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I'm having some trouble trying to visualize and physically understand what's happening with the gradient. I understand that the following is true:

  • The gradient of f (grad(f)) points in the direction of maximum increase of f

However, later on, we are told that a gradient of a surface f (grad(f)) gives us the surface normal vector (i.e pointing away). How can that be? From the first statement, I thought that the gradient must be pointing in the direction of maximum increase of f - surely the direction of maximum increase of f should be a vector pointing in some direction on f itself. How can it be pointing outward as a surface normal vector?

FShiwani
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  • I was asking myself the same question, and I noticed a lot of people have shared the same confusion. I think our multivariate calculus textbooks did not do a good job describing the two kinds of gradient vectors! – RobertF Oct 06 '20 at 14:15

1 Answers1

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They’re talking about gradients of two different functions. In the first case, you’ve got, for example, the graph of some function $f:\mathbb R^{n-1}\to\mathbb R$, i.e., the surface described by $y=f(\mathbf x)$. In this case, $\nabla f\in\mathbb R^{n-1}$ points in the direction of greatest increase. In the second case, you’ve got a level surface of a function, that is, some function $g:\mathbb R^n\to\mathbb R$ and the surface defined implicitly by $g(\mathbf x)=0$. In this second case, the gradient $\nabla g\in\mathbb R^n$ is normal to the tangent (hyper)plane, and so is a surface normal.

The two ideas are connected, of course. The surface $y=f(\mathbf x)$ can also be viewed as a level surface of the function $g:(\mathbf x;y)\mapsto f(\mathbf x)-y$, and so $\nabla g=(\nabla f;-1)$.

amd
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  • Thank you! Is there an easy way to figure out if a surface is a level surface or not? Also, in some parts of this book it says that the gradient points in the direction of maximum increase, and in other parts it says that the gradient is the rate of change of f(x,y,z) in any direction in space. Aren't those two contradictory statements? – FShiwani Oct 06 '17 at 23:37
  • @FShiwani Well, they’re all level surfaces of some function, as I noted at the end. However, it’s a matter of context. If the surface is described by an explicit function that, given $n-1$ coordinates, spits out the $n$th one, then it’s of the first type; if it’s given implicitly, then it’s the second. – amd Oct 07 '17 at 00:27
  • @FShiwani As for your other question, it sounds to me like you’ve misunderstood something. The gradient always points in a specific direction and its length gives the function’s rate of change in that direction (in the first type of gradient). If the function is differentiable at a point, however, you can use its gradient to compute the rate of change in any direction. – amd Oct 07 '17 at 00:30