math.harvard.edu/~ytzeng/worksheet/1121_sol.pdf
To use Stokes' Theorem, we need to first find the boundary C of S and figure out how it .... The plane z = x + 4 and the cylinder x2 + y2 = 4 intersect in a curve C.
Harvard University
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Pauls Online Notes : Calculus III - Stokes' Theorem
tutorial.math.lamar.edu/Classes/.../StokesTheorem.aspx
In this section we are going to take a look at a theorem that is a higher dimensional version of Green's Theorem. In Green's Theorem we related a line integral to ...
Lamar University
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[PDF]1 Statement of Stokes' theorem 2 Examples
www.math.uiuc.edu/.../Math241_168_Stokes...
Stokes' theorem claims that if we “cap off” the curve C by any surface S (with ..... Let S is the part of the cylinder of radius R around the z-axis, of height H,.
Department of Mathematics
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[PDF]The History of Stokes' Theorem
www.ingelec.uns.edu.ar/asnl/Materiales/Cap03Extras/Stokes-Katz.pdf
Gauss then denotes by d2 an element of the y -z plane and erects a cylinder .... The final theorem of our triad, Stokes' theorem, first appeared in print in 1854. ..... not again until Erich Kahler reintroduced it in his 1934 book Einfiihrung in die ...[PDF]The History of Stokes' Theorem
www.math.ups.edu/~martinj/courses/spring2005/.../stokeshistory_katz.pd...
Apr 9, 2005 - Theorems of Green, Gauss and Stokes ... element of the y -z plane and erects a cylinder above it, this cylinder intersecting the surface in ..... not again until Erich Kahler reintroduced it in his 1934 book Einfiihrung in die Theorie ...[PDF]Solved Problems - Applied Electromagnetics/6e by Ulaby ...
em.eecs.umich.edu/pdf/ulaby_solved_problems.pdf
by FT Ulaby - Cited by 663 - Related articles
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics ...... Verify Stokes's theorem for the vector field B = (r cosφ + ..... A cylinder-shaped carbon resistor is 8 cm in length and its circular cross ...Use Stoke's theorem to evaluate the integral? | Yahoo Answers
https://answers.yahoo.com/question/index?qid...
Apr 30, 2011 - Use Stoke's theorem to evaluate ∫∫S curlFdS where F(x,y,z) = -2yz i + ... z = x^2
Why are most of the fundamental laws in Physics second order degree differential equations?
If we look at the laws of Newton, Schroedinger, Einstein and others we can observe that they are all second order degree differential equations, ordinary or partial. Why such a coincidence? Is this an indicator that our projection of reality is just a linear projection or is it something deeper behind this universality of the 2nd degree?
http://en.wikipedia.org/wiki/Le_Sage%27s_theory_of_gravitation
Φ, ψ and φ are differentiable quaternionic functions (DQF's).
ALL continuous quaternionic functions that can be differentiated obey this coupling equation:
Φ = ∇ψ = m φ
∇={∂/∂τ, ∂/∂x, ∂/∂y, ∂/∂z}
∇ is the quaternionic nabla. m is the coupling factor.
||ψ||² =∫|ψ|² dV= 1
||φ||² =∫|φ|²dV= 1
||Φ||² =∫|Φ|² dV= ∫|∇ψ|² dV= m²
In quaternionic format the Dirac equation for the free electron runs ∇ψ = m ψ*
The Dirac equation for the free positron runs ∇*ψ* = m ψ
Φ = ∇ψ represents a differential continuity equation
Maxwell-Minkowski based approach versus Hamilton-Euclidean based approach
The difference between the Maxwell-Minkowski based approach and the Hamilton-Euclidean based approach will become clear when the difference between the coordinate time t and the proper time τ is investigated. This becomes difficult when space is curved, but for infinitesimal steps space can be considered flat. In that situation holds:
Coordinate time step vector = proper time step vector + spatial step vector
Or in Pythagoras format:
(∆t)² = (∆τ)² + (∆x)² + (∆y)² + (∆z)²
This influence is easily recognizable in the corresponding wave equations:
In Maxell-Minkowski format the wave equation uses coordinate time t. It runs as:
∂²ψ/∂t²−∂²ψ/∂x²−∂²ψ/∂y²−∂²ψ/∂z²=0
Papers on Huygens principle work with this formula or it uses the version with polar coordinates.
For 3D the general solution runs:
ψ =f(r−ct)/r, where c=±1; f is real
For 1D the general solution runs:
ψ =f(x−ct), where c=±1; f is real
For the Hamilton-Euclidean version, which uses proper time τ, we use the quaternionic nabla ∇:
∇={∂/∂τ, ∂/∂x, ∂/∂y, ∂/∂z}=∇₀+▽; ∇*=∇₀−▽
∇ψ = ∇₀ ψ₀ – (▽,Ψ) + ∇₀ Ψ + ▽ ψ₀ ± ▽ × Ψ
The ± sign reflects the choice between right handed and left handed quaternions.
In this way the Hamilton-Euclidean format of the wave equation runs:
∇*∇ψ = ∇₀∇₀ψ +(▽,▽)ψ =0
∂²ψ/∂τ²+∂²ψ/∂x²+∂²ψ/∂y²+∂²ψ/∂z²=0
Where ψ= ψ₀+Ψ
For the general solution holds: f= f₀+F
For the real part ψ₀ of ψ:
ψ₀ =f₀ (î r−c τ)/r, where c=±1 and î is an imaginary base vector in radial direction
For the imaginary part Ψ of ψ:
Ψ = F(î z−c τ), where c=±1 and î= î(z) is an imaginary base vector in the x,y plane
The orientation θ(z) of î(z) in the x,y plane determines the polarization of the 1D wave front.
Quaternionic coordinate data, which are based on proper time τ, fit as eigenvalues in Hilbert spaces. This is not the case for the spacetime coordinates that are based on coordinate time t. Hilbert spaces require that their eigenvalues are members of a division ring. Only three suitable division rings exist: real numbers, complex numbers and quaternions.
The formula:
(∆t)² = (∆τ)² + (∆x)² + (∆y)² + (∆z)²
indicates that the coordinate time step corresponds to the step of a full quaternion, which is a superposition of a proper time step and a spatial step.
An infinitesimal spacetime step ∆s is usually presented as an infinitesimal proper time step ∆τ.
(∆s)² = (∆t)² - (∆x)² - (∆y)² - (∆z)², with signature + - - - .
Above it is indicated that the coordinate time step ∆t corresponds to a quaternionic step. It mixes progression and 3D space. Proper time corresponds to pure progression.
Thus, if spacetime expands (proper time expands) than progression expands.
This does not say that space does not expand in the same way. (However, due to constant speed of information transfer c, it probably does expand in the same way)
In this way it might become clear that the choice for starting with Maxwell equations puts contemporary physics models in the direction of spacetime with Minkowski signature {(∆s)² = (∆t)² - (∆x)² - (∆y)² - (∆z)²}, while a choice for the quaternionic approach puts the model developer in the direction of a Euclidean space-progression model. {(∆t)² = (∆τ)² + (∆x)² + (∆y)² + (∆z)²}
This simple toy model has the advantage that could be reproduced at any scale from infinite small to infinite big in steps - at the upper scales at which the atoms (or galaxies) of the lower scale become "axions" (for the considered "upper scale").
"Antimatter" is just ordinary matter whose dynamics has opposite handedness. Dirac eq. shows the perfect symmetry but only we humans try to "spontaneously break it" :).
如果要改变真空中的光速，必须推翻电动力学的Maxwell方程组。这只有在“真空”成为“非线性介质”的时候才能做到。因为Maxwell方程组实际是“真空本底”上小的“起伏”（Fluctuation）之“线性描述”的最一般形式
物理学之《2012》——物理学基本规律的非线性？