A point charge q appears at the origin at time 0, in an infinite vacuum. What's the electric field a billion miles away at time 0?  I suspect the real answer is 0. But it seems to me that Maxwell's first equation gradient(E) = (charge density) give us the answer q/(a billion miles) * (unit vector pointing away from the origin). Did I go wrong somewhere, or is this an example of Maxwell's equations not modelling nature perfectly?
In an electromagnetic theory class, my professor introduced the concept of "action at a distance in physics". He said that:     If two charges are at some very large distance, and if any one of the charge moves, then the force associated with the charges changes instantaneously.   But according to Einstein, no information can travel faster than the speed of light.   So photons (the information carriers in electromagnetic force) cannot instantaneously deliver information.   So that we associate a field with the two charges and if any charge moves, there is a deformation in that field and this deformation travels with the speed of light and conveys the information.   If the field deformation information cannot travel more than the speed of light, how does the force instantaneously change at very large distances?
Any data for average number of papers per year at different career stages? How do I handle null or optional DLL struct parameters What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory?  Some subjects off the top of my head that probably need covering:   Differential geometry, Manifolds, etc. Lie groups, Lie algebras and their representation theory. Algebraic topology. $$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$  Obviously, one of my algebraic manipulations is not valid. I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea? This works:   This does not (blank image):   Notice the 1.1.1 -> 1.3.0 and SRS->CRS are the only changes.  Is this a problem with the service or is my second URL incorrect?  I've observed the same issue with other WMS services:  Works:   Does not work (blank image):   So I'm assuming it's an issue with my 1.3.0 URLs. I want to automatically split all lines in a QGIS layer at their vertices (). I've done a Google search and found , which is quite close but the workflow is complicated. Because I'll have to train my co-workers (who are unfamiliar with GIS) to use this tool, I want something as simple as a plugin or something that doesn't require GRASS. Since it's been a long time from 2010 - the time of , I wonder if there is any better way to achieve the same result? (a) Solve the equation $yu_x+xu_y=0$  with the condition $u(0,y) = e^{-y^2}$. (b) In which region of the xy plane is the solution uniquely determined?     I did the first part but I don't understand how to do part b.  I read   But I don't understand how the argument is based. A more clear explanation is appreciated Topological/Geometrical justification for $\text{CFT}_2$ being special It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n&gt;2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are from a much wider class of maps, holomorphic/antiholomorphic maps. I was wondering to know if there is any topological or geometrical description for this.   To show what I mean, consider this example: in $\mathbb{R}^n$ for $n&gt;2$ interchanging particles can only change the wave function to itself or its minus. It is related to the fundamental group of $\mathbb{R}^n-x_0$ ($x_0$ is a point in $\mathbb{R}^n$ and $\pi_1(\mathbb{R}^n-\{x_0\})=e$ for $n&gt;2$) but this is not true for $n=2$.   I want to know whether exists any topological invariant or just any geometrical explanation that is related to the fact that I mentioned about conformal maps on $\mathbb{R}^n$. How to get battery status? How do I get the battery status on an iPhone? When can we find holomorphic bijections between annuli? Is there an online course in Mathematical Methods for Physics, Covering Matrices and Vector Analysis? I am taking a course in Mathematical Methods for physicsPhysics (Junior Level).   We are working from   > Mathematical Methods for physicists, George B.Arfken .    I just need online resources covering ( Matrices, Determinants, Vector Analysis, Tensors and differential forms and Vector spaces...etc) to study and understand from. Why is $\frac{d^n}{dx^n}(y(x))$ the notation for the $n$th derivative of $y(x)$, instead of $\frac{d^n}{d^nx}(y(x))$? I've always wondered why the numerator is $d^n$ while the denominator is $dx^n$ instead of $d^nx$ like the numerator.  I must be missing something very obvious or fundamental. Is this notation derived from the chain rule in some way? Application_End global.asax Can anybody tell me when Application_End is triggered in a lifecycle of an application? When all sessions are ended, will Application_End be triggered automatically? + Are there any other reasons why Application_End could be triggered? Antonym of "deepen" Since there is no word "shallowen", is there a good antonym of "deepen"? The meaning of "deepen" that I am referring to here is "to cause to become deeper." It doesn't necessarily refer to digging a hole. For example, a temporary deformation in a soft surface:     Putting the anvil on the mattress deepens the depression in it, but removing the anvil ______ it.   I would also be happy with an antonym of the intransitive version "to become deeper." The [ifs] tag is used for 2 separate things I think  might correspond to IBM's  according to this .   But currently out of the 27 question tagged with , most of it have to do with the 's "Internal Field Separator".  What should be done here?   create a new tag for "Internal Field Separator" e.g. , and keep the IBM's under ? create a new tag for "Integrated File System" e.g. , and keep the other one under ? just remove the  tag from the  questions? Make display of reputation in revisions list consistent with the user card display Is there always an equivalent metric which is not complete? What is the best algorithm for overriding GetHashCode? In .NET, the  is used in a lot of places throughout the .NET base class libraries. Implementing it properly is especially important to find items quickly in a collection or when determining equality.  Is there a standard algorithm or best practice on how to implement GetHashCode for my custom classes so I don't degrade performance? How much longer will my 10 year old frame last?