01 Chapter Map — Vector Calculus & Field Fundamentals

What this chapter covers

The mathematical toolkit for the entire course, plus the conceptual case for why field theory is needed at all. Coordinate systems and differential elements, the four vector operators, the two integral theorems and the two null identities, the field concept versus action-at-a-distance, the breakdown of circuit theory at high frequency, quasi-static conditions, and material classification.

---

📚 Study Notes Index (read in this order)

#	Note	What it gives you	PYQ weight
1	1.01 Spatial Coordinates & Differential Elements (Cartesian, Cylindrical, Spherical)	$d\vec{l}$, $d\vec{s}$, $dv$ in all three systems; metric coefficients	Low direct (2019 only) — but a prerequisite for everything
2	1.02 Vector Operators (Gradient, Divergence, Curl & Laplacian)	$\nabla V$, $\nabla \cdot \vec{A}$, $\nabla \times \vec{A}$, ${\nabla }^{2}V$; the repeated $\vec{A},\vec{B},\vec{H},\vec{J}$ numerical	Medium (2018, 2022, 2025)
3	1.03 Integral Theorems (Divergence Theorem & Stokes_s Theorem) & Identity Proofs	Divergence & Stokes theorems, null identities, field concept, circuit-theory inadequacy, quasi-static	High (2018, 2019, 2020, 2021, 2024)
4	1.04 Media Properties & Conductor-Insulator Behaviour	Homogeneous/linear/isotropic, loss tangent, complex permittivity, conductor vs insulator	High (2015, 2017, 2019, 2020, 2023, 2024)
✅	00 Chapter 1 Active-Recall Diagnostic Quiz	Test yourself before and after	—

---

🎯 Highest-Yield Items in This Chapter

Concept	Years asked	Where
Inadequacy of circuit theory / necessity of field concept	2018, 2019, 2021	1.03 Integral Theorems (Divergence Theorem & Stokes_s Theorem) & Identity Proofs
Define homogeneous, linear and isotropic media	2015, 2017, 2019, 2023, 2024	1.04 Media Properties & Conductor-Insulator Behaviour
$\vec{A},\vec{B},\vec{H},\vec{J}$ numerical at $P(3,-1,2)$	2022, 2025	1.02 Vector Operators (Gradient, Divergence, Curl & Laplacian)
Meaning of ‘field’ + significance for an ECE engineer	2024	1.03 Integral Theorems (Divergence Theorem & Stokes_s Theorem) & Identity Proofs
Good conductor vs good insulator conditions	2020	1.04 Media Properties & Conductor-Insulator Behaviour
Properties of curl / consequences of curl-free	2022	1.02 Vector Operators (Gradient, Divergence, Curl & Laplacian)
Quasi-static conditions	2020	1.03 Integral Theorems (Divergence Theorem & Stokes_s Theorem) & Identity Proofs
Physical significance of divergence	2018	1.02 Vector Operators (Gradient, Divergence, Curl & Laplacian)
Define metric coefficient	2019	1.01 Spatial Coordinates & Differential Elements (Cartesian, Cylindrical, Spherical)

---

🔗 How This Chapter Feeds Forward

graph TD
    A["Divergence theorem"] --> B["Gauss's law integral form"]
    D["Stokes's theorem"] --> E["Ampere's circuital law"]
    F["Curl of gradient = 0"] --> G["Scalar potential V exists"]
    H["Divergence of curl = 0"] --> I["Vector potential A exists"]
    J["dl, ds, dv elements"] --> K["Every field integral in the course"]
    L["Laplacian"] --> M["Poisson / Laplace equations, capacitance"]

---

🔗 Related Resources

• Course teaching plan: ECE 2105 - Electromagnetic Fields and Waves
• Full checklist with PYQ years: field checklist
• Verbatim question bank: ece 2105 field pyq
• Definitions-only bank: pyq definations
• Next chapter: 02 Chapter Map - Electrostatics & Boundary Conditions