====== Gradient ======

<tabbox Intuitive> 


<blockquote>The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that

  * Points in the direction of greatest increase of a function (intuition on why)
  * Is zero at a local maximum or local minimum (because there is no single direction of increase)


The term "gradient" is typically used for functions with several inputs and a single output (a scalar field). Yes, you can say a line has a gradient (its slope), but using "gradient" for single-variable functions is unnecessarily confusing. Keep it simple.
<cite>[[https://betterexplained.com/articles/vector-calculus-understanding-the-gradient/|Vector Calculus: Understanding the Gradient]] by Kalid Azad</cite></blockquote>

  
<tabbox Concrete> 

  * [[https://betterexplained.com/articles/vector-calculus-understanding-the-gradient/|Vector Calculus: Understanding the Gradient]] by Kalid Azad
  * [[https://betterexplained.com/articles/understanding-pythagorean-distance-and-the-gradient/|Understanding Pythagorean Distance and the Gradient]] by Kalid Azad
 
<tabbox Abstract> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

  
<tabbox Why is it interesting?> 
The gradient, denoted $\nabla$ (spoken "Nabla"), is a tool that enables us to calculate how much a given function changes in different directions. A "normal" function $f(x)$, lives in a boring one-dimensional space, and the derivative is just another function: $\partial_x f(x)$. This function is the rate of change of $f(x)$. Our real world is three-dimensional and hence in physics we often encounter functions that depend on all spatial directions: $f(x,y,z)$. The gradient of such a function: 

$$\nabla f(x,y,z) =  \begin{pmatrix} \partial_x f(x,y,z) \\ \partial_y f(x,y,z)\\ \partial_z f(x,y,z) \end{pmatrix}$$

is a vector and tells us how much $f(x,y,z)$ changes in each direction. 

<tabbox FAQ> 

--> Why should we view the derivative as a vector?#

See http://mathinsight.org/gradient_vector
  
<--


</tabbox>