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Suppose, I have a multivariable function: $f(x, y) = 2x + 10y + 5$.

The gradient vector of this function would be:

$ \text{gradient of } f = \nabla f = \begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}\end{bmatrix}$

What does a gradient vector represent if $f$ is considered as a surface in 3D?

As far as I understand:

A column vector of values:
1. instantaneous change of the surface represented by f in the x-direction,
2. instantaneous change of the surface represented by f in the y-direction

So, what is the intuition here if x represents, say, force, and y represents time?

Also, how does the intuition change if the number of variables increases?

user366312
  • 1,641
  • $\nabla f(x,y)$ is a vector that points in the direction of steepest ascent for $f$ at $(x,y)$. The magnitude of $\nabla f(x,y)$ tells you how fast $f$ increases when you move in the direction of steepest ascent. – littleO Dec 26 '18 at 19:18
  • Looking at the surface $z=f(x,y)$ in 3D is not a fruitful way of understanding the gradient. You should think about the level sets $f(x,y)=C$ in 2D instead. – Hans Lundmark Dec 26 '18 at 21:05
  • How exactly are you considering $f$ as a surface in 3D? – amd Dec 27 '18 at 06:46
  • @amd, I think it should have been written as $z = f(x, y)$. – user366312 Dec 27 '18 at 11:17
  • See https://math.stackexchange.com/q/2459214/265466 and related questions, of which this one might be a duplicate. TL;DR: it’s the projection of a surface normal and points in the direction of greatest increase of $f$. – amd Dec 27 '18 at 20:14

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