What Does the Gradient Vector Mean Intuitively?
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- เผยแพร่เมื่อ 6 ก.พ. 2025
- What Does the Gradient Vector Mean Intuitively?
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Thank you:)
![Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]](http://i.ytimg.com/vi/lKXW7DRyyro/mqdefault.jpg)
Sometimes the most simple explanations for the most complex things are the best . Cheers!
I agree but whould be even more interesting if he would add the case: what if f(x,y) = z is constant :)
I have struggled so many days understanding graident wrt to machine learning, but now it makes total sense to me. Thannks a lot!!
The best explanation ever on what's gradient vector. Thank you!
Wow thank you! Your comment means a lot:)
Yes I also would like to know the why . And would you mind doing an intuitive version video of directional derivatives using the gradient ? I really want to understand that one as well . Thank you !!
Sure good idea!
Love it, thanks! Wish I had videos like this back in college 35 years ago, would have cleared up a lot of confusion.
Make so much sense! Thank you so much
The best explanation I've ever heard, thanks a lot.
It does make sense, but for those who are not familiar with the gradient, they need more graphic illustration. Most students including myself were just told that the gradient is a vector that points in the steepest ascent of any 3-D surface and how to compute it given the surface function, but we were never told or shown what the gradient vector looks like any given point on the surface and whether such vector lies in the 2-D or the 3-D space, and why it points in the direction of the steepest ascent. The easiest example was the paraboloid surface. The steepest ascent on any point on the surface is always in the direction of radial away from the vertex. I think the gradient concept needs more illustration and justification. Any video that is proactive that allows for inputting the function, computes the gradient and graphs the gradient might be a good idea. Thanks.
yes this, Im still in the learning stage (like 30 minutes ago), have you ever found anything that would help visualize it?
@@lubululu3232 well, the only thing I know is how to compute a gradient, like I said before, so given a scalar function f(x,y,z) apparently in 3D, we operate the Nebla on the function and get the gradient, which is (df/dx)i +(df/dy)j +(df/dz)k. This is a real vector in 3D with three components in the Cartesian coordinate system. Just like any other vector in 3D, we could graph this vector pointing out from the origin. Could we ascertain that the direction of such vector points in the steepest ascent of the surface (x,y,z)? That is the whole point of the comment I previously made. Students are not typically shown how to verify that visually.
Thank you, for the video!
Thank you! Great help for my vector cal / EM review, especially cause it helps me picture what is orthogonal to it!
Nice and easy explanation. Thank you!
Eureka! Thank you very much!
skilled as always, thanks you so much!!!
That was great! Thanks!
Thanks for this. Very well explained!
youre THE best, MS !
Awesome explanation 🙂
Holy smokes. Me: *pushes play* You: Hi. *teaches instantly* Me: *begins learning*
No title card or intro to skip. Appreciated!
Would be nice if you graphically showed why it ascends as fast as possible.
Why is the magnitude of the gradient vector said to be the RATE of maximum ascent? When I see "rate", I think slope. Why isn't the rate of ascent simply the partial of y divided by the partial of x.? Isn't this the slope of the gradient....i.e. change in y over the change in x? What am I missing? thanks
thank you ❤
This was great
Jeff Bezos with long hair is teaching me mathematica.
Thank you!!
you are welcome!
the explanation is great, but wtf is a rolling plane?
thank you
Immediately googles “rolling plains” bc I’m so cultured.
Thanks!
this dude's hair style reminds me of sir issac newton
But why does it do that :( I don't understand the reasoning behind that
The fastest way for variation
He said to climb as fast as possible
The slope is always given as dz / sqrt(dx^2+dy^2), but we know from the chain rule dz = δx z dx + δy z dy. We can rewrite that as dz = grad(z) • , and sub it into the slope formula to get m=grad(z) • / || this shows that your slope will be increasing the most when dx and dy are in the direction of the gradient, because of the formula for dot product
its the slope the more slanted it is the faster you climb. Imagine a 2d graph in x and y, and say thats a representation of speed, what happens when the slope gets steeper ? It means the car is moving very fast from point a to to point b. What does it mean when the slope is not steep? well it means the car is going slowly
We can use directional derivative. Take any unit vector v. The directional derivative of f at P with respect to v is the dot product of the gradient of f and v. This is less than or equal to the magnitude of the gradient of f.
Let v be the unit vector with the same direction as the gradient of f at P. Then the directional derivative of f at P with respect to v is equal to the gradient of f.
Therefore the gradient of f at P represents the direction in which f increases the most quickly.
goated video
But then when we evaluate it at a given point what does the resultant number say? Is it the total increase in magnitude given that we've moved towards that point?
In one dimension, it is the only way how f goes. In two dimensions or more, it is only a description and an assumption about how it goes.
Ayooo jeeff bezooooozooss is that u ?
fx(f(x,y)) gives the as fast as possible changing x direction and fy(f(x,y)) gives the as fast as possible y direction and vector of these gradient vector am i correct
and what if the 3d surface is totally flat or if the surface is equally increasing or decreasing in all directions?? does it point anywhere or does it just become 0?
thx bro
Thank you. My teacher is a potato. This helped
in short: ∇ =
when did Jeff Bezos grow hair and star explaining calculus??
is gradient somewhat like google maps showing directions??
you look like jef bezos
🥇👏👏👏👏👏👏🥇
Not really. I don't get how a 2D plane gives you a direction to go.
because you have a function of x and y, there are only 2 input variables, hence 2 axis for you to go
If you call the direction on the xy-plane by some vector or line, then points on the line which get mapped onto the 3d surface will give you a path on that surface, and that is the path you take for fastest increase.
You should find better realistic explanation
Kon kon india se hai🙌🙌☺️🇮🇳🇮🇳
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The Math Fraud 🤕
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