The Gradient Vector Field
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- Опубліковано 10 жов 2024
- One prominent example of a vector field is the Gradient Vector Field. Given any scalar, multivariable function f: R^n\to R, we can get a corresponding vector field that has a precise geometrical meaning: the vectors point in the direction of maximal increase of the function.
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1:37 "as you might recall from multivariable calculus"
me trying to use this to study for multivariable calculus: "wait a minute"...
haha oops:D Check out my multivariable playlist as well as the vector calculus playlist, some instructors bundle those into the same course.
your doing the lords work. learning this in a second language has been a grind, but you put it so eloquently im almost embarrassed to have not understood it before!
English is my first language and i STILL feel like i'm listening to a different language when people explain math lol
the lords homework lol
Such a great visualization of the gradient vector field! Thank you sir!
Greatest lectures of vector calculus 👍👍👍👍
Leaving a comment in every video of this playlist since now to help you. Good job bro keep doing it. Im shure your job will be apreciated by more people soon. Thank you.
hey I really appreciate that!
I'm really enjoying your videos with clear explanations of advanced concepts. I just finished Calc 3, in pursuit of my Engineering Degree... Seeing instructors like you enthusiastically teaching math helps me find more enjoyment and appreciation for math.
Congrats on finishing calc3 and thanks for your kind words!
Professor, you are real gem 💎.your each and every video are very conceptual and nicely explained.
You‘re explaining really good and I like the way you talk
Hats off to you sir.... your videos are very helpful to me for building an understanding on these complicated topics....thank you sir.
Glad they are helping!!
Firstly, it is a wonderful representation and thank you for that sir. I believe that 1 minute example for every video(I think you can put them between videos) would make this subjects more understandable.
you are the only one who always tells us geometrical meaning of mathematics . thanks sir from my heart.
So, I have been watching your videos for the past few months and I must say you are an amazing teacher. I have only question: Do you recommend any resources for practicing the concepts we learn in your videos, like a book or a site? Something that poses a harder challenge than, say, Thomas/Stewart Calculus, but not as difficult as Putnam & Beyond. Once again, thanks for doing this. Your videos are one of, if not the most, helpful set of lectures currently on UA-cam.
Thank you! A good text in between those you mentioned might be the one by Advanced Calculus by Folland. It does a bit of sophistication first (this playlist is meant to pair more with thomas/stewart levels) like introducing some basic topology of R^n to help prove things, but you might like it.
@@DrTrefor Thank you for your reply.
Thankyou sir , you are just amazing!
They way you explain the topics it's amazing. Love from India 🇮🇳.
Thanks so much
You're too great. I hope I had seen your videos when I was learning these topics. Too good...too good. :) Continue this work !!
Thank you, I will!
Wish you an extremely healthy and happy life dear sir and may you live long thank you for all these wonderful videos.
First video I’ve seen in ages without a single dislike. Deservedly so!
You've explained it so beatifully I can even visualize this: i.e. the curved surface area of this figure is actually the gradient of the vector field and the vector field itself is a function of x,y ( x&y as input and an output drawn in terms of arrows on the x,y plane. whereas the F( x,y) is the projection of the circle which satisfies the z direction, If the gradient of the vector field was not of the question, we might would have got a cyclinder, who's curved surface area could have been calculated by simpliy the line integral of the circle curve obtained for each F (x,y) in the z- direction. In more specific can't we, say its better to have a gradient of the vector field, but my question is whether this gradient will meet the circumference of the projected circle at infinity or, is the domain of the gradient defined ?? could you please help me with this, as F(x,y) should have a definite range depending on the x and y input, so shall we consider the figure obtained having a intesection point of the gradient line of the vector field at the circumference of the circle projected by specific input of F(x,y). Its more like obtained a figure out of the vector field without making a curve on the the x-y plane, vector field is just awesome
Thank you sir
nice video, rapid, but important :)
Sir! you should mention x and y and function with the corresponding axis.
congratulation for 100k
Thanks a lot sir 🔥🔥🔥
Thank you very much for your videos!
Excellent💯💯
This is awesome. Is it possible to give a visual example where f itself is a vector field?
Thank you brother
thanks
Thanks alot
Hey sir i have this one question thats been bugging me for like months when i first learned about vector calculus.
The question is how do i visualize a gradient vector for a function f(x,y,z)?
ik this is a simple question (some might even say lame) but if you help me i would really appreciate it thanks ❤❤
Great
Great!!!
sir, I HAVE DOUBT ON trajectory of vector field. kindly make video on trajectory of vector field.
What would the gradient look like for a function of 3 variables?
so if you want to go in the direction of steepest decline would that be the negative of the gradient?
👍
Can you share the algorithm that you used to plot the graphs.
Thanks.
These are done in MATLAB
@@DrTrefor That was not my question, but thanks.
I am cansused that you used the same function to define the Cone (x^2 +y^2 ) in the ' Describing Surfaces' video and you are using the same function for this shape ??? though the difference is not that much but the cone has corners at the bottom and this surface doesn't. Please help me understanding that.
Why is a gradient vector field always conservative and vice versa? I get why some gradient fields are conservative, like gravity, but for some reason I can't see why ALL gradient vector fields are conservative. Like if you have any vector field F, can't you find a function such that the gradient of that function gives you F? And in that case....every vector field is a gradient field? Maybe I dont understand gradients or gradient fields... :(
It is not quite if and only if in general but it does go both ways if you assume that F is sufficiently nice, I believe it needs to be smooth. I cover this more in my video on conservative fields
me too , I still have the same question and dont know the reason
Excuse me professors, is gradient vector , perpendicular on plot??
What does magnitude of gradient vector represent?
Do you have any concepts that tackle pre-calculus :0
My professors aren't cutting it for me D;
Not a whole lot, have you checked out Khan Academy yet?
@@DrTrefor I have, but they aren't covering the material for what I' covering...if anything. only some (not alot..maybe 2 topics) that are covered in class are actually labeled under their Calculus section and not pre-calc
I got confused about direction of the gradient. . Plz help
If you were standing on the side of a hill, the gradient is the direction where the hill has the steepest slope.
Can you please give code for it 🙏
He is a mathematician not a coder😂
Sorry, the correlation isn't clear
The vectors show the direction of every point in which the function increases the most. To find the incline of the curve, we use the derivative at each point to calculate the speed at which the function changes.
Sir can u send me your valuable suggestions on whatsapp
you are using too many words.
hey
could you please tell me why gradient is normal vector to surface,i have seen the video where you prove its normal to level curves, but how is it normal to the surface
Thanks!