Oxford Calculus: Gradient (Grad) and Divergence (Div) Explained

Check your working using the Maple Calculator App – available for free on Google Play and the App Store.

We begin with the formal definition of the gradient vector (Grad) and a visualisation of what it represents for a multivariable function. We then look at some examples with explicit calculation and 3D plots.

The Divergence (Div) of a vector function is then introduced - both as an equation and via the physical interpretation of what it represents. We calculate the divergence for several vector fields and then show where the notation 'Grad Dot F' comes from with a derivation.

Finally, the link between Grad, Div and the Laplacian is explored.

Don’t forget to check out the other videos in the ‘Oxford Calculus’ series – all links below.










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[davidrakic]

Hands down, you're the coolest university professor in the world. Keep up the amazing work. I love this channel.

[Numberkeijidefender]

My sister got an autograph from u on her calculator today!!! Tyyy

[kramlyn]

It is way way too clear thank you tomrocks pls do more of this one!
Edit:you already unfolding what we so called Spacetime and it's nature

[myounuskamran]

Please upload more videos on more subjects. I am graduating in Physics and I have know Maths very well. Trying. By the way thank for being so usefull

[duxxgd]

great video I love listening to you this is math ASMR!

[eisbarpolar]

Hey @Tom Rocks Maths
I'm in 11th grade and I got the grade one in math.
At present I started to train Math at a university level.
How good should i get to have a good chance of getting into oxford university?🤔
(My english is not the best but I'm trying to become better😅)

[arts]

Would you also tell us about the gradient of a vector function?

[florianzellmer]

Hey Tom,
I am currently studying this kind of stuff in uni. All of my lecturers and tutors are very keen on clear notation of what is a scalar vs what is a vector. So I am wondering, is there a specific reason, you are not underlining the gradient operator (thus notating it is a vector)? I got used to doing it this way, but looking at basically any literature, I am always wondering why noone does that.

[aamirismail]

Hi tom to be honest you are coolest teacher of this era and the main thing the way you said the English is very lovely and nice format of teaching so iam very glad to find you by chance through you tube so please let me know how can i check my answer of your work sheet of lecture Gradient and Divergence if i solve by my self and telly my answer from where and once again thanks a lot to share your knowledge with every one

[Catalinawolff]

Are There free Solutions to the Maple Papier too ? Thanks for explaining ❤

[TrongNguyen-szwr]

Thanks so much, it very helpful for me🎉

[SkatersCH]

Just started advance e&m course at my college, this is fucking amazing !!!!

[bboywaters]

Thanks for the great explanations. But I think you meant the Electric field from electric charges as sources or sinks, not the Magnetic field. The Divergence of magnetic fields is zero because they form closed loops. They can hence be described by curl instead of divergence.

[pneptun]

I'm a bit confused. In the examples the div is 2 (or -2) EVERYWHERE, at every point. However, clearly the point [0, 0] is special. It's the true sink or source in these examples. How can one see that fact from the div if it's constant throughout? 🤔

[deltalima]

If I understand this correctly, that plane-in-3D grad graph is highly misleading. F(x, y, z) has to be 4D since x, y, z, and f(x, y, z) each need thier own dimensions. Then you can take partial. F(x, y, z) with respect to just dz or dy or dx. Otherwise it looks like you are taking dz with respect to dx, dz with respect to dy, and there is nothing but confusion about what to take with respect to dz.

Do I have that right?

[DJsTeLF]

I think a bigger portion of your audience would understand this better if you worked with a modern pro in 3d visuals

[mmelmiri]

you yap way too much for a grad and div vid