Vector Calculus: Understanding the Gradient

The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that

This is a companion discussion topic for the original entry at

i like it… well explained.


You are the man! Nice work!

Thanks, glad it was helpful for you.

i was always looking for conceptual and practical examples and yes i finally got.


well you made a good explanation, that even a not-so-smart guy gets it, but i think you missed the obvious -> WHY does gradient show the direction of the greatest increase.

I think that the principle of the gradient is quite easy, but understanding why does it work the way it does is a bit tricky and you should have focued on it more.

It would be interesting if you would somehow add it to this good article. Inspiration

good luck !

Hi Palo, that’s a great point! I’ve been feeling a bit guilty, if you can imagine it, because I’ve lacked that explanation :slight_smile:

I’m probably going to do a separate article on the reason why the gradient points in the direction of greatest increase – I have another explanation that it works well with. Thanks for the link and feedback!

Your introduction is not quite correct:

You claim: “Points in the direction of greatest increase of a function”.

Why? It can also point in the direction of greatest decrease of a function.

A gradient is one or more directional derivatives. These derivatives are considered in a particular direction. In the case of single variable calculus, we generally talk about a directional derivative when we consider multiples of the x unit vector, i.e. k*(1,0). To consider the y unit vector, we deal with the partial derivatives with respect to y in a given direction. In three dimensions, the 3 partial derivatives form what we now call a ‘gradient’.

So in fact it is incorrect to call this a slope or anything else except to say that it describes the partial derivatives of a point in the direction of a given vector in space.

Does this make sense? Please visit my blog for some more interesting reading.

Hi John, thanks for writing. You’re right, the formal definition of a gradient is a set of directional derivatives.

But when thinking about the intuitive meaning, I think it’s ok to consider the gradient as a vector that “points” in the direction of greatest increase (i.e. if you follow that direction your function will tend towards a local maximum).

Unless I’m mistaken, the gradient vector always points in the direction of greatest increase (greatest decrease would be in the opposite direction).

What I was saying is that it points either one way or the other, it is not restricted to the direction of greatest increase. As a simple example, consider what happens when you differentiate a parabola: You set the derivative equal to 0 and then you determine that it has either a maximum or a minimum at its turning point. It is not always a maximum just as it is not always a minimum. Think I have explained this correctly now.

good john you have done a great job.

Hi John, thanks for the clarification. I’d still politely disagree and say that in general, the gradient points in the direction of greatest increase :).

In the case of 2 dimensions, the gradient/slope only gives a forward or backward direction. A positive slope means travel “forward” and a negative slope means travel “backwards”.

Consider f(x) = x^2, a regular parobola. The gradient is zero at the minimum (x=0), and there is no single direction to go. At x = -1, the slope is negative, which means travel “backwards” (to x = -2) to increase your value. Similarly, at x = 1, you travel forward (to x = 2) to increase your value.

But, as you mention, strange things can happen when the derivative = 0. It can mean you are at a local maximum (no way to improve), or at a local minimum (no single direction to improve your position – forward or back will help). I consider the corner case of zero an exception to the general rule / intuition that the gradient is “the direction to follow” if you want to improve your function.

Wonderful explanation!

Thanks Vidhya, glad you liked it.

hi john keep it up you done a great job

Thanks a bunch! I didn’t think it could be this simple to find the maximum increase at a point, so I thought I’d look it up. Thanks to your great explaination, it turn out it was as easy as it seemed it should be. Great job! Thanks!


Awesome, glad it worked for you :slight_smile: