10.41.2 Greatest hits from previous weeks

10.41.2.1 Multivariable Taylor expansion does not work as expected

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=15023

The basic multivariable Taylor expansion formula around a point is as follows: 
 
$$ 
f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot \nabla)... 
[calculus-and-analysis] [series-expansion]
 

asked by matheorem https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=4742 Score of 77

answered by Jens https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=245 Score of 87

10.41.2.2 Get a "step-by-step" evaluation in Mathematica

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=148

Is it possible in Mathematica to get a step-by-step evaluation of some functions; that's to say, outputting not only the result but all the stages that have led to it? If so, how does one do it? 
 
... 
[evaluation] [interactive] [faq]
 

asked by Cydonia7 https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=44 Score of 147

answered by J. M.'s missing motivation https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=50 Score of 121

10.41.2.3 What do the X and Y axis stand for in the Fourier transform domain?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=85139

In Wolfram Mathematica the function  Fourier  has the following declaration 
 
 Fourier[list] 
 
 
And after a list is given to the ... 
[numerics] [fourier-analysis] [faq]
 

asked by user253956 https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=29958 Score of 33

answered by Hugh https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=12558 Score of 80

10.41.2.4 What does "Tag Integer is Protected" message mean?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=6738

(Newbie question) 
 
For every function I define I get this message: 
When I close and reopen Mathematica, and re-enter the function definition, I don't get the message anymore. What does it mean? 
[warning-messages]
 

asked by stevenvh https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=1450 Score of 16

answered by J. M.'s missing motivation https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=50 Score of 17

10.41.2.5 Mathematica vs. Python - how does Mathematica compare to Python's scientific computing suite?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=86058

Python's family of packages for scientific computing has matured rapidly. I can pretty much replicate all of Mathematica's functionalities, but with production level and open-source code using the ... 
[python] [system-performance]
 

asked by M.R. https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=403 Score of 65

answered by Jens https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=245 Score of 42

10.41.2.6 How to add a vertical line to a plot?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=3561

In the plot below I would like to add two vertical lines at $x = \frac{\pi}{15} \pm \frac{1}{20}$. 
How can I do that? 
 
... 
[plotting]
 

asked by sjdh https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=515 Score of 87

answered by Ajasja https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=745 Score of 102

10.41.2.7 Placing text within a plot

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=71776

I want to place text within a plot. The text is always centered relative to the coordinates. How can I left-align or right-align the text relative to the coordinates? For example: 
 
... 
[plotting] [graphics] [text]
 

asked by Anthony Mannucci https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=24733 Score of 29

answered by m_goldberg https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=3066 Score of 13