Differential Notation Magic in Integration by u-Substitution I'm really confused now. I always thought that the differential notation $\frac{df}{dx}$ was just that, a notation.  But somehow when doing integration by u-substitution I'm told that you can turn something like this $\frac{du}{dx} = 2x\;$ into this $\;du = 2x\ dx$.  But how is that even possible? I understand that the notation comes from the fact that $\frac{du}{dx}$ actually means the limit of the difference in $u$ over the difference in $x$, with $\Delta x$ approaching $0$.  $$u'(x) = \frac{du}{dx} = \frac{du(x)}{dx} = \lim_{\Delta x\to 0} \frac{u(x+\Delta x)\ -\ u(x)}{(x+\Delta x) - x} = \lim_{\Delta x\to 0} \frac{u(x+\Delta x)\ -\ u(x)}{\Delta x}$$  So if $\frac{df}{dx}$ is just a notation for the limit mentioned above, then what is the underlying argument to say that you can treat $\frac{du}{dx}$ as if it were an actual fraction?  Appreciate the help =)
Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $dy = f'(x)dx$ we are able to plug in values for $dx$ and calculate a $dy$ (differential). Then if we rearrange we get $dy/dx$ which could be seen as a ratio.   I wonder if the author says this because $dx$ is an independent variable, and $dy$ is a dependent variable, for $dy/dx$ to be a ratio both variables need to be independent.. maybe?
What are some CD Ripping Programs you can use on Ubuntu? What are some CD ripping programs for Ubuntu? Can you list the Pros? What are the Cons? For each program there should be a screenshot, sources to install, and instructions to install and use. Is there a way to get the count of rows in all tables in a MySQL database without running a SELECT count() on each table? FFmpeg lossy compression problems How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$? cannot start Tomcat 7 server - java.net.BindException: Address already in use Is this group solvable? Let $G=H\times K$ (a direct product of $H , K$) where $H$ is an abelian 2-group and $K$ is a non-abelian simple group. Is $G$ solvable? why?  These three answers are true, and I thought so. But in a book I found a remark that made me confused: T.M. Gagen, Topics in Finite Groups, London Math. Soc. Lecture Note Ser., vol. 16, Cambridge Univ. Press, Cambridge, 1976, Remark, Theorem A on p. $40 $ and Definition $11.3$ on p. $39$. Please see it and say me your ideas. How to refer Javascript cloud page in my email in content builder? I got a visa with my mother when I was 17 years old. On it was stated "Not valid unless accompanied by my mother".    Can I travel to the UK alone now that I am 18? Show that any bounded linear functional on a normed linear space is continuous.   Can we say that it is uniformly continuous ?   Also, is it true, if we reverse the statement    any continuous  linear functional on a normed linear space is bounded ?? What are the advantages and disadvantages of using ARC? Is their a way to move multiple vertices gradually at a time? I have moved multiple vertices gradually but a set of vertices at a time not all at a time, as shown in the animation below (First I moved the four vertices from top, then three, then two and at last the last vertex).  I wanted to do the same thing as shown in the animation below but all at a time. I am carrying out research to look at differences in fish density and fish species richness when using two different underwater visual census methods. My data was originally count data but then typically this is changed to fish density but I have still decided to use a Poisson GLM, which I hope is right.   model1 &lt;- glm(g_den ~ method + site + depth, poisson)   My 3 predictor variables are method, site and depth which I ordered as factors when I input them.   My response variables are grouper species richness, grouper density and the same for other fish groups. I am aware that density is not an integer and it is numerical data e.g 1.34849. I am now however getting this error:      In dpois(y, mu, log = TRUE) : non-integer x = 0.037500   I've been reading up and many people suggest using an offset, is this the most advisable thing to do? Align cell height of the tabularx rows The following latex code prints rows one at a time. However all the cells of the rows don't come out aligned. How can I fix it.   \documentclass{beamer}     \usepackage{tabularx}    \begin{document}  \begin{frame}{}  \begin{tabularx}{3.50in}{lXX}           \hline           &amp; Observation &amp; Conclusion \\           \hline             \uncover&lt;2-&gt;{1.} &amp; \uncover&lt;2-&gt;{Observation 1 } &amp; \uncover&lt;2-&gt;{Conclusion 1} \\   \uncover&lt;3-&gt;{2.} &amp; \uncover&lt;3-&gt;{Observation 2} &amp; \uncover&lt;3-&gt;{Conclusion 2} \\           \end{tabularx}      \end{frame}   \end{document}   The output looks as follows :    As we can see cells in first column do not align with the rest of the rows. Will appreciate any input as to how to fix this. Beamer: bold math no longer working How to manipulate a CSV file with sed or awk? How can I do the following to a CSV file using sed or awk?   Delete a column Duplicate a column Move a column   I have a big table with over 200 rows, and I'm not that familiar with sed. How do I set the default program? How to find a program which is preventing sleeping? I am noticing that recently my laptop does not go to sleep/lockout after its configured X minutes of inactivity like it did in the past. I suspect some program (like media player does) is preventing it from going to sleep. I do not have any media player running, so how can I find which program is preventing it from sleeping?  It does go to sleep if I close the lid. But I would also like for it to go to sleep after X minutes of inactivity.  Update: Upgraded to Ubuntu 18.04 with default Gnome and Wayland enabled. Show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$. The problem is:  Suppose that $f(z)$ is continuous on a domain $D$ that contains the unit circle, and that $f(z)$ satisfies: $$\vert f(e^{i\theta})\vert \leq M \; \forall \theta \, \in [0, 2\pi )$$ and $$\bigg \vert \int _{\vert z \vert=1} f(z) \text{d}z \bigg \vert =2\pi M $$ show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$.  I've tried lots of different approaches such as defining a function $g(z)=\frac{f(z)}{\bar{z}}$ and trying to show that $g'(z)=0$ so that $g$ is constant on the circle, or trying to use some bounding theorems for analytic functions to show that $g$ is constant but this requires $g$ to be analytic. Some other methods were using the Cauchy-Riemann equations but I couldn't find any reason why $g$ should be analytic.  Any help would be greatly appreciated, thanks! Anti-Matter Black Holes Assuming for a second that there were a pocket of anti matter somewhere sufficiently large to form all the type of object we can see forming from normal matter - then one of these objects would be a black hole.  Question is, would there be a difference between an anti matter black hole and a normal matter black hole - in terms of how would the matter/anti-matter make the black hole different, or would they be the same?  I would expect the answer to be that the black hole formed from anti matter would retain the anti-matter properties in such a way that if it was to merge with a black hole at same size formed from normal matter that they would annihilate  each other con convert into pure energy - Would that be a correct understanding? Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$ I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a series of algebraic manipulations that can lead from $\sum\limits_{i=0}^n \binom{n}{i}$ to $2^n$.