I found it super surprising that the gradient determines the value of every directional derivative. Like, really?
When reading this comment, I was surprised for a moment, too, but now that you mention it—it’s because if the function is smooth at the point where you’re taking the directional derivative, then it has to locally resemble a plane, just like a how a differentiable function of a single variable is said to be “locally linear”. If the directional derivative varied in any other way, then the surface would have to have a “crinkle” at that point and it wouldn’t be differentiable. Right?
I have since learned that there are functions which do have all partial derivatives at a point but are not smooth. Wikipedia’s example is f(x,y)=y3x2+y2 with f(0,0)=0. And in this case, there is still a continuous function ϕ:S2→R that maps each point to the value of the directional derivative, but it’s ϕ(x,y)=y3, so different from the regular case.
So you can probably have all kinds of relationships between direction and {value of derivative in that direction}, but the class of smooth functions have a fixed relationship. It still feels surprising that ‘most’ functions we work with just happen to be smooth.
When reading this comment, I was surprised for a moment, too, but now that you mention it—it’s because if the function is smooth at the point where you’re taking the directional derivative, then it has to locally resemble a plane, just like a how a differentiable function of a single variable is said to be “locally linear”. If the directional derivative varied in any other way, then the surface would have to have a “crinkle” at that point and it wouldn’t be differentiable. Right?
That’s probably right.
I have since learned that there are functions which do have all partial derivatives at a point but are not smooth. Wikipedia’s example is f(x,y)=y3x2+y2 with f(0,0)=0. And in this case, there is still a continuous function ϕ:S2→R that maps each point to the value of the directional derivative, but it’s ϕ(x,y)=y3, so different from the regular case.
So you can probably have all kinds of relationships between direction and {value of derivative in that direction}, but the class of smooth functions have a fixed relationship. It still feels surprising that ‘most’ functions we work with just happen to be smooth.