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Physics 3 - Maxwell
Note from the Author
Table of Contents
What is this book
Historical facts
New aspects
Fully erroneous
Incorrectly interpreted

Physics 3 - Chapter 1
Equations
Complex vectors form
The Most General form
The General Solution

Physics 3 - Chapter 2
Solutions
Initial conditions
Non-homogeneous equation
Solution for three-directions
The four laws

Physics 3 - Supplement
Fermat's proof
Beal's conjecture
Pythagorean triples
Inertial mass
Gravity constans big G
What does the Moon look at?

Physics 3 - Final notes
Final notes

Physics 4 - New book
Entry

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2.2.3. Solution for nonhomogeneous equation Nonhomogeneous one-direction equivalent of equation (2.3) is expressed as:
$\frac{\partial^{2} F (x, τ)}{\partial x^{2}} - \frac{\partial^{2} F (x, τ)}{\partial τ^{2}} = f (x, τ)$	(2.32)
Textbooks ([L2]–Marcinkowska, [L3]–Krzyzanski) contain solution of this equation. Applying notations used in this book, it is written as:
$F (x, τ) = g_{1} (x, τ) + g_{2} (x, τ)$ ,	(2.33)
where $g_{1} (x, τ)$ and $g_{2} (x, τ)$ represent solutions derived from initial conditions (2.16-2.28) and related forcing
$g_{1} (x, τ) = \frac{1}{2} \cdot [F (x - τ, 0) + F (x + τ, 0)] + \frac{1}{2} \cdot \int_{x - τ}^{x + τ} \frac{\partial F (x, 0)}{\partial τ} dx$	(2.34)
$g_{2} (x, τ) = - \frac{1}{2} \cdot \int_{0}^{τ} (\int_{x - (τ - q)}^{x + (τ - q)} f (r, p) dr) dq$	(2.35)
This is correct solution; however, it cannot be applied for solving Maxwell’s equations. This is due to assumed conditions (used in above textbooks) which were required to prove correctness of the solution, but at the same time imposing significant limitations affecting its application. The deciding factor determining its unsuitability comes from the assumption that forcing $f (x, τ)$ and then only from this point the solution is determined. Such assumptions are justified when performing analysis of already mentioned above string equation, but for Maxwell’s equations we should search for a solution with leading forcing function without limitations for analyzed time point. Textbook [L2] by Marcinkowska applies additional requirement for limited and enclosed space (x, τ) , which further limits application of this solution. It should be noted that for Maxwell’s equations, solution derived from initial conditions in some considerations may become irrelevant. This is because initial conditions in Maxwell’s equations represent a wave created by forcing function from another position in space, thus being a component of a solution itself. Due to limitations mentioned above, it was necessary to derive restriction-free solution. Presented solution is easy to understand since it is based on elementary knowledge regarding differential equations without need of additional knowledge about other theorems. After substitution of variables in equation (2.32) we obtain
$4 \cdot \frac{\partial^{2} Fo (u, v)}{\partial u \partial v} = f (x, τ) = f (\frac{u + v}{2}, \frac{u - v}{2})$	(2.36)
and we arrive at final formal solution
$Fo (u_{0}, v_{0}) = \frac{1}{4} \cdot \iint_{(u_{C}, v_{C})}^{(u_{0,} v_{0})} f (\frac{u + v}{2}, \frac{u - v}{2}) du dv + Fo (u_{C}, v_{C})$	(2.37)
However, deriving practical equations from this is not simple, so we will follow an easier way. The drawing below (Fig.2) clarifies meaning of individual variables and operations performed.

x = x₀

τ = τ₀

τ = τ_C

O(u₀,v₀)= O(x₀, τ₀)

A(u_A,v₀)

C(x₀,τ_c)

B(u₀,v_B)

v_C

v_B

u_C

u_A

Fig.2
Drawing legend: – $u, v, x, τ$ denote axis of coordinates, – $Fo (u_{0}, v_{0})$ represents value of the function in point $(u_{0}, v_{0})$ , i.e. $(x_{0}, τ_{0})$ marked on drawing as $O (x_{0}, τ_{0}) = O (u_{0}, v_{0})$ , – $Fo (u_{c} = x_{0} + τ_{c}, v_{c} = x_{0} - τ_{c})$ represents value in point $(x_{0}, τ_{c})$ marked on drawing as $C (x_{0}, τ_{c})$ , which we can consider as an initial value of function $F ()$ . – The wave moving in the direction of increasing values of $x$ is denoted by red line and respectively blue represents the wave moving towards decreasing values of $x$ . Integral $\frac{1}{4} \cdot \iint_{(u_{C}, v_{C})}^{(u_{0,} v_{0})} f (\frac{u + v}{2}, \frac{u - v}{2}) du dv$ represents change of a value of function $F (x, τ)$ while traveling from point $C (x_{0}, τ_{c})$ , to point $O (x_{0}, τ_{0})$ i.e. change of value of function $F (x, τ)$ in point $x_{0}$ from time = $τ_{c}$ to time = $τ_{0}$ , while $τ_{c} ⋜ τ_{0}$ . For traveling from point $C$ to point $O$ (as marked on Fig.2), we will apply the following relation:
$Fo (u_{0}, v_{0}) = F (x_{0,} τ_{0}) = F (x_{0,} τ_{C}) + \int_{τ_{C}}^{τ_{0}} \frac{\partial F (x_{0}, τ)}{\partial τ} d τ$	(2.38)
In further calculations, we will be frequently performing various substitutions of functions and variables. In order to facilitate reader with easier following of these calculations, basic relations are being repeated below:
$u = x + τ$ , $v = x - τ$ ,	(2.39)
$x = \frac{u + v}{2}$ , $τ = \frac{u - v}{2}$ ,	(2.40)
$u -v=2\cdotτ$ , $u +v=2\cdotx$ , $u -v=2\cdotτ$ , $u +v=2\cdotx$ ,	(2.41)
$F (x, τ) = Fo (u, v)$ ,	(2.42)
$F (x, τ) = F (v + τ, τ) = F (u - τ, τ) = F (\frac{u + v}{2}, \frac{u - v}{2})$ ,	(2.43)
$Fo (u, v) = Fo (x + τ, x - τ)$ ,	(2.44a)
$Fo (u, v) = Fo (u, 2 x - u) = Fo (2 x - v, v)$ ,	(2.44b)
$Fo (u, v) = Fo (u, u - 2 τ) = Fo (v + 2 τ, v)$ .	(2.44c)
Consequently; the following relations apply:
$\frac{\partial F (x, τ)}{\partial τ} = \frac{\partial Fo (u, v)}{\partial τ}$ ,	(2.45)
$\frac{\partial F (x, τ)}{\partial x} = \frac{\partial Fo (u, v)}{\partial x}$ ,	(2.46)
$\frac{\partial F (x, τ)}{\partial u} = \frac{\partial Fo (u, v)}{\partial u}$ ,	(2.47)
$\frac{\partial F (x, τ)}{\partial v} = \frac{\partial Fo (u, v)}{\partial v}$ ,	(2.48)
$\frac{\partial F (x, τ)}{\partial x} = \frac{\partial Fo (u, v)}{\partial u} + \frac{\partial Fo (u, v)}{\partial v}$ ,	(2.49)
$\frac{\partial F (x, τ)}{\partial τ} = \frac{\partial Fo (u, v)}{\partial τ} = \frac{\partial Fo (u, v)}{\partial u} - \frac{\partial Fo (u, v)}{\partial v}$ ,	(2.50)
$\frac{\partial Fo (u, v)}{\partial u} = \frac{1}{2} \frac{\partial Fo (u, v)}{\partial x} + \frac{1}{2} \frac{\partial Fo (u, v)}{\partial τ}$ ,	(2.51)
$\frac{\partial Fo (x, τ)}{\partial v} = \frac{1}{2} \frac{\partial Fo (u, v)}{\partial x} - \frac{1}{2} \frac{\partial Fo (u, v)}{\partial τ}$ ,	(2.52)
$\frac{\partial^{2} Fo (u, v)}{\partial u \partial v} = \frac{1}{4} f (x, τ) = \frac{1}{4} f (\frac{u + v}{2}, \frac{u - v}{2})$ ,	(2.53)
$\frac{\partial^{2} Fo (u, v)}{\partial u \partial v} = \frac{1}{4} f (v + τ, τ) = \frac{1}{4} f (u - τ, τ)$ ,	(2.54)
Formula (2.50) can be applied in w (2.38), thus we obtain
$Fo (u_{0}, v_{0}) - F (x_{0,} τ_{C}) = \int_{τ_{C}}^{τ_{0}} (\frac{\partial Fo (u, v_{0})}{\partial u} - \frac{\partial Fo (u_{0}, v)}{\partial v}) d τ$ .	(2.55)
The right-side components in (2.55) can be easily determined for fixed $τ$ and for selected point $x_{0}$ on $x$ axis, from following relations:
$\frac{\partial Fo (u, v_{0})}{\partial u} - \frac{\partial F (x_{0}, τ)}{\partial u} = \int_{v (x_{0}, τ)}^{v_{0}} \frac{\partial}{\partial v} (\frac{\partial F (v + τ, τ)}{\partial u}) dv$ ,	(2.56)
$\frac{\partial Fo (u_{0}, v)}{\partial v} - \frac{\partial F (x_{0}, τ)}{\partial v} = \int_{u (x_{0}, τ)}^{u_{0}} \frac{\partial}{\partial u} (\frac{\partial F (u - τ, τ)}{\partial v}) du$ ,	(2.57)
and applying (2.54) we have
$\frac{\partial Fo (u, v_{0})}{\partial u} - \frac{\partial F (x_{0}, τ)}{\partial u} = \frac{1}{4} \int_{v (x_{0}, τ)}^{v_{0}} f (v + τ, τ) dv = \frac{1}{2} \int_{x_{0}}^{x_{0} + (τ - τ_{0})} f (x, τ) dx$ ,	(2.58)
$\frac{\partial Fo (u_{0}, v)}{\partial v} - \frac{\partial F (x_{0}, τ)}{\partial v} = \frac{1}{4} \int_{u (x_{0}, τ)}^{u_{0}} f (u - τ, τ) du = \frac{1}{2} \int_{x_{0}}^{x_{0} - (τ - τ_{0})} f (x, τ) dx$ .	(2.59)
Based on (2.40)
$\frac{\partial x}{\partial u} = \frac{\partial x}{\partial v} = \frac{1}{2}$	(2.60)
therefore
$\frac{\partial F (x_{0}, τ)}{\partial u} = \frac{\partial F (x_{0}, τ)}{\partial x} \cdot \frac{\partial x}{\partial u} = \frac{1}{2} \frac{\partial F (x_{0}, τ)}{\partial x}$ ,	(2.61)
$\frac{\partial F (x_{0}, τ)}{\partial v} = \frac{\partial F (x_{0}, τ)}{\partial x} \cdot \frac{\partial x}{\partial v} = \frac{1}{2} \frac{\partial F (x_{0}, τ)}{\partial x}$	(2.62)
and consequently
$\frac{\partial Fo (u_{0}, v)}{\partial v} = \frac{1}{2} \int_{x_{0}}^{x_{0} - (τ - τ_{0})} f (x, τ) dx + \frac{1}{2} \frac{\partial F (x_{0}, τ)}{\partial x}$ ,	(2.63)
$\frac{\partial Fo (u, v_{0})}{\partial u} = \frac{1}{2} \int_{x_{0}}^{x_{0} + (τ - τ_{0})} f (x, τ) dx + \frac{1}{2} \frac{\partial F (x_{0}, τ)}{\partial x}$ .	(2.64)
Inserting these values into do (2.55), we finally obtain:
$Fo (u_{0}, v_{0}) = F (x_{0,} τ_{C}) - \frac{1}{2} \int_{τ_{C}}^{τ_{0}} (\int_{x_{0} + (τ - τ_{0})}^{x_{0} - (τ - τ_{0})} f (x, τ) dx) d τ$	(2.65)
This is equivalent of formula (2.35) and it was derived without any limitations for value of time $τ_{c}$ , which can be extended even to $-∞$ . Constant $F (x_{0}, τ)$ results from integral of function $f (x, τ_{c})$ for time periods before $τ_{c}$ . For deriving formula (2.35), it was assumed that for $τ < τ_{c}$ function $f (x, τ_{c}) = 0$ , thus integral of this function (our value of $F (x_{0,} τ_{c})$ also equals zero. For solving Maxwell’s equations, there is no need to consider solutions derived from initial conditions $F (x, τ_{c})$ and $\partial F (x, τ_{c}) / \partial τ$ , these can be considered for other than Maxwell’s equations while relating these conditions to sources $f (x, τ)$ . Variable $τ_{0}$ was introduced to demonstrate equivalency of formulas (2.35) and (2.65). Since this variable represents actual time, we can present concise and clear formulas (without losing generality), by selecting $τ_{0} = 0$ . In this case, formula (2.65) is expressed as follows:
$Fo (u_{0}, v_{0}) = F (x_{0,} τ_{C}) - \frac{1}{2} \int_{τ_{C}}^{0} (\int_{x_{0} + τ}^{x_{0} - τ} f (x, τ) dx) d τ$ .	(2.66)
By extending time $τ_{c}$ to negative infinity, we can eliminate the constant $F (x_{0,} τ_{c})$ , therefore:
$Fo (u_{0}, v_{0}) = - \frac{1}{2} \int_{- \infty}^{0} (\int_{x_{0} + τ}^{x_{0} - τ} f (x, τ) dx) d τ$ ,	(2.67)
$Fo (u_{0}, v_{0}) = \frac{1}{2} \int_{0}^{- \infty} (- \int_{x_{0}}^{x_{0} + τ} f (x, τ) dx + \int_{x_{0}}^{x_{0} - τ} f (x, τ) dx) d τ$ .	(2.68)
Integration constants (2.61) and (2.62) in formulas (2.63) and (2.64) are identical; they are also present in formula (2.65) but with opposite signs, so they eliminate each other. The values of $\partial Fo (u, v_{0}) / \partial u$ and $\partial Fo (u_{0,} v) / \partial v$ as shown on Fig.2 represent values of these derivatives in points where time axis $τ$ crosses lines corresponding to variables $u_{0}$ and $v_{0}$ . These variables represent respectively a wave traveling towards decreasing $x$ values and a wave traveling towards increasing $x$ values. As in homogenous equation, these waves can be separated and named $Fu (u_{0}, v)$ and $Fv (u, v_{0})$ . Therefore
$Fo (u_{0,} v_{0}) = Fu (u_{0}, v_{0}) + Fv (u_{0}, v_{0})$ ,	(2.69)
$Fv (u_{0}, v_{0}) = - \frac{1}{2} \int_{0}^{- \infty} (\int_{x_{0}}^{x_{0} + τ} f (x, τ) dx) d τ$ ,	(2.70)
$Fu (u_{0}, v_{0}) = \frac{1}{2} \int_{0}^{- \infty} (\int_{x_{0}}^{x_{0} - τ} f (x, τ) dx) d τ$ .	(2.71)
Integration in respect to $τ$ , in formula (2.70) is performed along straight line $v_{0}$ which means that $v = x - τ = v_{0} = x_{0} - τ_{0} = x_{0}$ , herefore $τ = x_{0} - x$ and $d τ = d x$ . In order to distinguish variable $x$ derived from substitution of $τ$ (based on $d τ = d x$ ) we will denote it with apostrophe $x'$ so equation (2.70) can be expressed as:
$Fv (u_{0}, v_{0}) = - \frac{1}{2} \int_{x' = x_{0}}^{- \infty} (\int_{x = x_{0}}^{x'} f (x, x' - x_{0}) dx) dx'$	(2.72)
Similarly, integration in (2.71) is performed along straight line $u_{0}$ u_o , which means that $u = x + τ = u_{0} = x_{0} + τ_{0} = x_{0}$ , herefore $τ = x_{0} - x$ and $d τ = - d x$ and formula (2.71) takes following form
$Fu (u_{0}, v_{0}) = - \frac{1}{2} \int_{x_{0}}^{\infty} (\int_{x_{0}}^{x'} f (x, x_{0} - x') dx) dx'$	(2.73)
The formulas become even more readable when applying additional auxiliary variable $r$ .
$Fv (u_{0}, v_{0}) = \frac{1}{2} \int_{r = 0}^{\infty} (\int_{x_{0}}^{x_{0} - r} f (x, - r) dx) dr$	(2.74)
$Fu (u_{0}, v_{0}) = - \frac{1}{2} \int_{r = 0}^{\infty} (\int_{x_{0}}^{x_{0} + r} f (x, - r) dx) dr$	(2.75)
Formulas (2.69), (2.74) and (2.75) represent solutions of general form (2.32) of one-direction nonhomogeneous Maxwell’s equation.