Gradient: Normal vs Direction of Increase

In summary, the gradient of a scalar field represents the rate of change of the field in the direction of the vector v with the magnitude of the most rapid increase. It is perpendicular to the level sets of a function and can be used to find the normal of a surface.
  • #1
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?
 
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  • #2
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.
 
  • #3
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, [itex]S=F^{-1}(c)[/itex] 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 [itex]v=\gamma'(0)[/itex] for some curve [itex]\gamma:]-1,1[\rightarrow S[/itex] on S with [itex]\gamma(0)=x[/itex]. Notice that for [itex]v=\gamma'(0)[/itex] a tangent vector to S at x, the derivative of F at x in the direction v vanishes:

[tex]DF(x)v=DF(x)\gamma'(0)=\frac{d}{dt}_{t=0}(F\circ\gamma)(t)=0[/tex]

The second equality is the chain rule and the third equality is because the map [itex](F\circ\gamma)(t)[/itex] is the map [itex]t\mapsto c[/itex].
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.
 

1. What is a gradient?

A gradient is a measure of the rate of change of a function at a particular point. It represents the direction and magnitude of the steepest ascent or descent of the function.

2. What is the normal of a gradient?

The normal of a gradient is a vector that is perpendicular to the direction of the gradient at a particular point. It represents the direction of maximum change or the direction in which the function is flattest.

3. What is the difference between gradient and direction of increase?

The direction of increase is the direction in which the function value is increasing, while the gradient is the direction of steepest ascent or descent. The direction of increase is always parallel to the gradient, but the gradient may have a different magnitude and direction.

4. How are gradients and directional derivatives related?

Gradients and directional derivatives are both measures of the rate of change of a function at a particular point. The gradient represents the maximum rate of change in all directions, while the directional derivative represents the rate of change in a specific direction.

5. Can the gradient be negative?

Yes, the gradient can be negative. A negative gradient indicates a decrease in the function value. The magnitude of the gradient represents the steepness of the decrease, while the direction of the gradient indicates the direction of the decrease.

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