Vector analysis and coordinate systems

Why it matters

Every law in electromagnetics — Coulomb, Gauss, Ampere, Faraday, Maxwell — is written in the language of vectors and vector calculus. If you can pick the coordinate system that matches the symmetry of a problem and handle dot products, cross products, gradient, divergence and curl without slips, most field problems reduce to a few lines of algebra. Instrumentation work (capacitive and inductive sensors, coaxial cables, Hall probes) uses exactly these tools.

Key ideas

Scalars and vectors. A scalar has magnitude only (charge, potential V, energy). A vector has magnitude and direction (E, D, H, B, force). A unit vector a_A = A / |A| has magnitude 1 and carries only direction.

Dot product gives a scalar: the product of one vector's magnitude and the projection of the other onto it. A·B = 0 means the vectors are perpendicular. It is used for work (F·dl), flux (D·dS) and the component of a vector along a direction (A·a_n).

Cross product gives a vector perpendicular to both, with direction from the right-hand rule: fingers along A, curl them towards B, the thumb points along A × B. It is anti-commutative: B × A = −(A × B). A × B = 0 for parallel vectors. |A × B| is the area of the parallelogram formed by A and B. Used for force on a current (I dl × B), torque (r × F) and the Biot–Savart law.

Coordinate systems. Choose the one whose surfaces match the geometry:

• Cartesian (x, y, z): planes, sheets of charge, rectangular plates.
• Cylindrical (ρ, φ, z): line charges, wires, coaxial cables. ρ is the perpendicular distance from the z-axis, φ is measured from the x-axis.
• Spherical (r, θ, φ): point charges, spheres. r is the distance from the origin, θ is the polar angle from the +z axis (0 to π), φ the azimuth (0 to 2π).

Unit vectors a_x, a_y, a_z are constant everywhere; a_ρ, a_φ, a_r, a_θ change direction from point to point, so they cannot be pulled outside an integral without first converting to Cartesian components.

Differential elements. Line, surface and volume elements carry the scale factors (1, ρ, r, r sin θ). Forgetting the ρ in ρ dφ or the r² sin θ in the spherical volume element is the most common integration error.

Vector calculus.

• Gradient ∇V: a vector pointing in the direction of the fastest increase of a scalar field; its magnitude is that maximum rate. In electrostatics E = −∇V.
• Divergence ∇·A: a scalar giving the net outward flux per unit volume at a point (a source or sink density). ∇·D = ρv.
• Curl ∇×A: a vector giving the circulation per unit area about a point (rotation). ∇×H = J. A field with zero curl everywhere is conservative (irrotational); a field with zero divergence is solenoidal.
• Divergence theorem: ∮ A·dS = ∫ (∇·A) dv — converts a closed-surface flux into a volume integral.
• Stokes' theorem: ∮ A·dl = ∫ (∇×A)·dS — converts a closed-line integral into a surface integral.
• Two identities that hold for any smooth field: ∇×(∇V) = 0 and ∇·(∇×A) = 0.

Formulas

• A·B = |A||B| cos θ = AxBx + AyBy + AzBz — θ is the angle between A and B; the result is a scalar.
• A × B = |A||B| sin θ a_n = (AyBz − AzBy) a_x + (AzBx − AxBz) a_y + (AxBy − AyBx) a_z — a_n is the unit normal given by the right-hand rule.
• ρ = √(x² + y²), φ = tan⁻¹(y/x), z = z — Cartesian to cylindrical (pick the quadrant of φ from the signs of x and y).
• r = √(x² + y² + z²), θ = cos⁻¹(z/r), φ = tan⁻¹(y/x) — Cartesian to spherical.
• dv = dx dy dz = ρ dρ dφ dz = r² sin θ dr dθ dφ — volume elements (m³).
• ∇V = ∂V/∂x a_x + ∂V/∂y a_y + ∂V/∂z a_z (Cartesian); ∇V = ∂V/∂ρ a_ρ + (1/ρ) ∂V/∂φ a_φ + ∂V/∂z a_z (cylindrical); ∇V = ∂V/∂r a_r + (1/r) ∂V/∂θ a_θ + (1/(r sin θ)) ∂V/∂φ a_φ (spherical).
• ∇·A = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z (Cartesian); ∇·A = (1/ρ) ∂(ρAρ)/∂ρ + (1/ρ) ∂Aφ/∂φ + ∂Az/∂z (cylindrical); ∇·A = (1/r²) ∂(r²Ar)/∂r + (1/(r sin θ)) ∂(sin θ Aθ)/∂θ + (1/(r sin θ)) ∂Aφ/∂φ (spherical).
• ∇×A = (∂Az/∂y − ∂Ay/∂z) a_x + (∂Ax/∂z − ∂Az/∂x) a_y + (∂Ay/∂x − ∂Ax/∂y) a_z (Cartesian).
• ∮ A·dS = ∫ (∇·A) dv (divergence theorem); ∮ A·dl = ∫ (∇×A)·dS (Stokes' theorem) — both need A and its derivatives continuous in the region.

Worked examples

Example 1 (standard). Given A = 3a_x + 4a_y + 5a_z and B = 2a_x − a_y + 3a_z, find A·B, A × B, the angle between them and the scalar component of A along B.

1. A·B = AxBx + AyBy + AzBz = (3)(2) + (4)(−1) + (5)(3) = 6 − 4 + 15 = 17.
2. A × B: x-component = (4)(3) − (5)(−1) = 17; y-component = (5)(2) − (3)(3) = 1; z-component = (3)(−1) − (4)(2) = −11. So A × B = 17a_x + a_y − 11a_z.
3. |A| = √(9 + 16 + 25) = √50 = 7.071; |B| = √(4 + 1 + 9) = √14 = 3.742.
4. cos θ = A·B / (|A||B|) = 17 / (7.071 × 3.742) = 0.6425, so θ = 50.0°.
5. Scalar component of A along B = A·B / |B| = 17 / 3.742 = 4.54. Check: (A × B)·A = 51 + 4 − 55 = 0, so the cross product is perpendicular to A, as it must be.

Example 2 (GATE level). D = ρ² a_ρ C/m² in cylindrical coordinates. Verify the divergence theorem for the closed cylinder ρ ≤ 2 m, 0 ≤ z ≤ 1 m.

1. Divergence: ∇·D = (1/ρ) ∂(ρ·ρ²)/∂ρ = (1/ρ)(3ρ²) = 3ρ C/m³.
2. Volume integral: ∫ 3ρ · ρ dρ dφ dz over ρ from 0 to 2, φ from 0 to 2π, z from 0 to 1 = 3 × (2³/3) × 2π × 1 = 16π = 50.27 C.
3. Surface integral. The top and bottom faces have normals ±a_z, and D has no z-component, so they contribute zero. On the curved face ρ = 2 m, dS = ρ dφ dz a_ρ, so ∮ D·dS = ρ² × ρ × 2π × h = 2³ × 2π × 1 = 16π = 50.27 C.
4. Both sides agree, so the theorem is verified; by Gauss's law, 50.27 C is also the charge enclosed.

Example 3 (conversion). Express the point P(1, √3, 2) in cylindrical and spherical coordinates.

1. ρ = √(1 + 3) = 2 m; φ = tan⁻¹(√3/1) = 60°; z = 2 m → P(2, 60°, 2).
2. r = √(1 + 3 + 4) = √8 = 2.828 m; θ = cos⁻¹(2/2.828) = 45°; φ = 60° → P(2.828, 45°, 60°).

Common mistakes

• Writing ∇·A in cylindrical or spherical form as ∂Aρ/∂ρ + … without the scale factors (1/ρ)∂(ρAρ)/∂ρ, (1/r²)∂(r²Ar)/∂r.
• Treating a_ρ or a_r as constants and taking them outside an integral — for example, integrating a_ρ around a full circle gives zero, not 2π a_ρ.
• Getting the cross-product sign wrong: a_x × a_y = a_z, a_y × a_z = a_x, a_z × a_x = a_y (cyclic order); reversing the order flips the sign.
• Using the wrong quadrant for φ = tan⁻¹(y/x) when x < 0.
• Confusing θ (spherical polar angle, from +z) with φ (azimuth, from +x).
• Forgetting ρ in the surface element ρ dφ dz of a cylinder, or r² sin θ in the spherical surface element.

For GATE IN

Expect short MCQ/NAT questions on: divergence or curl of a given field (often in cylindrical or spherical form), checking whether a field is conservative or solenoidal, the gradient of a potential at a point, unit normals, the angle between vectors, and applying the divergence or Stokes' theorem to save effort on a flux or circulation integral. Practise the three del-operator forms until you can write them from memory, and always look for symmetry before integrating.

Quick check

1. What is the dot product of two perpendicular vectors?
2. What is ∇·(ρ a_ρ) in cylindrical coordinates?
3. What is the curl of the gradient of any scalar field?
4. Is B × A equal to A × B?
5. What is |A × B| for A = 3a_x + 4a_y and B = 5a_z?

Answers: 1. Zero. 2. 2. 3. Zero (the null vector). 4. No, B × A = −(A × B). 5. 25.