Learn Gradient Intuition: A Beginner's Guide

[cmcraes]

Hi there, I just started to learn about gradients. I can calculate them with ease; but I don't think I really understand them conceptually. I understand the usual example of the temperature scalar field where the temperature in a room is a function of your position T(x, y, z). But when it comes to generic functions, I don't know what I am doing! And it drives me crazy! Any/all help is welcomed! :)

 

[SteamKing]

The gradient of a function of several variables is similar to the derivative of a function of a single variable.

The gradient represents the rate of change of a multi-variable function with respect to one of the variables.

The steeper the gradient, the more rapidly the function is changing with a given change in a particular variable.

 

[HallsofIvy]

The gradient vector of function f(x, y, z) always points in the direction of fastest increase and its length is that maximum increase.

 

[Chestermiller]

The (vector) gradient of a scalar function enables you to determine the rate of change of the function in any arbitrary spatial direction. To do this, you just dot the gradient with a unit vector in the arbitrary direction. If you want to determine the change in the function between two neighboring spatial points joined by a differential position vector, you just dot the gradient with the differential position vector. Thus, the vector gradient of a function is a very useful, and frequently used, tool to quantify directional changes in a function.

Chet

 

[verty]

Think of the gradient as an n-vector when there are n variables. For intuition, consider the function f(x,y) = x^2. Can you interpret the gradient vector (sorry, I should say gradient vectors) as pointing in the direction of quickest increase? What about g(x,y) = x^2 + y^2?

 

[economicsnerd]

[A caveat: All the responses seen so far are true as long as the function (say f) whose gradient you're computing is well-behaved enough, e.g. if the function (x,y,...,z)\mapsto \nabla f(x,y,...,z) is continuous.]

Spatial metaphors are a very good way to get an intuition for these formal objects. Your temperature metaphor is a good one, and can serve your intuition even in situations where f isn't a temperature.

My analogy of choice: You're standing on a hill, and at coordinates (i.e. latitude and longitude) \vec x, the height of the hill is f(\vec x). At any location \vec x, the gradient \nabla f(\vec x) somehow describes the "slope in all directions" all at once. How does it encode this? Well, consider any unit vector \vec u, thought of as a direction in which one could walk. If you walk a small amount in direction \vec u from location \vec x, then \vec u \cdot f(\vec x) is the slope of the incline you're walking. Again, this intuition might be helpful even if points \vec x in the domain of f are 19-dimensional instead of 2-dimensional.

 

[cmcraes]

Okay its starting to come together now! Thanks everyone. On another note, if its not to much trouble; Could someone explain to me the uses of curl and divergence? Obviously theyre useful in someway but I can't really wrap my head around what they mean. Thanks again everyone!

 

[economicsnerd]

If you walk a small amount in direction \vec u from location \vec x, then \vec u \cdot f(\vec x) is the slope of the incline you're walking.

Whoops, typo. The corrected version (with an added \nabla) is below.

If you walk a small amount in direction \vec u from location \vec x, then \vec u \cdot \nabla f(\vec x) is the slope of the incline you're walking.