Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Recap

Field point $P (x, y, z)$ , source point $Q (x^{'}, y^{'}, z^{'})$ (primed). $R = r - r^{'}$ points from source to field point, $R = ∣ R ∣$ , $a_{R} = R / R$ , and $\nabla (1/ R) = - a_{R} / R^{2}$ , $\nabla^{'} (1/ R) = - \nabla (1/ R)$ .
Orthogonal coordinates: $\partial r / \partial u_{i} = h_{i} a_{i}$ , so $d ℓ = \sum_{i} h_{i} d u_{i} a_{i}$ , $d S_{1} = a_{1} h_{2} h_{3} d u_{2} d u_{3}$ (cyclic) and $d V = h_{1} h_{2} h_{3} d u_{1} d u_{2} d u_{3}$ .
Cartesian: $h_{x} = h_{y} = h_{z} = 1$ , unit vectors constant, $d ℓ = d x a_{x} + d y a_{y} + d z a_{z}$ , $d V = d x d y d z$ .
Line integral: $\int_{C} A \cdot d ℓ$ with $d ℓ$ always written with plus signs. The direction of travel is set by the limits only. Flux: $\int_{S} A \cdot d S$ , orientation $a_{n}$ stated.
Cylindrical: $x = ρ cos ϕ$ , $y = ρ sin ϕ$ ( $ϕ$ from both $cos ϕ$ and $sin ϕ$ ). Scale factors $h_{ρ} = 1$ , $h_{ϕ} = ρ$ , $h_{z} = 1$ , so $d ℓ = d ρ a_{ρ} + ρ d ϕ a_{ϕ} + d z a_{z}$ and $d V = ρ d ρ d ϕ d z$ . The unit vectors $a_{ρ}, a_{ϕ}$ depend on $ϕ$ , so never add them at different points.
Components convert by the rotation matrix (2.3.28), with the angle $ϕ$ of the point where the vector is located. Then substitute the coordinates (Table 2.3.3).
Spherical coordinates $(r, θ, ϕ)$ , $0 \leq r < \infty$ , $0 \leq θ \leq π$ , $0 \leq ϕ < 2 π$ : $x = r sin θ cos ϕ$ , $y = r sin θ sin ϕ$ , $z = r cos θ$ , and conversely $r = x^{2} + y^{2} + z^{2}$ , $θ = arccos (z / r)$ , $ϕ$ from $cos ϕ$ and $sin ϕ$ . $θ$ is the colatitude, $ϕ$ the (east) longitude.
Coordinate surfaces: spheres ( $r$ ), cones ( $θ$ ), half-planes ( $ϕ$ ), meeting at right angles.
Unit vectors, Eqs. (2.3.34) to (2.3.36), from $\partial r / \partial u_{i} = h_{i} a_{i}$ , with scale factors $h_{r} = 1$ , $h_{θ} = r$ , $h_{ϕ} = r sin θ$ (radius of the circle each angle moves along). The triple $(a_{r}, a_{θ}, a_{ϕ})$ is orthonormal and right-handed, and $r = r a_{r}$ .
The curvilinear unit vectors depend on position, Eq. (2.3.38): convert them to Cartesian ones before integrating a vector, and never add unit vectors of different points.
$d ℓ = d r a_{r} + r d θ a_{θ} + r sin θ d ϕ a_{ϕ}$ , $d V = r^{2} sin θ d r d θ d ϕ$ , and on a sphere $d S = r^{2} sin θ d θ d ϕ$ . In any orthogonal system $d V = h_{1} h_{2} h_{3} d u_{1} d u_{2} d u_{3}$ (Table 2.3.4).
Component transformations Cartesian ↔ spherical, Eq. (2.3.44), and cylindrical ↔ spherical, Eq. (2.3.46), are orthogonal matrices (inverse = transpose) whose entries depend on the position. Convert components and coordinates.
Sphere of radius $b$ : area $4 π b^{2}$ , volume $\frac{4}{3} π b^{3}$ .
The operators are defined by their meaning: $df = \nabla f \cdot d ℓ$ , divergence = outward flux per unit volume, Eq. (2.3.51), and curl = circulation per unit area, Eq. (2.3.55). Applied to small curvilinear boxes and loops, these give the formulas in any orthogonal system (Section 2.3.10).
Gradient: divide each derivative by its scale factor, e.g. $\nabla f = \partial_{r} f a_{r} + \frac{1}{r} \partial_{θ} f a_{θ} + \frac{1}{r s i n θ} \partial_{ϕ} f a_{ϕ}$ .
Divergence and curl: the factors such as $r^{2}$ , $sin θ$ and $ρ$ account for the changing size of the faces and sides of the curvilinear elements, e.g. the radial part $\frac{1}{r ^{2}} \partial_{r} (r^{2} A_{r})$ .
Laplacian $Δ f = \nabla \cdot \nabla f$ . The functions $1/ r$ and $ln ρ$ are harmonic (away from $r = 0$ , $ρ = 0$ ).
Useful results: $\nabla (1/ r) = - a_{r} / r^{2}$ and $\nabla \cdot r = 3$ . Away from the origin/axis, $\nabla \cdot (a_{r} / r^{2}) = 0$ and $\nabla \cdot (a_{ρ} / ρ) = 0$ . Radial fields $A_{r} (r) a_{r}$ are curl-free, and $\nabla \times (ω ρ a_{ϕ}) = 2 ω a_{z}$ .

2.3.11 Recap