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Real parts of $e^{- i ω t}$ phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system

Statement

Let $D \subseteq R^{3}$ be open, $c > 0$ and $ω > 0$ , and let $\tilde{E}, \tilde{B} : D \to C^{3}$ . Define $E, B : D \times R \to R^{3}$ by $E (x, t) : = Re (\tilde{E} (x) e^{- i ω t})$ and $B (x, t) : = Re (\tilde{B} (x) e^{- i ω t})$ componentwise (The complex exponential by its power series, Real and imaginary parts, complex conjugation, and modulus; $C^{k}$ Euclidean maps and diffeomorphisms for the classes). Then: (a) [Calculus] if $\tilde{E}, \tilde{B}$ are $C^{1}$ , then $E, B$ are $C^{1}$ and at every $(x, t) \in D \times R$ : $\partial_{t} E (x, t) = Re (- i ω \tilde{E} (x) e^{- i ω t})$ , and likewise for $B$ ; for each $j \in {1, 2, 3}$ , $\partial_{j} E (x, t) = Re (\partial_{j} \tilde{E} (x) e^{- i ω t})$ , and likewise for $B$ ; and with the divergence and curl of Divergence and curl of a $C^{1}$ vector field acting on the spatial variable, $div E (x, t) = Re ((div \tilde{E}) (x) e^{- i ω t})$ , $curl E (x, t) = Re ((curl \tilde{E}) (x) e^{- i ω t})$ , and likewise for $B$ . (b) [Injectivity] if $w \in C^{3}$ and $Re (w e^{- i ω t}) = 0$ for all $t \in R$ , then $w = 0$ ; more generally, if $w : D \to C^{3}$ is continuous and $Re (w (x) e^{- i ω t}) = 0$ for all $(x, t) \in D \times R$ , then $w \equiv 0$ on $D$ . (c) [Equivalence] if $\tilde{E}, \tilde{B}$ are $C^{1}$ , the phasor system $div \tilde{E} = 0, div \tilde{B} = 0, curl \tilde{E} = i ω \tilde{B}, curl \tilde{B} = - i ω c^{- 2} \tilde{E}$ holds on $D$ if and only if the real system $div E = 0, div B = 0, curl E = - \partial_{t} B, curl B = c^{- 2} \partial_{t} E$ holds on $D \times R$ .

Facts & Assumptions

Given: An open set $D \subseteq R^{3}$ , reals $c > 0$ and $ω > 0$ , complex fields $\tilde{E}, \tilde{B} : D \to C^{3}$ , and the associated real fields $E, B : D \times R \to R^{3}$ of the Statement. Components are indexed by $k, j \in {1, 2, 3}$ .

[F1]

Complex exponential and Euler form: $\exp$ is entire with $\exp^{'} = \exp$ ; for real $θ$ one has $e^{i θ} = \cos θ + i \sin θ$ , so $e^{- i ω t} = \cos ω t - i \sin ω t$ and $e^{- i π / 2} = \cos (π / 2) - i \sin (π / 2) = - i$ ; $\sin^{'} = \cos$ and $\cos^{'} = - \sin$ on $R$ (The complex exponential by its power series, The complex exponential is entire and its complex derivative is itself, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The derivatives of sine and cosine are cosine and minus sine, Quarter-turn values and shifts by pi/2 and pi).

[F2]

Real and imaginary parts: $Re (a + b i) = a$ , $Im (a + b i) = b$ ; $Re$ and $Im$ are $R$ -linear; $(a + b i) (s + t i) = (a s - b t) + (a t + b s) i$ ; in particular $Re ((a + b i) (c - i s)) = a c + b s$ and $Re (- i ω (a + b i) (c - i s)) = ω b c - ω a s$ for real $a, b, ω, c, s$ (Real and imaginary parts, complex conjugation, and modulus, The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ).

[F3]

Complex differentiability and real differentiability: complex differentiability at a point is equivalent to real total differentiability there with the derivative given by multiplication by the complex derivative; hence $\exp$ is real totally differentiable everywhere with real derivative the multiplication map $h \mapsto e^{z} h$ , and the chain rule for total derivatives applies to real-differentiable compositions (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations, The complex exponential is entire and its complex derivative is itself, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F4]

One-variable and componentwise rules: sums, scalar multiples and products of differentiable real functions are differentiable with the usual derivatives (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ); partial derivatives are coordinate-line directional derivatives, so $\partial_{j} g (x)$ is the derivative at $0$ of $h \mapsto g (x + h e_{j})$ (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ); a map $U \to R^{q}$ on an open $U$ is $C^{k}$ when every component's iterated partials up to order $k$ exist and are continuous, and a complex field is $C^{1}$ exactly when its six real coordinate maps $Re {\tilde{E}}_{k}, Im {\tilde{E}}_{k}$ are $C^{1}$ , with complex partials $\partial_{j} {\tilde{E}}_{k} = \partial_{j} (Re {\tilde{E}}_{k}) + i \partial_{j} (Im {\tilde{E}}_{k})$ ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

The divergence of a $C^{1}$ real field $F$ on open $U \subseteq R^{n}$ is $div F = \sum_{k} \partial_{k} F_{k}$ , and the curl on $U \subseteq R^{3}$ is $curl F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x})$ ; both are linear in $F$ with real scalars, and on complex fields both are applied componentwise to real and imaginary parts (Divergence and curl of a $C^{1}$ vector field, Divergence and curl are linear and satisfy the scalar product rules).

[F6]

Continuity toolkit: continuity is the metric notion of Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, and on $R^{m}$ the Euclidean topology agrees with the product-of- $R$ topology (For $n \geq 1$ the product topology on $n$ copies of the usual topology of $R$ is the metric topology of $d_{\infty}$ on $R^{n}$ , and hence also of $d_{1}$ and $d_{2}$ , so $R^{n}$ as a product and $R^{n}$ as a metric space are one space); projections from a product are continuous and a map into a product is continuous exactly when all its components are (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice); finite sums, products, maxima, minima and quotients of continuous real-valued maps on a topological space are continuous (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined); and a function differentiable at a point of its domain is continuous there (A function differentiable at $c$ is continuous at $c$ ). For the pullbacks used here, fix $(x_{*}, t_{*})$ . The Euclidean product distance bounds both $∣ x - x_{*} ∣$ and $∣ t - t_{*} ∣$ ; choosing the $δ$ from continuity of $g : D \to R$ at $x_{*}$ or of $h : R \to R$ at $t_{*}$ therefore proves directly that $(x, t) \mapsto g (x)$ and $(x, t) \mapsto h (t)$ are continuous. Finite sums and products of these pullbacks are continuous by the cited algebra lemma.

Proof

technique · direct; expand the encoding in real components, differentiate along the time and coordinate directions with the one-variable rules, read the divergence and curl off the componentwise formulas, and obtain the equivalence from the injectivity of the real-part evaluation proved at the two phases $t=0$ and $t=\pi/(2\omega)$

1.1givenF1F2F4

[Encoding] For clauses (a) and (c), assume the stated $C^{1}$ hypotheses. For every $k$ and every $(x, t) \in D \times R$ , [F1] and [F2] give $E_{k} (x, t) = Re ({\tilde{E}}_{k} (x) e^{- i ω t}) = (Re {\tilde{E}}_{k} (x)) \cos ω t + (Im {\tilde{E}}_{k} (x)) \sin ω t$ , and the same display with ${\tilde{B}}_{k}$ in place of ${\tilde{E}}_{k}$ gives $B_{k} (x, t)$ ; moreover, since $\tilde{E}$ and $\tilde{B}$ are $C^{1}$ , their six real coordinate maps and all first partials are continuous on $D$ and the complex partials are $\partial_{j} {\tilde{E}}_{k} = \partial_{j} (Re {\tilde{E}}_{k}) + i \partial_{j} (Im {\tilde{E}}_{k})$ , by [F4].

1.2givenF1F2

[Injectivity] Let $w \in C^{3}$ satisfy $Re (w e^{- i ω t}) = 0$ for every real $t$ . At $t = 0$ , [F1] gives $e^{0} = 1$ , so $Re w = 0$ ; at $t = π / (2 ω)$ one has $e^{- i π / 2} = - i$ by [F1], so $0 = Re (- i w) = Im w$ by [F2]; hence $w = Re w + i Im w = 0$ . If $w : D \to C^{3}$ satisfies $Re (w (x) e^{- i ω t}) = 0$ for all $(x, t)$ , applying the preceding conclusion at each fixed $x \in D$ gives $w (x) = 0$ for every $x$ , so $w \equiv 0$ ; no continuity of $w$ is needed for this conclusion.

2.1step 1.1F1F2F3F4

[Time derivative] Fix $x$ and $k$ . The map $t \mapsto {\tilde{E}}_{k} (x) e^{- i ω t}$ is the composition of the real-linear map $t \mapsto - i ω t$ (which is differentiable on $R$ with derivative $- i ω$ and real-linear, so totally differentiable with that derivative) with $\exp$ , which is real totally differentiable everywhere with derivative the multiplication $h \mapsto e^{z} h$ by [F3], followed by multiplication by the constant ${\tilde{E}}_{k} (x)$ ; the chain and product rules of [F3], [F4] give derivative $- i ω {\tilde{E}}_{k} (x) e^{- i ω t}$ . Taking real parts by [F2] and inserting the display of step 1.1, $\partial_{t} E_{k} (x, t) = Re (- i ω {\tilde{E}}_{k} (x) e^{- i ω t}) = ω (Im {\tilde{E}}_{k} (x)) \cos ω t - ω (Re {\tilde{E}}_{k} (x)) \sin ω t$ for every $(x, t)$ ; the same computation with $\tilde{B}$ gives $\partial_{t} B_{k} (x, t) = Re (- i ω {\tilde{B}}_{k} (x) e^{- i ω t})$ .

2.2step 1.1F2F4

[Spatial derivatives] Fix $(x, t)$ and indices $j, k$ . The map $h \mapsto E_{k} (x + h e_{j}, t)$ equals $h \mapsto Re ({\tilde{E}}_{k} (x + h e_{j}))$ times $\cos ω t$ plus $h \mapsto Im ({\tilde{E}}_{k} (x + h e_{j}))$ times $\sin ω t$ on a neighbourhood of $0$ small enough that $x + h e_{j} \in D$ , by step 1.1. By [F4] the coordinate-line maps $h \mapsto Re {\tilde{E}}_{k} (x + h e_{j})$ and $h \mapsto Im {\tilde{E}}_{k} (x + h e_{j})$ are differentiable at $h = 0$ with derivatives $\partial_{j} Re {\tilde{E}}_{k} (x)$ and $\partial_{j} Im {\tilde{E}}_{k} (x)$ , so the sum, scalar-multiple and product rules of [F4] give $\partial_{j} E_{k} (x, t) = (\partial_{j} Re {\tilde{E}}_{k} (x)) \cos ω t + (\partial_{j} Im {\tilde{E}}_{k} (x)) \sin ω t$ , which by [F2] and step 1.1 equals $Re ((\partial_{j} {\tilde{E}}_{k}) (x) e^{- i ω t})$ ; the same computation with $\tilde{B}$ gives $\partial_{j} B_{k} (x, t) = Re ((\partial_{j} {\tilde{B}}_{k}) (x) e^{- i ω t})$ .

3.1step 2.2F2F4F5

[Divergence and curl] By [F4] the divergence and curl act componentwise on real and imaginary parts of complex fields; hence, summing step 2.2 over $k$ and using $R$ -linearity of $Re$ [F2], $div E (x, t) = \sum_{k} \partial_{k} E_{k} = Re ((div \tilde{E}) (x) e^{- i ω t})$ , and for the third curl coordinate, in the naming of [F5], the alternating sum of the displayed partials and [F2] give $(curl E)_{z} (x, t) = \partial_{x} E_{y} - \partial_{y} E_{x} = Re ((\partial_{x} {\tilde{E}}_{y} - \partial_{y} {\tilde{E}}_{x}) (x) e^{- i ω t}) = Re ((curl \tilde{E})_{z} (x) e^{- i ω t})$ , the other two coordinates being identical by cyclic relabelling of the indices; the linearity of [F5] organizes these componentwise identities into the displayed formulas, and the same argument with $\tilde{B}$ gives those for $B$ .

4.1step 2.1step 2.2step 3.1F1F4F6

[Regularity] Each of the displayed first partials of $E$ and of $B$ in steps 2.1, 2.2 and 3.1 is a finite sum of products of the maps $(x, t) \mapsto g (x)$ for $g$ among $Re {\tilde{E}}_{k}, Im {\tilde{E}}_{k}, \partial_{j} Re {\tilde{E}}_{k}, \partial_{j} Im {\tilde{E}}_{k}$ (and the $\tilde{B}$ analogues), each continuous on $D \times R$ by [F6] since its one-variable factor is continuous on $D$ by [F4], together with the maps $(x, t) \mapsto \cos ω t$ and $(x, t) \mapsto \sin ω t$ , continuous on $D \times R$ by [F1], [F6]; the fields $E_{k}, B_{k}$ themselves are of the same form by step 1.1. By the sum-and-product clause of [F6], all these functions are continuous on $D \times R$ ; hence every first partial of $E$ and $B$ exists and is continuous there, so $E$ and $B$ are $C^{1}$ by [F4], with the partials displayed in steps 2.1, 2.2 and 3.1. This proves (a).

4.2step 2.1step 3.1givenalgebra

[(i) implies (ii)] Assume the phasor system of (c) holds on $D$ . Then for every $(x, t)$ : $div E = Re ((div \tilde{E}) e^{- i ω t}) = 0$ and $div B = 0$ by step 3.1 and $div \tilde{E} = div \tilde{B} = 0$ ; $curl E = Re ((curl \tilde{E}) e^{- i ω t}) = Re (i ω \tilde{B} e^{- i ω t}) = - \partial_{t} B$ by steps 3.1, 2.1 and $curl \tilde{E} = i ω \tilde{B}$ ; and $curl B = Re ((curl \tilde{B}) e^{- i ω t}) = Re (- i ω c^{- 2} \tilde{E} e^{- i ω t}) = c^{- 2} \partial_{t} E$ by steps 3.1, 2.1 and $curl \tilde{B} = - i ω c^{- 2} \tilde{E}$ . Hence the real system holds on $D \times R$ .

4.3step 1.2step 2.1step 3.1givenalgebra

[(ii) implies (i)] Assume the real system of (c) holds on $D \times R$ . The fields $u : = div \tilde{E}$ , $v : = div \tilde{B}$ , $w : = curl \tilde{E} - i ω \tilde{B}$ and $z : = curl \tilde{B} + i ω c^{- 2} \tilde{E}$ are continuous maps on $D$ , with $u, v$ scalar-valued and $w, z$ valued in $C^{3}$ by step 1.1 and [F4], [F5], and steps 2.1 and 3.1 give $Re (u (x) e^{- i ω t}) = div E = 0$ , $Re (v (x) e^{- i ω t}) = div B = 0$ , $Re (w (x) e^{- i ω t}) = curl E + \partial_{t} B = 0$ and $Re (z (x) e^{- i ω t}) = curl B - c^{- 2} \partial_{t} E = 0$ for all $(x, t)$ ; the two-phase calculation of step 1.2, also valid for a single complex scalar, then forces $u = v = w = z = 0$ on $D$ , which is exactly the phasor system.

5.1step 1.2step 2.1step 2.2step 3.1step 4.1step 4.2step 4.3given∎

[Conclusion] Steps 2.1, 2.2, 3.1 and 4.1 prove the calculus and $C^{1}$ clauses of (a); step 1.2 proves the injectivity clause (b); steps 4.2 and 4.3 prove both directions of the equivalence (c). All assertions are pointwise identities on the open domain $D \times R$ , no existence theory for partial differential equations, no boundary condition and no choice principle are used, and the argument rests only on the cited complex, real-analysis and topological continuity suppliers.

Remarks

The two evaluation points $t = 0$ and $t = π / (2 ω)$ are the whole content of injectivity; no integration or orthogonality argument is needed, and the field version of (b) does not use the continuity hypothesis, which is recorded only because the Statement carries it.
The equivalence (c) concerns the four source-free equations only. It says nothing about boundary conditions, energy, uniqueness of solutions or existence; those belong to the consuming items.
The proof uses no choice principle: all steps are pointwise computations with explicit finite data, and the continuity conclusion only composes explicitly given continuous maps.

Real parts of $e^{- i ω t}$ phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Real parts of e−iωte−iωt phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Real parts of $e^{- i ω t}$ phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system