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Difference between revisions of "Calc3.8"
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=The Maxwell Equations=
  
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=Generalized Integration=
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One the most important applications of the concepts you have learned are the equations which describe classical electricity and magnetism.  When we set <math>\mathbf{E} \ </math> as the [[electric field]],  <math>\mathbf{B} \ </math> as the [[magnetic field]], <math>\mathbf{D} \ </math> as the [[electric displacement field]], and <math>\mathbf{H} \ </math> as the [[magnetizing field]], then the equations describing these quantities are
  
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=The Line Integral=
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<math>\nabla \cdot \mathbf{D} = \rho</math>
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==Definition==
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We've integrated single variable functions along lines, and two variable functions over regions of the plane, and even three variable functions over regions of space.  But lines, and curves, can be contained in the plane, and surfaces can be contained in space.  Why have we been restricting ourselves to integrating functions over regions which have the same dimensions as their domains?  
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<math>\nabla \cdot \mathbf{B} = 0</math>  
  
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Well, because it is simpler, and useful to have under our belts before we expand our repetoire. But there is no reason we can't integrate a function over a region in its domain which is a lower dimension than the domain itself. 
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<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>  
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<math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>
  
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We'll start with the line integral.  Technically, it should be called the curve integral, but let's let that slide for now and get to the definition.  We're trying to integrate over a curve, say, ''C''.  At first, we'll let this curve lie in the ''xy'' plane - it's not difficult to generalize to three dimensions.  So we proceed as before: we divide this region into ''n'' pieces <math>C_k, 1\leq k \leq n</math>, and we pick a point <math>(x_k,y_k)\in C_k</math> for each piece.  Letting the length of <math>C_k</math> be <math>L_k</math>.  The definition follows as expected:
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We can integrate all these equations to get the '''integral forms''':
  
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<math>\int_C f(x,y)dL = \lim_{L_k\to 0, n\to\infty} \sum_{k=1}^n f(x_k,y_k)L_k </math>
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<math>\oint_S  \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V</math>
  
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==Computation==
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<math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>
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[[Image:Arc length.jpg|right|197px|thumb|As the length goes to zero, the Pythagorean theorem provides a more and more precise result for the length element ''dL''.]]
 
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As before, we're not going to be using the Riemann sum definition to compute line integrals.  Let <math>x=x(t),y=y(t)  </math> be a parametric description of <math>C </math>.  This allows us to compute the length element, ''dL'':
 
  
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<math>dL = \sqrt{dx^2+dy^2} </math>
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<math>\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l}  = -  \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A} </math>
  
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But we don't have any formulas for ''dx'' or ''dy''!  The closest we've got are formulas for <math>dx/dt, dy/dt</math>; we'll have to plug these in and see if we can patch things up:
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<math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
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\int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math>
  
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<math>= \sqrt{ \left({  \frac{dx}{dt} }\right)^2 + \left({  \frac{dy}{dt} }\right)^2 } \sqrt{dt^2} </math>
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The four laws are called, respectively, Coulombs/Gauss' Law, absence of magnetic monopoles, Faraday's/Induction Law, and Ampère's Law. Together, they are called the [[Maxwell equations]], after [[James Clerk Maxwell]].
  
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Now we can just plug in our formula for ''dL'' into the integral to get an easy-to-use formula for a line integral:
 
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<math>\int_C f(x,y)dL = \int_{t_0}^{t_1} f(x,y)\sqrt{ \left({  \frac{dx}{dt} }\right)^2 + \left({  \frac{dy}{dt} }\right)^2 } dt </math>
 
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=Green's Theorem in the Plane=
 
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=The Surface Integral=
 
  
 
{{MVCalc}}
 
{{MVCalc}}

Latest revision as of 15:01, July 14, 2016

The Maxwell Equations

One the most important applications of the concepts you have learned are the equations which describe classical electricity and magnetism. When we set   as the electric field,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} \ } as the magnetic field,   as the electric displacement field, and   as the magnetizing field, then the equations describing these quantities are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{D} = \rho}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{B} = 0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}}

We can integrate all these equations to get the integral forms:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}}

The four laws are called, respectively, Coulombs/Gauss' Law, absence of magnetic monopoles, Faraday's/Induction Law, and Ampère's Law. Together, they are called the Maxwell equations, after James Clerk Maxwell.


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