The Gradient Vector Field

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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[robertoberidojr.]

It's so amazing that we can all get this much of information without paying a single penny

[BraveGisgo]

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!

[sachidb]

1:37 "as you might recall from multivariable calculus"

me trying to use this to study for multivariable calculus: "wait a minute"...

[proudaojiao]

Such a great visualization of the gradient vector field! Thank you sir!

[TheFpsPlayer]

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.

[justicelowman]

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.

[scholar-mjom]

Greatest lectures of vector calculus 👍👍👍👍

[domibrandt]

You‘re explaining really good and I like the way you talk

[santoshkharel]

Professor, you are real gem 💎.your each and every video are very conceptual and nicely explained.

[mohdalshad]

you are the only one who always tells us geometrical meaning of mathematics . thanks sir from my heart.

[yutalas]

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.

[arandomghost]

Hats off to you sir.... your videos are very helpful to me for building an understanding on these complicated topics....thank you sir.

[Soji_Jacob]

You're too great. I hope I had seen your videos when I was learning these topics. Too good...too good. :) Continue this work !!

[Rajankumar-ukoq]

Thankyou sir , you are just amazing!
They way you explain the topics it's amazing. Love from India 🇮🇳.

[manujakirinde]

Wish you an extremely healthy and happy life dear sir and may you live long thank you for all these wonderful videos.

[mihirrao]

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 YouTube.

[griffinbur]

First video I’ve seen in ages without a single dislike. Deservedly so!

[sachinsahani]

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

[PinakiSwain]

This is awesome. Is it possible to give a visual example where f itself is a vector field?

[Fireflyy]

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.