Question

Hello, what is the intuition behind the definition of the gradient?

Answer 1

I know what it means, but not why - what am I missing that's obvious?

Answer 2

By gradient are you referring to slopes? @inkyvoyd

Answer 3

by gradient I mean ∇f(X)

Answer 4

do you know how to find the gradient ∇f(x) of a two-variable function?

@inkyvoyd

Answer 5

yeah, read the wiki definition along wiht my textbooks - partials with a variable multiplied by that unit vector. I was told by khan academy that the direction of the vector was the path of greatest ascent, but I'm not really seeing immediately why adding up these vectors yields path of greatest ascent. PS: sorry for hte wait

Answer 6

@genius12

Answer 7

wiki said a bunch of jumble about dot product, so if it has to do with that, where's the cosine @.@

Answer 8

What do you mean by cousin?

Answer 9

there is a cosine in the dotproduct for the angle theta in between the two vectors I believe

Answer 10

Oh

Answer 11

The gradient tells you, for example, the direction a ball would roll when released from the side of a hill. If you look at topographical maps, contours indicate where altitudes are equal. Take any one of these lines. If you released a ball at any point along this line, the ball would roll perpendicular to this line. This perpendicular is the gradient. We have the same thing in electric fields. Contours of equal potential are perpendicular to the electric field. So if an electron were to be released at any of theses equipotentials it would begin to move perpendicular to these contours. This is the gradient.

Answer 12

The gradient takes in a vector input, and then gives you the slope of the tangent line in the direction of that vector.

Answer 13

And by slope, I mean the instantaneous change of the output of the function over the change in the instantaneous change ind the direction or said vector.

Answer 14

@wio what is the nature of this vector input and what's the meaning of hte vector's direction?

Answer 15

The \(v_1, v_2, v_3\) are example vectors.
Unlike with single variable functions, there are multiply directions you could travel, and the change in our multi variable function is going \(f(x,y)\) is going to be different depending.