Mathematics Yoshio Sep 8th, 2023 at 8:00 PM 8 0
電磁學 Electromagnetics
ElectromagneticsCoulomb's Law
\(k_e=8.988\times 10^{9}\dfrac{N\cdot m^{2}}{C^{2}}\approx 8.99\times 10^{9}\dfrac{N\cdot m^{2}}{C^{2}}.\)
For convenience we write \(k\) in terms of the permittivity of free space (\(ϵ_0\)) as
\(k_e=\frac1{4\pi\varepsilon_0}\)
The value of \(ϵ_0\) is \(8.854\times10^{-12}N^{-1}m^{-2}C^2\)
Coulomb's Formula
\(F=\frac1{4\pi\varepsilon_0\varepsilon_r}\cdot\frac{q_1q_2}{d^2}=\frac1{4\pi\varepsilon_0K}\cdot\frac{q_1q_2}{d^2}\)
\(F=\frac1{4\pi\varepsilon}\cdot\frac{q_1q_2}{d^2}\)
In short, \(F\propto\frac{q_1\times q_2}{d^2}\)
Where,
\(ε\) is absolute permittivity,
\(K\) or \(ε_r\) is the relative permittivity or specific inductive capacity
\(ε_0\) is the permittivity of free space.
\(K\) or \(ε_r\) is also called a dielectric constant of the medium in which the two charges are placed.
Gauss's Law to Coulomb's Law
\(\nabla grad, \nabla\cdot div, \nabla\times curl\)
Mass, Momentum, Energy
Gradient operator \(\nabla\)
gradient, in mathematics, a differential operator applied to a three-dimensional vector-valued function to yield a vector whose three components are the partial derivatives of the function with respect to its three variables. The symbol for gradient is \(\nabla\).
gradient : \(\nabla f=\frac{\partial f}{\partial x}\overset\rightharpoonup i+\frac{\partial f}{\partial y}\overset\rightharpoonup j+\frac{\partial f}{\partial z}\overset\rightharpoonup k\)
divergence : \(\nabla\cdot f=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}\)
curl : \(\nabla\times f=\left(\partial f_z\partial y-\partial f_y\partial z\right)\overset\rightharpoonup i+\left(\partial f_x\partial z-\partial f_z\partial x\right)\overset\rightharpoonup j+\left(\partial f_y\partial x-\partial f_x\partial y\right)\overset\rightharpoonup k\)
\(\nabla\times\overset\rightharpoonup F=\begin{vmatrix}\widehat i&\widehat j&\widehat k\\\frac\partial{\partial x}&\frac\partial{\partial y}&\frac\partial{\partial z}\\F_x&F_y&F_z\end{vmatrix}\)
Laplacian : \(\triangle f=\nabla^2f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}+\frac{\partial^2f}{\partial z^2}\)
\(\nabla^{2}=\vec{\nabla}\cdot\vec{\nabla}\)
\(\nabla^2f(x)=\lim_{dx\rightarrow0}6\frac{{\left\langle f\right\rangle}_{around}-f(x_0)}{dx^2}\)
Closed surface
A closed surface contains a volume of space, enclosed from all directions; It consists of one connected, hollow piece that has no holes and doesn’t intersect itself.
Closed Surface Examples
Closed surfaces (L to R): Torus, Sphere, Polyhedron, Cube.
Examples of closed surfaces:
- Cube,
- Polyhedron,
- Sphere,
- Torus (an inflated inner tube).
- Gaussian surface: any closed surface through which an electric field passes.
Contour
The outline of a figure or body; the edge or line that defines or bounds a shape or object.
Flux
Electric flux
For a closed Gaussian surface, electric flux is given by:
\(\Phi_E=\iint_sE\cdot dS=\frac Q{\varepsilon_0}\)
Q is the total electric charge inside the surface S,
\({\varepsilon_0}\) is the electric constant (a universal constant, also called the "permittivity of free space")
\(\varepsilon_0\approx8.854187817\times10^{-12}\;F/m\)
Magnetic flux
For a closed Gaussian surface, magnetic flux is given by:
\(\Phi_B=\iint_sB\cdot dS=0\)
Faraday's law of induction
A change in the magnetic flux passing through a loop of conductive wire will cause an electromotive force, and therefore an electric current, in the loop. The relationship is given by Faraday's law:
\(\oint_c\overset\rightharpoonup E\cdot d\overset\rightharpoonup l=-\frac\partial{\partial t}\iint_s\overset\rightharpoonup B\cdot d\overset\rightharpoonup s\)
Ampère's circuital law
\(\oint_c\overset\rightharpoonup B\cdot d\overset\rightharpoonup l=\mu_0I+\mu_0\varepsilon_0\frac\partial{\partial t}\iint_s\overset\rightharpoonup E\cdot d\overset\rightharpoonup s\)
\(\oint_c\) is the closed line integral around the closed curve C
\(\iint_s\) denotes a 2-D surface integral over S enclosed by C
\(\mu_0\) is the magnetic constant (vacuum magnetic permeability)
Since the redefinition of SI units in 2019, \(\mu_0\) is an experimentally determined constant
\(\mu_0=1.25663706212(19)\;\times\;10^{-6}N/A^2\)
(1 henry per metre = 1 newton per square ampere = 1 tesla metre per ampere)
Ampère–Maxwell equation
\(\oint_c\overset\rightharpoonup B\cdot d\overset\rightharpoonup l={\iint_s(\mu_0\overset\rightharpoonup J+\mu_0\varepsilon_0\frac{\partial\overset\rightharpoonup E}{\partial t})}\cdot d\overset\rightharpoonup s\)
\(\overset\rightharpoonup J\) is current density. It includes magnetization current density as well as conduction and polarization current densities
The total current density \(J\) due to free and bound charges is then:
\(J=J_f+J_M+J_P\)
\(J_f\) is the "free" or "conduction" current density
\(J_M\) is magnetization current density
\(J_P\) is polarization current density
the wave equation for the electric field that can be derived from Maxwell's equations in the absence of charges and currents:
\(\nabla^2\overset\rightharpoonup E-\mu_0\varepsilon_0\frac{\partial^2\overset\rightharpoonup E}{\partial t^2}=0\)
same as magnetic field
\(\nabla^2\overset\rightharpoonup B-\mu_0\varepsilon_0\frac{\partial^2\overset\rightharpoonup B}{\partial t^2}=0\)
Electromagnetic waves speed
The speed of any electromagnetic waves in free space is the speed of light \(c=3\times10^8\frac ms\). Electromagnetic waves can have any wavelength \(\lambda\) or frequency \(f\) as long as \(\lambda f=c\).
\(\nu=\frac1{\sqrt{\varepsilon_0\mu_0}}=c\)
\(\varepsilon_0\) is the permittivity of free space, \(\mu_0\) is the permeability of free space, \(\nu\) is the speed of light.
James Clerk Maxwell discovered the speed is as light speed. Heinrich Hertz proved that electromagnetic waves travel at the speed of light.
Tangent vector, Normal vector, and Binormal vector
Stokes - Tangent Vector
\(\nabla\times\overset\rightharpoonup v=\frac1{\triangle A}\oint\overset\rightharpoonup v\cdot\overset\rightharpoonup t\operatorname dB\\\)
Gauss - Normal Vector
\(\nabla\cdot\overset\rightharpoonup v=\frac1{\triangle A}\oint\overset\rightharpoonup v\cdot\overset\rightharpoonup n\operatorname dB\\\)
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function \(f\) of several variables is the vector field (or vector-valued function) \(\nabla f\) whose value at a point \(p\) is the "direction and rate of fastest increase".
Sphere
Surface area: \(4\pi r^2\)
Volume: \(\frac43\pi r^3\)
Cartesian coordinates
A three dimensional Cartesian coordinate system, with origin \(O\) and axis lines \(X, Y\) and \(Z\), oriented as shown by the arrows.
Cylindrical coordinates
A cylindrical coordinate system with origin \(O\), polar axis \(A\), and longitudinal axis \(L\). The dot is the point with radial distance \(ρ = 4\), angular coordinate \(φ = 130°\), and height \(z = 4\).
Spherical coordinates
Cartesian coordinates \(( x , y , z )\) and Spherical coordinates \(( ρ , θ , φ )\) of a point are related as follows: \(x = ρ sin φ cos θ\) These equations are used to convert from \(y = ρ sin φ sin θ\) Spherical coordinates to \(z = ρ cos φ\) Cartesian coordinates.
Representation of the same vector in three different systems of coordinates: Cartesian (x, y, z), Cylindrical (r, φ, z), and Spherical (r, φ, θ)
Converting among Spherical, Cylindrical, and Cartesian Coordinates
Cartesian coordinates \((x,y,z)\) and Spherical coordinates \((ρ,θ,φ)\) of a point are related as follows:
\(\begin{array}{cccccc}x&=&\rho\hspace{0.2em}sin\hspace{0.2em}\varphi\hspace{0.2em}cos\hspace{0.2em}\theta&&&\text{Thesee quations are used to convert from}\\y&=&\rho\hspace{0.2em}sin\hspace{0.2em}\varphi\hspace{0.2em}sin\hspace{0.2em}\theta&&&\text{Spherical coordinates to Cartesian}\\z&=&\rho\hspace{0.2em}cos\hspace{0.2em}\varphi&&&\text{coordinates.}\\&and&&&&\\\rho^2&=&x^2+y^2+z^2&&&\text{These equations are used to convert from}\\tan\hspace{0.2em}\theta&=&\frac yx&&&\text{Cartesian coordinate sto Spherical}\\\varphi&=&arccos{(\frac z{\sqrt{x^2+y^2+z^2}}).}&&&\text{coordinates.}\end{array}\)
If a point has Cylindrical coordinates \((r,θ,z)\), then these equations define the relationship between Cylindrical and Spherical coordinates.
\( \begin{array}{cccccc}r&=&\rho\hspace{0.2em}sin\hspace{0.2em}\varphi&&&\text{These equations are used to convert from}\\\theta&=&\theta&&&\text{Spherical coordinates to Cylindrical}\\z&=&\rho\hspace{0.2em}cos\hspace{0.2em}\varphi&&&\text{coordinates.}\\&and&&&&\\\rho&=&\sqrt{r^2+z^2}&&&\text{These equations are used to convert from}\\\theta&=&\theta&&&\text{Cylindrical coordinates to Spherical}\\\varphi&=&arccos{(\frac z{\sqrt{r^2+z^2}})}&&&\text{coordinates.}\end{array}\)
these equations define the relationship between Cylindrical and Cartesian coordinates.
\( \begin{array}{cccccc}x&=&r\hspace{0.2em}cos\hspace{0.2em}\theta&&&\text{These equations are used to convert from}\\y&=&r\hspace{0.2em}\sin\hspace{0.2em}\theta&&&\text{Cylindrical coordinates to Cartesian}\\z&=&z&&&\text{coordinates.}\\&and&&&&\\r&=&\sqrt{x^2+y^2}&&&\text{These equations are used to convert from}\\\theta&=&\tan^{-1}(\frac yx)&&&\text{Cartesian coordinates to Cylindrical}\\z&=&z&&&\text{coordinates.}\end{array} \)
Divergence
Divergence is an operation on a vector field that tells us how the field behaves toward or away from a point. Locally, the divergence of a vector field \(\mathrm F\) in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) at a particular point P is a measure of the “outflowing-ness” of the vector field at P. If \(\mathrm F\) represents the velocity of a fluid, then the divergence of \(\mathrm F\) at P measures the net rate of change with respect to time of the amount of fluid flowing away from P (the tendency of the fluid to flow “out of” P). In particular, if the amount of fluid flowing into P is the same as the amount flowing out, then the divergence at P is zero.
Maxwell's Equations
Divergence itegral theorem vs Curl integral theorem
\(\int_V(\nabla\cdot\boldsymbol F)\operatorname d\boldsymbol v=\oint_A\boldsymbol F\cdot\operatorname d\boldsymbol a\)
\(\Phi=\oint_A\boldsymbol F\cdot\operatorname d\boldsymbol a\)
\(\Phi_e=\oint_A\boldsymbol E\cdot\operatorname d\boldsymbol a\)
\(\Phi_m=\oint\boldsymbol B\cdot\operatorname d\boldsymbol a\)
\(\nabla\cdot\boldsymbol F\left(x,\;y,\;x\right)\;>\;0\)
\(\Phi\;>\;0\) The source of flux
\(\Phi\;<\;0\) The sink of flux
Refer to Stokes' theorem
\(\int_A\left(\nabla\times\boldsymbol F\right)\cdot\operatorname d\boldsymbol a=\oint_L\boldsymbol F\cdot d\boldsymbol l\)
\(U_e=\oint_L\boldsymbol E\cdot d\boldsymbol l\) Electrical voltage
\(U_m=\oint_L\boldsymbol B\cdot d\boldsymbol l\) Magnetic voltage
\({\boldsymbol F}_m=q\boldsymbol v\times\boldsymbol B\)
\(U_e=-\frac\partial{\partial t}\oint_A\boldsymbol B\cdot\operatorname d\boldsymbol a=-\frac{\partial\Phi_m}{\partial t}\)
\(\oint_L\boldsymbol E\operatorname d\boldsymbol l=-\frac\partial{\partial t}\oint_A\boldsymbol B\cdot\operatorname d\boldsymbol a\)
\(\oint_L\boldsymbol E\operatorname d\boldsymbol l=\oint_A(\nabla\times\boldsymbol E)d\boldsymbol a\)
\(\oint_A(\nabla\times\boldsymbol E)d\boldsymbol a=-\frac\partial{\partial t}\oint_A\boldsymbol B\cdot\operatorname d\boldsymbol a\)
\(\oint_A(\nabla\times\boldsymbol E)d\boldsymbol a=-\oint_A\frac{\partial\boldsymbol B}{\partial t}\cdot\operatorname d\boldsymbol a\)
\(\nabla\times\boldsymbol E=-\frac{\partial\boldsymbol B}{\partial t}\)
\(\oint_L\boldsymbol B\cdot\operatorname d\boldsymbol l=\mu_0\boldsymbol I+\mu_0\varepsilon_0\frac\partial{\partial t}\oint_A\boldsymbol E\cdot\operatorname d\boldsymbol a\)
\(U_m=\mu_0\boldsymbol I+\mu_0\varepsilon_0\frac\partial{\partial t}\oint_A\boldsymbol E\cdot\operatorname d\boldsymbol a\)
\(\oint_L\boldsymbol B\cdot d\boldsymbol l=\oint_A\left(\nabla\times\boldsymbol B\right)\cdot d\boldsymbol a\)
\(\oint_A\left(\nabla\times\boldsymbol B\right)\cdot d\boldsymbol a=\mu_0\boldsymbol I+\mu_0\varepsilon_0\frac\partial{\partial t}\oint_A\boldsymbol E\cdot\operatorname d\boldsymbol a\)
\(\boldsymbol j=\frac{\boldsymbol I}{\boldsymbol A}\)
\(\int_A\boldsymbol j\cdot\operatorname d\boldsymbol a=\boldsymbol I\)
\(\oint_A\left(\nabla\times\boldsymbol B\right)\cdot d\boldsymbol a=\mu_0\int_A\boldsymbol j\cdot\operatorname d\boldsymbol a+\mu_0\varepsilon_0\frac\partial{\partial t}\oint_A\boldsymbol E\cdot\operatorname d\boldsymbol a\)
\(\oint_A\left(\nabla\times\boldsymbol B\right)\cdot d\boldsymbol a=\int_A(\mu_0\boldsymbol j+\mu_0\varepsilon_0\frac{\partial\boldsymbol E}{\partial t})\cdot\operatorname d\boldsymbol a\)
\(\nabla\times\boldsymbol B=\mu_0\boldsymbol j+\mu_0\varepsilon_0\frac{\partial\boldsymbol E}{\partial t}\)
A circulating magnetic field is produced by an electric current and by an electric field that changes with time.
\(j_f\) is the free current caused by moving charges
\(j_p=\frac{\partial p}{\partial t}\) is the polarization or bound current, where \(p\) is the electric polarization resulting from bound charges in dielectrics
\(j_m=\nabla\times m\) is the magnetization current, that is, the currents needed to generate the magnetization \(m\)
The 4 Maxwell Equations. Get the Deepest Intuition! Pretty good lesson! https://www.youtube.com/watch?v=hJD8ywGrXks
Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve \(C\) to a double integral over the plane region \(D\) bounded by \(C\). It is the two-dimensional special case of Stokes' theorem.
\(\oint_{\partial D}\overset\rightharpoonup F\cdot d\overset\rightharpoonup S=\iint_D(\nabla\times\overset\rightharpoonup{F)}\cdot d\overset\rightharpoonup A\)
The left side is a line integral and the right side is a surface integral.
Proof of Green’s Theorem
If \(P(x,y),\;Q(x,y)\) is a continuous and integrable function around the closed curve \(C\) and over the plane region \(R\).
\(\oint_CP(x,y)dx+Q(x,y)dy=∯_R(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA\)
\(R=\left\{\left(x,\;y\right):a\leq x\leq b\;and\;g_1(x)\leq y\leq g_2(x)\right\}\;\)
\(and\;R=\left\{\left(x,\;y\right):c\leq y\leq d\;and\;h_1(y)\leq x\leq h_2(y)\right\}\)
\(\oint\limits_CPdx+Qdy=\iint\limits_R\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA\)
We consider the following vector field:
\(\overrightarrow f(x,\;y)=P(x,\;y)\widehat i+Q(x,\;y)\widehat j\)
We can now calculate the line integral as follows.
\(d\overrightarrow r=d\overrightarrow x+d\overrightarrow y=dx\widehat i+dy\widehat j\)
\(\oint_C\overrightarrow f(x,\;y)d\overrightarrow r=\oint_C\left(P(x,\;y)\widehat i+Q(x,\;y)\widehat j\right)\left(dx\widehat i+dy\widehat j\right)\)
\(=\oint_C\left[P(x,\;y)dx+Q(x,\;y)dy\right]\)
Integration with respect to \(x\) component
\(\oint_CP(x,\;y)dx\)
\(=\int_a^bP\left[x,g_1(x)\right]dx+\int_b^aP\left[x,g_2(x)\right]dx\)
\(=\int_a^bP\left[x,g_1(x)\right]dx-\int_a^bP\left[x,g_2(x)\right]dx\)
\(=\int_a^b\left\{P\left[x,g_1(x)\right]-P\left[x,g_2(x)\right]\right\}dx\)
Using the fundamental theorem of calculus, we get the following.
\(=\int_a^b\int_{g_2(x)}^{g_1(x)}\frac{\partial P}{\partial y}dydx\)
\(=-\int_a^b\int_{g_1(x)}^{g_2(x)}\frac{\partial P}{\partial y}dydx\)
Based on the increasing direction of the \(x\) and \(y\) axes and Fubini’s theorem
\(=-\iint_R\frac{\partial P}{\partial y}dxdy\)
Integration with respect to \(y\) component
\(\oint_CQ(x,\;y)dy\)
\(=\int_d^cQ\left[h_1(y),\;y\right]dy+\int_c^dQ\left[h_2(y),\;y\right]dy\)
\(=-\int_c^dQ\left[h_1(y),\;y\right]dy+\int_c^dQ\left[h_2(y),\;y\right]dy\)
\(=\int_c^d\left\{Q\left[h_2(y),\;y\right]-Q\left[h_1(y),\;y\right]\right\}dy\)
\(=\int_c^d\int_{h_1(y)}^{h_2(y)}\frac{\partial Q}{\partial x}dxdy\)
\(=\iint_R\frac{\partial Q}{\partial x}dxdy\)
Combining the results obtained for \(x\) and \(y\) components
\(\oint_C\overrightarrow f(x,\;y)d\overrightarrow r=\iint_R\frac{\partial Q}{\partial x}dxdy-\iint_R\frac{\partial P}{\partial y}dxdy\)
\(=\iint_R\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy\)
This completes the proof of Green’s theorem.
\(\oint_C\overrightarrow F(x,\;y)d\overrightarrow r=\iint_R\nabla\times\overrightarrow F(x,\;y)dA\) Green's Theorem
Stokes’ theorem
The Stoke’s theorem states that “the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface. Green's theorem is the two-dimensional special case of Stokes' theorem.”
\(\oint_C\overrightarrow F\cdot d\overrightarrow r=\iint_S\nabla\times\overrightarrow F\cdot d\overrightarrow S\)
Let \(\Sigma\) be a smooth oriented surface in \(\mathbb{R}^3\) with boundary \(\partial\Sigma\equiv\Gamma\) If a vector field
\(\boldsymbol F(x,\;y,\;z)=\left[F_x(x,\;y,\;z),\;F_y(x,\;y,\;z),\;F_z(x,\;y,\;z)\right]\)
is defined and has continuous first order partial derivatives in a region containing \(\Sigma\), then
\(\iint_\Sigma(\boldsymbol\nabla\times\boldsymbol F)\cdot\operatorname d\boldsymbol\Sigma=\oint_{\partial\Sigma}\boldsymbol F\cdot\operatorname d\boldsymbol\Gamma\)
More explicitly, the equality says that
\(\iint_\Sigma((\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z})dydz+(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x})dzdx+(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y})dxdy)\)
\(=\oint_{\partial\Sigma}(F_xdx+F_ydy+F_zdz)\)
Proof:
Let \(F(x,y,z) = \langle P,Q,R \rangle\) be a vector field with component functions that have continuous partial derivatives
We take the standard parameterization of \(S \, : \, x = x, \, y = y, \, z = g(x,y)\). The tangent vectors are \(t_x = \langle 1,0,g_x \rangle\) and \(t_y = \langle 0,1,g_y \rangle\), and therefore \(The\;vector\;\perp(t_x,t_y)\;=\;t_x\times t_y=\langle-g_x,\,-g_y,\,1\rangle\)
\(\iint_Scurl\,\overset\rightharpoonup F\cdot d\overset\rightharpoonup S=\iint_D\lbrack-(R_y-Q_z)z_x-(P_z-R_x)z_y+(Q_x-P_y)\rbrack\,dA\)
where the partial derivatives are all evaluated at \((x,y,g(x,y))\) making the integrand depend on x and y only. Suppose \(\langle x (t), \, y(t) \rangle, \, a \leq t \leq b\) is a parameterization of \(C'\).
Then, a parameterization of \(C\) is \(\langle x (t), \, y(t), \, g(x(t), \, y(t))\rangle, \, a \leq t \leq b\).
Armed with these parameterizations, the Chain rule, and Green’s theorem, and keeping in mind that \(P\), \(Q\), and \(R\) are all functions of \(x\) and \(y\), we can evaluate line integral
\(\int_C\overset\rightharpoonup{\boldsymbol F}\cdot d\overset\rightharpoonup{\boldsymbol r}\)
\(=\int_a^b(Px'(t)+Qy'(t)+Rz'(t))dt\)
\(=\int_a^b\left[Px'(t)+Qy'(t)+R(\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt})\right]dt\)
\(=\int_a^b\left[(P+R\frac{\partial z}{\partial x})x'(t)+(Q+R\frac{\partial z}{\partial y})y'(t)\right]dt\)
\(=\int_{C'}(P+R\frac{\partial z}{\partial x})dx+(Q+R\frac{\partial z}{\partial y})dy\)
\(=\iint_D\left[\frac\partial{\partial x}(Q+R\frac{\partial z}{\partial y})-\frac\partial{\partial y}(P+R\frac{\partial z}{\partial x})\right]dA\)
\(=\iint_D(\frac{\partial Q}{\partial x}+\frac{\partial Q}{\partial z}\frac{\partial z}{\partial x}+\frac{\partial R}{\partial x}\frac{\partial z}{\partial y}+\frac{\partial R}{\partial z}\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}+R\frac{\partial^2z}{\partial x\partial y})-(\frac{\partial P}{\partial y}+\frac{\partial P}{\partial z}\frac{\partial z}{\partial y}+\frac{\partial R}{\partial y}\frac{\partial z}{\partial x}+\frac{\partial R}{\partial z}\frac{\partial z}{\partial y}\frac{\partial z}{\partial x}+R\frac{\partial^2z}{\partial y\partial x})dA\)
\(=\iint_D(\frac{\partial Q}{\partial x}+\frac{\partial Q}{\partial z}\frac{\partial z}{\partial x}+\frac{\partial R}{\partial x}\frac{\partial z}{\partial y})-(\frac{\partial P}{\partial y}+\frac{\partial P}{\partial z}\frac{\partial z}{\partial y}+\frac{\partial R}{\partial y}\frac{\partial z}{\partial x})dA\)
By Clairaut’s theorem,
\(\dfrac{\partial^2 z}{\partial x \partial y} = \dfrac{\partial^2 z}{\partial y \partial x} \nonumber\)
Therefore, four of the terms disappear from this double integral, and we are left with
\(\iint_D [- (R_y - Q_z)Z_x - (P_z - R_x) z_y + (Q_x - P_y)] \, dA,\)
which equals
\(\iint_S\nabla\times\overrightarrow F\cdot d\overrightarrow S\)
Divergence theorem (aka Gauss's divergence theorem)
The divergence theorem is named after the German mathematician Carl Friedrich Gauss (1777-1855).
The divergence theorem states that the surface integral of the normal component of a vector point function \(\overset\rightharpoonup F\) over a closed surface \(S\) is equal to the volume integral of the divergence of \(\overset\rightharpoonup F\) taken over the volume \(V\) enclosed by the surface \(S\).
Proof:
For the vector field \(\boldsymbol F(x,\;y,\;z)=F_1\boldsymbol i+F_2\boldsymbol j+F_3\boldsymbol k\), let \(V\) be the area of the space surrounded by the smooth closed surface \(S\), and let \(\widehat n\) be the normal vector. Then
\(∯_S(\overset\rightharpoonup F\cdot\widehat n)dS=∰_V(\nabla\cdot\overset\rightharpoonup F)dV\)
\(∯_S(F_1dydz+F_2dzdx+F_3dxdy)=∰_V(\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z})dxdydz\)
If you think of \(S\) and \(V\) as variables，then
\(\int_S\boldsymbol F\boldsymbol\cdot\boldsymbol n\;dS=\int_V\boldsymbol\nabla\boldsymbol\cdot\boldsymbol F\;dV\)
First, suppose \(V\) is sandwiched between two curved surfaces \(S_{1}\), \(S_{2}\) from the bottom and top.
Also, \(S_{1}\) is given by \(z=f_1(x,y),\;(x,y)\in\Omega\), \(S_{2}\) is given by \(z=f_2(x,y),\;(x,y)\in\Omega\). Then
\(∰_V\frac{\partial F_3}{\partial z}dV\)
\(=∰_V\frac{\partial F_3}{\partial z}dzdydx\)
\(=∯_\Omega\lbrack\int_{z=f_1(x,y)}^{f_2(x,y)}\frac{\partial F_3}{\partial z}dz\rbrack dydx\)
\(=∯_\Omega\lbrack F_3(x,y,f_2(x,y))-F_3(x,y,f_1(x,y))\rbrack dydx\)
For the surface \(S_{2}\)，normal unit vector for curvilinear coordinates \((x, y)\)
\(\frac{r_x\times r_y}{\left\|r_x\times r_y\right\|}\)
\(∯_\Omega\lbrack F_3(x,y,f_2(x,y))dxdy=∯_{S_2}F_3dxdy\)
\(-∯_\Omega\lbrack F_3(x,y,f_1(x,y))dxdy=∯_{S_1}F_3dxdy\)
Therefore，
\(∰_V\frac{\partial F_3}{\partial z}dV=∯_{S_2}F_3dxdy+∯_{S_1}F_3dxdy=∯_SF_3dxdy\)
Similarly, by projecting $S$ onto another plane, we can show
\(∰_V\frac{\partial F_1}{\partial z}dV=∯_SF_1dxdy\)
\(∰_V\frac{\partial F_2}{\partial z}dV=∯_SF_2dxdy\)
Adding each, we have
\(∰_{\mathbf V}\boldsymbol\nabla\boldsymbol\cdot\boldsymbol FdV=∯_S\boldsymbol F\boldsymbol\cdot\boldsymbol ndS\)
Proof 2:
Consider a cube with measures \(\mathrm x\in\lbrack\mathrm a,\;\mathrm b\rbrack,\;\mathrm y\in\lbrack\mathrm c,\;\mathrm d\rbrack,\;\mathrm z\in\lbrack\mathrm e,\;\mathrm f\rbrack\).
The total flux regarding a vector field \(\overset\rightharpoonup{\mathrm E}\) then is
\(\Phi=\Phi_x+\Phi_y+\Phi_z=\oint\overset\rightharpoonup Ed\overset\rightharpoonup A\)
Let us examine the x-component first:
\(\Phi_x=\int\overset\rightharpoonup E(a,\;y,\;z)d\overset\rightharpoonup A+\int\overset\rightharpoonup E(b,\;y,\;z)d\overset\rightharpoonup A\)
Since both surfaces’ normal vectors are opposing each other we can write
\(\Phi_x=\int\overset\rightharpoonup E(a,\;y,\;z)(-{\widehat e}_x)dA+\int\overset\rightharpoonup E(b,\;y,\;z){\widehat e}_xdA\)
\(\Phi_x=\int E_x(b,\;y,\;z)dA-\int E_x(a,\;y,\;z)dA\)
Now make use of the rule \(f(b)-f(a)=\int_a^b\frac{df(x)}{dx}dx\)
\(\Phi_x=\int\int_a^b\frac{\partial E_x}{\partial x}dxdA\;with\;dA=dydz\)
\(\Phi_x=\int_e^f\int_c^d\int_a^b\frac{\partial E_x}{\partial x}dxdydz\)
\(\Phi_x=\int\frac{\partial E_x}{\partial x}dV\)
\(\Phi=\Phi_x+\Phi_y+\Phi_z=\int\frac{\partial E_x}{\partial x}dV+\int\frac{\partial E_y}{\partial y}dV+\int\frac{\partial E_z}{\partial z}dV\)
\(\Phi=\int(\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{\partial E_z}{\partial z})dV\)
\(\Phi=\int{(\frac\partial{\partial x}\;\frac\partial{\partial y}\;\frac\partial{\partial z})}^T\cdot{(E_x\;E_y\;E_z)}^TdV\)
this yields the divergence theorem
\(\oint\overset\rightharpoonup Ed\overset\rightharpoonup A=\int\overset\rightharpoonup\nabla\overset\rightharpoonup EdV\)
Left: The Divergence Theorem
\(\overset\rightharpoonup F=F_1\boldsymbol i+F_2\boldsymbol j+F_3\boldsymbol k\)
\(∯_S(\overset\rightharpoonup F\cdot\widehat n)dS=∰_V(\nabla\cdot\overset\rightharpoonup F)dV\)
Right: Note on Stokes’ Theorem
\(∯_{S_1}(\nabla\times\overrightarrow F)\cdot{\widehat n}_1ds=∯_{S_2}(\nabla\times\overrightarrow F)\cdot{\widehat n}_2ds\)
For two different surfaces \(S_1\) and \(S_1\) with the same orientation and with the same boundary \(C\), they are the same value.
Conservative field
A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative.
\(\int_{C_1}\mathbf F⋅d\mathbf s=\int_{C_2}\mathbf F⋅d\mathbf s\)
\(\nabla\times\overset\rightharpoonup F=\begin{vmatrix}\widehat i&\widehat j&\widehat k\\\frac\partial{\partial x}&\frac\partial{\partial y}&\frac\partial{\partial z}\\F_x&F_y&F_z\end{vmatrix}\)
\(=(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z})\widehat i-(\frac{\partial F_z}{\partial x}-\frac{\partial F_x}{\partial z})\widehat j+(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y})\widehat k\)
if \(\overset\rightharpoonup F\) is conservative, then its curl must be zero, as \(\nabla\times\overset\rightharpoonup F=0\)
Divergence of curl is zero
\(\overset\rightharpoonup\nabla\cdot(\overset\rightharpoonup\nabla\times\overset\rightharpoonup F)=0\)
Curl of a gradient is the zero vector
\(\overset\rightharpoonup\nabla\times(\overset\rightharpoonup\nabla f)=0\)
The slope of a tangent line at a point is its derivative at that point.
Continuous Charge Distribution - Charge Density
Volume Charge density \(\rho\;=\lim_{\triangle V\rightarrow0}\triangle q/\triangle V\) with units \(\frac C{m^3}\)
Surface Charge density \(\rho_s\;=\lim_{\triangle A\rightarrow0}\triangle q/\triangle A\) with units \(\frac C{m^2}\)
Linear Charge density \(\rho_l\;=\lim_{\triangle l\rightarrow0}\triangle q/\triangle l\) with units \(\frac C{m}\)
For the volume \(dV:\;dq=\rho dV\)
For the surface \(dA:\;dq=\rho_sdA\)
For the line segment \(dl:\;dq=\rho_ldl\)
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Quasistatic Laws
Maxwell's Equations model for the electromagnetic fields
Equation law | Flux equation |
Faraday's law - net magnetic flux | \(\Phi_{B,S}=\oint_S\boldsymbol B\cdot\operatorname d\boldsymbol A\) |
Ampere's law - net electric flux | \(\Phi_{D,S}=\oint_S\boldsymbol D\cdot\operatorname d\boldsymbol A\) |
Maxwell's Equations in general
From the viewpoint of theoretical physics, the equations can be expressed in a form which is always valid, in vacuum or in material.
differential form | integral form | |
Gauss's law for electric field | \(∇·\vec{E} = \frac{ρ}{ε_0}\) | \(∯_{Surface} \vec{E} · d\vec{a} = ∰_{Volume} \frac{q}{ε_0} d\vec{v}\) |
Gauss's law for magnetic field | \(∇·\vec{B} = 0\) | \(∯_{Surface} \vec{B}·d\vec{a} = 0\) |
Faraday's law of induction | \(∇×\vec{E} = -\frac{∂\vec{B}}{∂t}\) | \(\oint_{Contour}\vec E\cdot d\vec l=-\frac\partial{\partial t}∯_{Surface}\vec B\cdot d\vec a\) |
Ampère-Maxwell's circuital law | \(∇×\vec{B} = μ_0 · \vec{J} + μ_0 · ε_0 · \frac{∂\vec{E}}{∂t}\) | \(\oint_{Contour}\vec B\cdot d\vec l=\mu_0\cdot I+\mu_0\cdot\varepsilon_0\cdot\frac\partial{\partial t}∯_{Surface}\vec E\cdot d\vec a\) |
where: \(ρ\) - electric charge density (\(C/m^3\)), \(ε_0\) - electric permittivity of vacuum (\(F/m\)), \(q\) - electric charge (\(C\)), \(l\) - increment of path for integral (\(m\)), \(a\) - increment of surface for integral (\(m^2\)), \(μ_0\) - magnetic permeability of vacuum (\(H/m\)), \(J\) - electric current density (\(A/m^2\)), \(I\) - electric current (\(A\)), \(S\) - closed surface (region of integral), \(C\) - closed curve (path of integral)
Maxwell's Equations in matter
In matter, there are localised magnetic moments which respond to the magnetic field penetrating the matter. It is possible to express the response of the matter as a vector field which is averaged (smoothed out) over the whole volume of the material, so that the vector field is expressed in effect as a macroscopic quantity, rather than microscopic variation (which can very wildly).
differential form | integral form | |
Gauss's law for electric field | \(∇·\vec{D} = ρ_v\) | \(∯_{Surface} \vec{D} · d\vec{a} = Q_{Enc}\) |
Gauss's law for magnetic field | \(∇·\vec{B} = 0\) | \(∯_{Surface} \vec{B}·d\vec{a} = 0\) |
Faraday's law of induction | \(∇×\vec{E} = -\frac{∂\vec{B}}{∂t}\) | \(\oint_{Contour}\vec E\cdot d\vec l=-\frac\partial{\partial t}∯_{Surface}\vec B\cdot d\vec a\) |
Ampère-Maxwell's circuital law | \(∇×\vec{H} = \vec{J} + \frac{∂\vec{D}}{∂t}\) | \(\oint_{Contour}\vec H\cdot d\vec l=I_{Enc}+\frac\partial{\partial t}∯_{Surface}\vec D\cdot d\vec a\) |
where: \(ρ\) - electric charge density (\(C/m^3\)), \(a\) - increment of surface for integral (\(m^2\)), \(Q\) - electric charge (\(C\)), \(t\) - time (\(s\)), \(l\) - increment of path for integral (\(m\)), \(J\) - electric current density (\(A/m^2\)), \(I\) - electric current (\(A\)), \(S\) - closed surface (region of integral), \(C\) - closed curve (path of integral)
Where:
D = Electric flux density = ε0E
E = Electric field in Volts/meter
B = Magnetic flux density = µ0H
H = Magnetic field in Amps/meter
ε0 = Free space permittivity= 8.85 x 10-12
µ0 = Free space permeability= 4π x 10-7
What is the difference between the differential and integral form of Maxwell's equations?
They're just different ways of expressing the same relationships. It's the difference between, e.g. \(a^2\;+\;b^2\;=\;c^2\) and \(c=\pm\sqrt{a^2+b^2}\) or \(y=e^x\) and \(x=\ln\left(y\right)\). Integrals and derivatives are inverses, similar to how squaring and square roots and exponentials and logs are inverses. To take an even simpler example of what I mean, saying that \(2\;+\;5\;=\;7\) is equivalent to saying that \(7\;-\;5\;=\;2\). We've expressed the same relationship is two ways, first using addition and then subtraction.
Since you asked specifically about Maxwell's equations, let's look at one of them. I'm most familiar with Gauss’ law for electricity, so I'll use that. It can be expressed in integral form as \(\oint\overset\rightharpoonup E\cdot\operatorname d\overset\rightharpoonup A=\oint\nabla\cdot\operatorname d\overset\rightharpoonup V=\frac q{\varepsilon_0}=\int\rho\operatorname dV\) In English, this is saying that the sum of the electric field over a closed surface, \(A\), is equivalent to the sum of the divergence of the electric field over a closed volume, \(V\), both of which are proportional to the amount of charge, \(q\), enclosed by the surface or present in that volume, which is equivalent to the integral of the charge density, \(ρ\), w.r.t. volume. Wow, that sentence is a real mouthful! Now you see why we write it as an equation instead.
The differential form is \(\nabla\overset\rightharpoonup E=\frac\rho{\varepsilon_0}\). Notice that all that changed is that is that we dropped the integrals, or, more formally, we took the divergence of both sides of the equation.
The differential forms:
- Gauss’ law for electricity: \(\nabla\cdot\overset\rightharpoonup E=\frac\rho{\varepsilon_0}\) , where \(E\) is the electric field generated by a charge density of \(\rho\), \(\varepsilon_0\) is the permittivity of free space in vacuum, and \(\nabla\) is the vector differential operator.
- Gauss’ law for magnetism: \(\nabla\cdot\overset\rightharpoonup B=0\), where \(B\) represents the magnetic field.
- Maxwell-Faraday equation/Faraday’s law of induction: \(\nabla\times\overset\rightharpoonup E=-\frac{\operatorname d\overset\rightharpoonup B}{\operatorname dt}\) , where \(E\) is a time/space varying electric field and \(\frac{\operatorname d\overset\rightharpoonup B}{\operatorname dt}\) is the time derivative of a time-varying magnetic field.
- Ampere’s circuital law: \(\nabla\times\overset\rightharpoonup B=\mu_0\left(J+\varepsilon_0\frac{\operatorname d\overset\rightharpoonup E}{\operatorname dt}\right)\), where \(\overset\rightharpoonup B\) is the magnetic field, \(J\) is the displacement current, \(\mu_0\) is the permeability of free space in vacuum, and \(\frac{\operatorname d\overset\rightharpoonup E}{\operatorname dt}\) is the time derivative of the electric field.
The integral forms:
- Gauss’ law for electricity: \(\oint\overset\rightharpoonup E\cdot\operatorname d\overset\rightharpoonup A=\frac q{\varepsilon_0}\), where \(\oint\overset\rightharpoonup E\cdot\operatorname d\overset\rightharpoonup A\) is the surface integral of the electric field over a closed charged surface and \(q\) is the total charge contained in that closed surface.
- Gauss’ law for magnetism: \(\oint\overset\rightharpoonup B\cdot\operatorname d\overset\rightharpoonup A=0\) , where \(\oint\overset\rightharpoonup B\cdot\operatorname d\overset\rightharpoonup A\) gives the net magnetic flux of any closed surface.
- Maxwell-Faraday equation/Faraday’s law of induction: \(\oint\overset\rightharpoonup E\cdot\operatorname d\overset\rightharpoonup s=-\frac{\operatorname d\Phi B}{\operatorname dt}\) , where \(\oint\overset\rightharpoonup E\cdot\operatorname d\overset\rightharpoonup s\) is the line integral of the electric field around a closed loop and \(\frac{\operatorname d\Phi B}{\operatorname dt}\) is the rate of change of the magnetic flux through the area enclosed by that loop.
- Ampere’s circuital law: \(\oint\overset\rightharpoonup B\cdot\operatorname d\overset\rightharpoonup s=\mu_0i+\frac1{c^2}\frac\partial{\partial t}\int\overset\rightharpoonup E\cdot d\overset\rightharpoonup A\) , where \(\oint\overset\rightharpoonup B\cdot\operatorname d\overset\rightharpoonup s\) is the line integral of the magnetic field around a closed loop and the terms in the R.H.S relate to the electric current flowing through the loop.
Curl
Curl measures the extent of rotation of the field about a point. Suppose that \(\overset\rightharpoonup F\) represents the velocity field of a fluid. Then, the curl of \(\overset\rightharpoonup F\) at point P is a vector that measures the tendency of particles near P to rotate about the axis that points in the direction of this vector. The magnitude of the curl vector at P measures how quickly the particles rotate around this axis. In other words, the curl at a point is a measure of the vector field’s “spin” at that point.
Area Vector
Magnetic field:
A vector field produced by current, moving charges, or magnetic materials is known as magnetic field. The magnetic field is always bipolar, consisting of two poles, north and south. Magnetic field
lines move from north to south. Symbol for magnetic field:
James Clerk Maxwell, a Scottish scientist, introduced the idea of recognizing a few physical quantities as vector fields and gave their notations. In 1873, he wrote a book named “A treatise on
Electricity and Magnetism”. In this book, he randomly assigned alphabets to vectors. He assigned A for vector potential, B for the magnetic field, E as electric field, and so on. Hence, the symbol for
the magnetic field is ‘B’.
Maxwell's equations integral form in the avbsence of magnetic or polarizable media:
\(I.\) Gauss' law for electricity \(\int_S\overset\rightharpoonup E\cdot\operatorname d\overset\rightharpoonup A=\frac q{\varepsilon_0}\)
\(II.\) Guass' law for magnetism \(\oint\overset\rightharpoonup B\cdot d\overset\rightharpoonup A=0\)
\(III.\) Faraday's law of induction \(\oint\overset\rightharpoonup E\cdot d\overset\rightharpoonup S=-\frac{\partial\overset\rightharpoonup B}{\partial t}\)
\(IV.\) Ampere's law \(\oint\overset\rightharpoonup B\cdot d\overset\rightharpoonup S=\mu_0i+\frac1{c^2}\int\frac{\partial\overset\rightharpoonup E}{\partial t}\cdot d\overset\rightharpoonup A\)
Directional Derivatives
Cauchy’s Integral Theorem (Cauchy–Goursat theorem)
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if \(f(z)\) is holomorphic in a simply connected domain Ω, then for any simply closed contour C in Ω, that contour integral is zero.
Cauchy’s Integral Formula
https://www.youtube.com/watch?v=iTZKKh9_CT8
https://www.youtube.com/watch?v=OvvyzITwiX8
https://www.youtube.com/watch?v=y4mM5acmQqw
https://www.youtube.com/watch?v=6NLxnnNk9FE
https://www.youtube.com/watch?v=b5VJVa5q3Oc
https://www.youtube.com/watch?v=pdy-0oLEIA4
https://www.youtube.com/watch?v=cwLzkbOC0gg
https://www.youtube.com/watch?v=b5VJVa5q3Oc
https://www.youtube.com/@MuPrimeMath/videos
https://www.youtube.com/watch?v=k2Hza-ku9Lw
The rotation problem and Hamilton's discovery of quaternions
https://www.youtube.com/watch?v=uRKZnFAR7yw