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

University of Oxford Mathematician Dr Tom Crawford explains the gradient vector (Grad) and the divergence (Div) for scalar and vector functions. Test yourself with this accompanying FREE worksheet in Maple Learn: learn.maplesoft.com/doc/ipube...
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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.
Full playlist: • Oxford Calculus
Finding critical points for functions of several variables: • Oxford Calculus: Findi...
Classifying critical points using the method of the discriminant: • Oxford Calculus: Class...
Partial differentiation explained: • Oxford Calculus: Parti...
Second order linear differential equations: • Oxford Mathematics Ope...
Integrating factors explained: • Oxford Calculus: Integ...
Solving simple PDEs: • Oxford Calculus: Solvi...
Jacobians explained: • Oxford Calculus: Jacob...
Separation of variables integration technique explained: • Oxford Calculus: Separ...
Solving homogeneous first order differential equations: • Oxford Calculus: Solvi...
Taylor’s Theorem explained with examples and derivation: • Oxford Calculus: Taylo...
Heat Equation derivation: • Oxford Calculus: Heat ...
Separable Solutions to PDEs: • Oxford Calculus: Separ...
How to solve the Heat Equation: • Oxford Calculus: How t...
Fourier Series Derivation: • Oxford Calculus: Fouri...
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Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
For more maths content check out Tom's website tomrocksmaths.com/
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[@TomRocksMaths]

Test your knowledge of Grad and Div with this accompanying FREE worksheet in Maple Learn: learn.maplesoft.com/doc/ipubec5cip/trm-grad-and-div-worksheet

[@davidrakic9086]

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

[@arts5852]

That's right definitely

[@larzcaetano]

Thanks for the video, Dr Tom!

[@josevellezcaldas]

High-quality content, thank you!

[@mpalin11]

Nice refresher 👍

[@kramlyn2412]

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

[@duxxgd2733]

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

[@fierydino9402]

Thank you!

[@Metanoia4417]

Very precise description 😇

[@johnchristian5027]

Nice video!

[@saeadabdoli]

Great video. Thank you

[@myounuskamran2485]

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

[@inkarathasaninkarathas6858]

It's clear now❤

[@TrongNguyen-sz4wr]

Thanks so much, it very helpful for me🎉

[@justaperson4952]

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

[@TomRocksMaths]

My pleasure :)

[@bilalabbad7954]

Thank you 🎉🎉🎉

[@aamirismail4701]

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

[@doaamohamedz9z9z9]

Thanks 💚

[@arts5852]

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

[@eisbar2polar246]

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😅)

[@SkatersCH]

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

[@harith9467]

Any utube channel do u want to recommend bro pls😢i also will be taking Em next sem

[@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.

[@vansf3433]

It's in Calculus 3 in the US

[@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? 🤔

[@minhuchoangtrong1049]

nice

[@Catalinawolff]

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

[@florianzellmer8735]

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.

[@mrmartin988]

Del is not strictly a vector, but rather a vector operator. It doesn't follow all of the rules of vectors (it violates commutativity i.e. Del.V != V.Del) The expression of Del as a vector is a useful shorthand. As for why texts don't treat it with formal vector notation? I'm not sure. I suspect it's either related to it not being a formal vector, or simply because the nabla symbol unambiguously refers to Del (I'm not aware of the symbol being used anywhere else in maths).

[@A-Karim]

you are my hero, thanks.

[@emilywong6923]

find curl

[@DJsTeLF]

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

[@deltalima6703]

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?

[@feilou8143]

Think about it this way: The tangent plane of a function z= f(x,y) (the function can be represented in 3D) at a point (x0,y0,z0) is given by: z-z0 = fx(x0,y0)*(x-x0) + fy(x0,y0)(y-y0) with fx being the partial derivative with respect to x and fy being the partial derivative with respect to y. We can rewrite the tangent plane equation to: 0 = fx(x0,y0)*(x-x0) + fy(x0,y0)(y-y0) -z-z0 Which is still a plane equation. Just to prove that, remember that fx(x0,y0) and fy(x0,y0) are just numbers, so multiplying them with the numbers x0 and y0 also gives numbers. We can afterwards add these numbers altogether and also include z0 to get: 0 = fx(x0,y0)*x + fy(x0,y0)*y -z +d with d being the summed numbers. Let's take fx(x0,y0) as a and fy(x0,y0) as b and we get: d = a*x + b*y -z This should be a known plane equation (so a plane can have 3 variables). Going back to the previous equation: 0 = fx(x0,y0)*(x-x0) + fy(x0,y0)(y-y0) -z-z0 We know that we can write a plane equation with respect to the normal vector of it... and the normal vector of that plane is what? (fx, fy, -1) also known as the gradient of the function F(x,y,z) = z- f(x,y) Thus, making the 3D-graph shown in the video the actual 3D-graph of the function F(x,y,z)=0 (iff F(x,y,z)= 0... if this is still confusing think of the 0 as setting z to a constant value).

[@deltalima6703]

Jesus. I have to retype that as dx = derivative with respect to x and f(x) as function with respect to x. As it is, its a mess and I am too tired to keep it straight at the moment

[@prakharjaiswal3890]

Div(grad.f)=0

[@cuteandfunny9154]

strongest anglo saxon lmao

[@Cdictator]

A math genius??

[@mmelmiri4625]

you yap way too much for a grad and div vid