Is this the correct intuition behind the covariant derivative? If not can you please show me how to covariant derivative is related to the directional derivative?  The covariant derivative can be thought of as a generalisation of the directional derivative to non Euclidean spaces. The directional derivative is a mapping which takes as input some tangent vector at a point $p\in \mathbb{R}^n$ and a scalar field $f(x^1, x^2,....,x^n)$ and returns a real number. Since we are working in $\mathbb{R}^n$ we can acknowledge that there exists a map $F: \mathbb{R}^n \to \mathbb{R}^n$ which is to be considered a vector field. Due to the geometry of $\mathbb{R}^n$ we can think of points as vectors and vectors as points and hence the directional derivative acting on scalar fields can naturally be thought of as acting on vector fields. However for a general manifold M, the notion of a derivative acting on vector fields does not arise naturally from that of a derivative which acts on functions, the reason being that we cannot assign to each point $p\in M$ a 'tangent vector in $M$' due to the fact that a vector field in one coordinate chart will not be consistent with a vector field in some other chart. In general the basis of tangent space at $p\in M$ is different to that of $q\in M$, a connection provides a way of relating the two tangent spaces in such a way that permits vector fields to be differentiated in a way that is consistent with notion of the directional derivative on $\mathbb{R}^n$.
I am curious how people picture geometrically how the covariant derivative acts with regards to vector fields. I understand that it is merely an object which has been defined for some use and obeys certain conditions but how would you picture such a thing geometrically? In other words, when someone asks you what the covariant derivative is, how do you explain it in absence of a formal definition?
Extracting numbers from vectors of strings Let $X,Y,Z$ Banach spaces and $A:X\rightarrow Y$ and $B:Y\rightarrow Z$ linear maps with $B$ bounded and injective and $BA$ bounded. Prove that $A$ is bounded as well.   If I knew that $B(Y)$ is closed I'd have a bounded linear map $B^{-1}:B(Y)\rightarrow Y$ by the bounded inverse theorem. Therefore $A=B^{-1}BA$ is bounded. How to prove the claim if $B(Y)$ is not closed. Why won't attribute modifiers add up on my armor? Is there any relation between the chromatic number of a graph $G$ and its complement $G'$ that are always true?  I saw these ones: $\chi(G)\chi(G')\geq n$ and $\chi(G)+\chi(G')\geq 2n$,  but I'm not pretty sure about them. Pandemic, Global Epidemic, Epidemic, Endemic I really want to hear from the experts the definitions of all 4 terms especially whether global epidemic is equivalent to pandemic. Or should this question be directed to the epidemiologist?   Pandemic Global Epidemic Epidemic Endemic   I have Googled the terms and I understand the definition of 1), 3) and 4). In short the definition is based on geographical location and scale of outbreak, Pandemic is an outbreak of disease on a global scale (e.g. HIV). Epidemic is an outbreak of disease on a localised area. Endemic is a disease in a localised area. However, a definition for Global Epidemic is not available online.  Recently, WHO mentioned that "“It is important to put this on the table: this virus(Covid19) may become just another endemic virus in our communities, and this virus may never go away,” WHO emergencies expert Mike Ryan told an online briefing." Isn't covid19 considered a pandemic since it is on a global scale and is spreading rapidly?  Another term that I come across often is Global Epidemic. For example in the WHO website, HIV is listed as Global Epidemic. Does that mean Pandemic and Global Epidemic are the same? Controlling colors of points in ListPlot I have video tutorials that I want to show in my presentation with beamer class. I don't want the video files get embedded to the PDF because they will make the PDF size very very huge.  If you know how the videos can be played on the HTML with a reference rather than embedding the video file to the HTML file, I want something like this to be applied to my PDF presentation. I just want to reference the video URL from my PDF presentation.   Shortly speaking, can we use media9 to show a video on a PDF but the video is not embedded to the PDF?  Uploading those videos to a web server might be difficult for me because internet connection is very slow and sometimes not available in the presentation room. :-) Rough Smart objects problem in Photoshop I am using Adobe CC(2015) latest version and the behaviour of the smart objects isn't normal. When I paste a vector from Illustrator to Photoshop, the vector looks low quality or rough.   ORIGINAL  IN PHOTOSHOP   However, if I click over the smart object layer(in Photoshop)and this is opened in Illustrator, the vectors looks as the original one.   VECTOR OPENED IN ILLUSTRATOR FROM PHOTOSHOP SMART OBJECT   How can I solve this? Position of "only" in “We 𝒐𝒏𝒍𝒚 have two possibilities” Is there any difference in meaning between the three following sentences?   We only have two possibilities. We have only two possibilities. We have two possibilities only.   I have already read  and I know that the position of only and the emphasis (written or oral) may change the meaning of some sentences, but I am not sure if it does here.  Note: It is clear that “only we have two possibilities” has a different meaning. My question is restricted to the sentences I’ve listed above. Stack Overflow in ES, PT, RU, JP have mixed-language descriptions terminator: permission denied for sudo Why does Ubuntu only support versions for a limited time (and what does it mean)? Help with Axis Placement when extruding I am just learning Blender. I am going through the .  The first part was good. I was able to follow it. Coming from SketchUp, it is taking some adjustment. But I am getting it.  I am just starting the second video. At the beginning, the instructor selects the top of the mug to extrude it. When he presses the E key and Enter, the red, green blue axis moves to the center of those vertices. That does not happen when I press the E key and Enter.   This is only 1:30 into the video.  How can I get the axis in the middle of this circle of vertices?  The next step is to scale. I need the reference point in the middle of this circle of vertices. I cannot left-click accurately enough to position the axis/reference point.  I am using 2.69. The video uses 2.64. How do I determine how many spell slots I have if I multiclass into more than one spellcasting class? For example, I have a character that is a level 3 Wizard, a level 5 Eldritch Knight Fighter, a level 4 Arcane Trickster Rogue, a level 4 Paladin, and a 4 Warlock. How many spell slots do I have and how do I determine this? Recovering lost text which was copy pasted from a website to MS Word Determine the Set of a Sum of Numbers I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.  $$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$  I want to check whether that would be: 0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.  Meaning that it would include 0, 3, 5, 8. Then from 9 and on, every Natural Number. But how would I explain it as a set? or prove that these are the numbers in the set? maximum of two uniform distributions I have a question. Let's suppose that the two random variables $X1$ and $X2$ follow two Uniform distributions that are independent but have different parameters:  $X1 \sim Uniform(l1, u1)$  $X2 \sim Uniform(l2,u2)$  If we define X3 as the maximum of X1, and X2, i.e., $X3 = max(X1, X2)$, what kind of distribution would it be? It is certainly not a uniform and I can calculate the cumulative probability, i.e., $p(X3 &lt;= a)$, because $p(X3 &lt;= a) = p(X1 &lt;= a) p(X2 &lt;= a)$. But, I wonder if there exists any density function that can model X3. Shining a laser onto a mirror Theoretically, if I shined a laser at a mirror at an angle of 0 degrees so that the light was perfectly reflected back to the light source, then I should not be able to see the light because it is not reflected to my eyes.  | |  me | |============ laser | | |   However this is evidently not the case, as I am able to see the laser point on the mirror. Why is this the case? My initial assumption is that it is due to diffuse reflection on the imperfect mirror's surface, but I am not sure. "Reality" of length contraction in SR WPF Binding to Grid Column Width I'm attempting to bind a DependancyProperty in one of my usercontrols to the Width property of a Column in a Grid.   I have code similar to this:  &lt;Grid x:Name="MyGridName"&gt;     &lt;Grid.ColumnDefinitions&gt;         &lt;ColumnDefinition x:Name="TitleSection" Width="100" /&gt;         &lt;ColumnDefinition Width="*" /&gt;     &lt;/Grid.ColumnDefinitions&gt;      &lt;Grid.RowDefinitions&gt;...&lt;/Grid.RowDefinitions&gt;      &lt;GridSplitter x:Name="MyGridSplitter" Grid.Row="0" Grid.Column="0" ... /&gt; &lt;/Grid&gt;   In a separate Usercontrol I have the followingDependancyProperty defined.  public static readonly DependencyProperty TitleWidthProperty = DependencyProperty.Register("TitleWidth", typeof(int), typeof(MyUserControl));  public int TitleWidth {     get { return (int)base.GetValue(TitleWidthProperty); }     set { base.SetValue(TitleWidthProperty, value); } }   I am creating instances of the Usercontrol in code, hence I have a binding statement similar to this :  MyUserControl Cntrl = new MyUserControl(/* Construction Params */); BindingOperations.SetBinding(Cntrl , MyUserControl.AnotherProperty, new Binding { ElementName = "objZoomSlider", Path = new PropertyPath("Value"), Mode = BindingMode.OneWay }); BindingOperations.SetBinding(Cntrl , MyUserControl.TitleWidthProperty, new Binding { ElementName = "TitleSection", Path = new PropertyPath("ActualWidth"), Mode = BindingMode.OneWay }); /* Other operations on Cntrl */   The first binding defined works fantastically, although that is binding to an actual UIElement (in this case a Slider), but the Binding to "TitleSection" (which is the ColumnDefinition defined in the Grid) fails. Putting a breakpoint in the code and doing a watch on "TitleSection" returns the expected object.   I am beginning to suspect that a x:Name'd ColumnDefinition can't be bound to. Can anyone suggest how I might be able to bind to the changing width of the first column in my grid?  EDIT #1 - To answer comments  The databinding 'fails' in the sense that with a breakpoint set on the setter for the TitleWidth property, and using the GridSplitter control to resize the first column, the breakpoint is never hit. Additionally, code I would expect to be fired when the DependancyProperty TitleWidth changes does not get executed.  The usercontrol is being created and added to a Stackpanel within the Grid in the Window_Loaded function. I would expect that the Grid has been rendered by the time the Usercontrols are being constructed. Certainly the x:Name'd Element TitleSection is watchable and has a value of 100 when they are being constructed / before the binding is happening.  EDIT #2 - Possibly something to do with this?  I've been having a sniff round the MSDN pages for the Grid ColumnDefinition documentation and have come across  but I can't get my head around how I can use this in a binding expression. I cannot use the associated GridLengthConverter as a converter in the binding code as it does not derive from IValueConverter.   I am leaning towards somehow binding to the ActualWidth property of one of the cells in the Grid object. It doesn't seem as clean as binding to the column definition, but at the moment I cannot get that to work.