What is up with the multiplication of unit vectors with partial derivatives?

Question

With gradient there's this combination of unit vectors with partial derivatives which when combined using dot product with small displacement say $dr$ gives rate of change of that function but regarding it's individual existence, I'm still not able to make sense of it. I know we call it gradient but somehow it's independent existence doesn't really make sense to me. You're multiplying a rate of change in whatever direction with unit vector in that direction given that it involves a function too. Can anyone tell me where I'm missing out on this one?

Answer 1

The following is couched in terms of a map $\mathbb{R}^2 \mapsto \mathbb{R}$, but the number of independent variables need not be two.

The gradient is a gadget which enters into this contract with you: "You tell me where on the map you are and which way you are driving and I'll tell you how fast your altitude is changing." It does this by the same method the derivative does: at each point, find the best linear approximation to the function. For each point, this linear approximation is a plane through the point, tangent to the topography of the map. The problem with that plane is that it tells you the rate of change of altitude in all directions at once. If you want to know the rate of change in a particular direction, you have to tell it which direction. Then it draws the ray on the plane corresponding to that direction and reports the rate of change of altitude along that ray.

(Image from Marty B.'s question.) (There are two ideas here I am going to use: the gridded plane is the tangent plane to the red surface and the "grad" vector is the gradient of the function having the colored surfaces for different level sets. I'll talk about these two ideas in this order.) This shows the plane tangent to the red surface. Notice if you go horizontally (in the screen coordinates) on the picture, the height of the plane is nearly constant, so the rate of change at that point in those two directions is nearly zero. If you go vertically (in the screen coordinates), the height of the plane is changing about as rapidly as it can, increasing in one direction and decreasing in the opposite direction. This corresponds to changes in height on the red surface if you start at the point where the plane is tangent to the surface and then pick a direction (in $(x,y)$ coordinates) to go on the red surface.

To get a rate of change (a number), we have to decide which direction we go along this plane, so we need to supply a direction vector. Since the result of taking the dot product of the gradient and the direction vector is scaled by the length of the direction vector, $ \nabla f \cdot (c\vec{v}) = c(\nabla f \cdot \vec{v}), $ we make that length $|\vec{v}| = 1$ so you get what you want instead of some $c$-multiple of what you want.

Exactly the same thing happens with the derivative. There are only two directions we can go: left or right. Suppose the slope of the tangent line is $1$ at the point $x$. If we go in the direction of the vector $(1)$ (so in the positive $x$-direction, the function is increasing at a rate of $1$ unit of $y$ per unit of $x$. If go in the direction $(-1)$ (so in the negative $x$ direction), the function is decreasing at the rate of $1$ unit of $y$ per unit of $x$. In this case, $\nabla f = (1)$ so that $\nabla f \cdot \vec{v}$ is $(1)\cdot(1) = 1$ in the "go right" case and is $(1) \cdot (-1) = -1$ in the "go left case". (These are dot products between $1$-dimensional vectors.) Notice that this $\nabla f = (1)$ "points" in the direction of maximal increase.

Now for the second idea from the image above. There are three independent variables, so this is a partial graph of a map $\mathbb{R}^3 \rightarrow \mathbb{R}$. The colored surfaces are collections of points all mapped to the same element of $\mathbb{R}$, i.e. are the same "height". The gradient vector shown points in the direction of maximal increase, just like the derivative did when there was one independent variable. For this function, if you start at the point where the "grad" vector meets the gridded plane and fly from that point on the same side of the plane as the "grad" vector, the function takes your position coordinates to increasing numbers. The more nearly you move in the direction of the "grad" vector the faster the increase. If you fly from that point on the opposite side of the plane as the "grad" vector, the function takes you to decreasing values. If you fly along the plane, the function values are nearly unchanged -- this is what we expect when we look at the surface: in an infinitesimal neighborhood of the point, the red surface runs off along that plane, but the red surface is a level surface, so the output of the function is constant along that surface.