalar, curl gives a vector. Putting a hat on a divergence answer, or omitting unit vectors from a curl, loses marks instantly.
2. The middle term of the curl determinant carries a minus sign. $-{\hat{a}}_y[\dots ]$. Half the wrong answers in this course come from dropping it.
3. Follow the chain in order: $\vec{B}=\nabla \times \vec{A}$, then $\vec{H}=\vec{B}/\mu$, then $\vec{J}=\nabla \times \vec{H}$. Do not try to get $\vec{J}$ directly from $\vec{A}$.
4. “Free space” is a hint, not decoration. It tells you $\mu ={\mu }_0$ and $ϵ={ϵ}_0$. Say so explicitly.
5. Never apply the Cartesian divergence/curl formula in cylindrical or spherical coordinates. The $1/r$ and $1/R^2$ factors are not optional.
6. When asked for “properties” of curl, list at least five. The two null identities plus Stokes’s theorem are free points.

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10. Self-Check

1. What does $\nabla \cdot \vec{B}=0$ physically state? (No magnetic monopoles; $\vec{B}$ lines close on themselves)
2. Name three consequences of $\nabla \times \vec{E}=0$. (Conservative; path-independent line integral; scalar potential $V$ exists)
3. Write ${\nabla }^{2}V$ in cylindrical coordinates.
4. For $\vec{A}=(3y-z){\hat{a}}_x+2xz{\hat{a}}_y$, what is $\vec{B}$ as a general function of position? ($-2x{\hat{a}}_x-{\hat{a}}_y+(2z-3){\hat{a}}_z$)

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Next: 1.03 Integral Theorems (Divergence Theorem & Stokes_s Theorem) & Identity Proofs