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In Progress
Skill 11 of 13
In Progress

Gradient

  • Your fellow hiker asks: ‘how steep is the path?’ He is of course referring to the steepness along the path (in the direction of the derivative $s’$). Nothing else would make sense. There is no ambiguity; there is only one slope to consider for a 1D path.
  • Then he asks: ‘how steep is the mountain actually right here?’. He is not talking about the path now. There are many directions to look along on a 2D mountain side, so which one is he referring to?

The steepest one, of course. Of the infinitely many directions you can look in, ‘steepness’ or ‘slope’ of a 2D landscape refers to the steepest steepness. The steepest steepness gives a feeling for the overall steepness at the point, whereas a steepness along any other direction doesn’t tell us much.

How do we find that single steepest direction?

  • In 1D we needed the derivative $s’$ to find steepness, and
  • in 2D we need two derivatives, namely the two partial derivatives, $s’_x$ and $s’_y$.[1,2]

They represent the steepness-value along either axis, and together as a “steepness” or “slope” vector $(s’_x,s’_y)$ they represent the full steepness and point up along the steepest direction. In other words, they represent the gradual change up along the hill in the very point you are standing in. Let’s name this “steepness” vector: gradient, and let’s invent a nice symbol for it such as the upside-down symbol for ‘change’, $\nabla$:1 2

$$\nabla s=\begin{pmatrix}s’_x\\s’_y\end{pmatrix}$$

Note: At any point on a 2D surface, the steepest direction upwards is of course exactly opposite to the steepest direction downwards. Their steepness is the same, just in opposite directions. We’ll keep in mind that the gradient always points along the steepest upwards direction (the “positive” direction).

So, when we see the symbol $s’$ or $\frac{\mathrm ds}{\mathrm dx}$ we have a 1D derivative, and when see the symbol $\nabla s$ or $\left(\frac{\mathrm ds}{\mathrm dx},\frac{\mathrm ds}{\mathrm dy}\right)$ we have a 2D “derivative” or gradient.3


References:

  1. Vector Calculus: Understanding the Gradient’ (web page), Kalid Azad, Better Explained, 2014, www.betterexplained.com/articles/vector-calculus-understanding-the-gradient (accessed Jun. 14th, 2020)
  2. Gradienter og tangentplaner’ (web page, in Danish), Karsten Schmidt, Technical University of Denmark, Institute for Mathematics and Computer Science, eNote 20, 2016, 01005.compute.dtu.dk/enotes/20_-_Gradienter_og_tangentplaner (accessed Jun. 14th, 2020), chapter 20
  3. Microfluidics: Modelling, Mechanics and Mathematics’ (book), Bastian E. Rapp, Elsevier, 2016, www.sciencedirect.com/science/article/pii/B9781455731411500071 (accessed Jun. 14th, 2020), ISBN 978-1-4557-3141-1, DOI 10.1016/B978-1-4557-3141-1.50007-1, chapter 7.1.3
  4. nabla’ (dictionary), Oxford English Dictionary, 3rd ed., 2005, www.oed.com (accessed Jun 14th, 2020)