Gradient: Normal vs Direction of Increase

[thepopasmurf]

Hi,

I'm having trouble understanding what exactly the gradient of a scalar field represents. According to wikipedia and the textbooks I have it points in the direction of greatest increase and has a magnitude of greatest increase. This by itself seems fine. However, I have also been using it to find the normal of a surface and I don't understand how it can be both.

Also, do the properties( eg, what it represents) of the gradient change as you change dimensions and how does it work on a simple 2d graph?

 

[Tinyboss]

The gradient is perpendicular to the level sets of a function. So if your surface is given by f(v)=c, the normal at v is given by the direction of the gradient at v.

 

[quasar987]

It all comes from the formula <v,w>=|v||w|cosθ where θ is the angle between v and w.

Let F:R³-->R be a function. By looking at the definition of DF(x)v for |v|=1 (the derivative of F at a point x in the direction of the vector v), we agree that this number represents the rate of change of F at x in the direction v. On the other hand, the gradient of F at x is defined as the (unique) vector ∇F(x) such that DF(x)v=<∇F(x),v> for all vectors v.

So to ask in which direction is F increasing the most rapidly at x is to ask which vector v of unit length (|v|=1) maximizes the value of DF(x)v. But DF(x)v=<∇F(x),v>=|∇F(x)||v|cosθ=|∇F(x)|cosθ, with cosθ taking values between -1 and 1. Clearly, |∇F(x)|cosθ is largest when cosθ=1; i.e. when θ=0. That is, when v points in the direction of ∇F(x)!

Now, consider S a surface in R³ that is realized as the level set F=c of F. That is, S=F^{-1}(c) for some constant c. Take x a point in S. By definition, a tangent vector to S at x is a vector v of the form v=\gamma&#039;(0) for some curve \gamma:]-1,1[\rightarrow S on S with \gamma(0)=x. Notice that for v=\gamma&#039;(0) a tangent vector to S at x, the derivative of F at x in the direction v vanishes:

DF(x)v=DF(x)\gamma&#039;(0)=\frac{d}{dt}_{t=0}(F\circ\gamma)(t)=0

The second equality is the chain rule and the third equality is because the map (F\circ\gamma)(t) is the map t\mapsto c.
Ok, so in terms of the gradient, what does this tells us? It tells us that 0=DF(x)v=<∇F(x),v>=|∇F(x)||v|cosθ, so cosθ=0, so θ=±90°. That is, ∇F(x) and v are perpendicular. By definition, this means ∇F(x) is perpendicular (or normal) to the surface S at the point x.