Explain why does gradient vector points in the direction of the steepest increase?

You can think of the gradient vector as a collection of the rates of change in each direction. The partial derivatives, when combined into a vector, indicate the direction and magnitude of the fastest increase in the function. The vector points in the direction where, if you were to move, you would experience the greatest increase in the function's value.

This is because the partial derivatives tell you how much the function changes as you move along each axis. The direction of the steepest increase is the one where the function increases the most rapidly when you move a small distance in that direction.

I can put it in a simpler way for you to better understand it. Imagine you're standing on a hilly landscape, and the landscape represents the values of your function $f(x,y)$. At each point on the landscape, the height represents the value of the function at that point.

Now, let's say you want to find the direction in which you should walk to experience the steepest increase in height (similar to your question). In other words, you want to find the direction where the function f increases the most rapidly. So, we have:

The gradient vector delta $f(x,y)$ at a specific point $(x,y)$ is formed by combining the partial derivatives $df/dx$ and $df/dy$. These partial derivatives tell you how the function changes as you move in the x and y directions from the point $(x,y)$.

When you have this gradient vector, it points in the direction of the steepest increase because it shows you the change in the function as you move a small distance in each direction. If you move a little bit in the direction of the gradient vector, you're essentially moving a bit in the x direction and a bit in the y direction. These changes in both directions combine to give you the steepest increase in the function's value from your current point.

To put it simply, the gradient vector guides you towards the path where the function f increases the most rapidly. It points in the direction of the steepest ascent on the "height" landscape of your function.

You can also assume you want to find out how fast your function will be changing with respect to some arbitrary direction.

Let v be a unit vector, it can be projected along this direction via the dot product

\[ \nabla f (a) \cdot v      \text{(directional derivative)}\]

Now in which direction this is maximal?

We know:

\[ \nabla f(a) \cdot v =|\nabla f(a)||v|\cos(\alpha)\] 
 

where $v$ is unit vector as we said above, so:

\[ |\nabla f|\cos(\alpha)   \text{is maximal when}   \cos(\alpha)=1.\]

In other words, when $v$ points in the same direction as $\nabla f(a)$, this is maximal.

If you liked the extra explanation, a tip is appreciated.

Thanks,
Kav10