The Gradient Operator in Vector Calculus: Directions of Fastest Change & the Directional Derivative

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But WHY is the directional derivative given by that dot product?

many thanks!!

New epsode at MathFlix! 😀

Keep it up. Awesome series

Now when I need to write gravity I can turn it into del mass instead.

Can I use the gradient to model the gradient of sand density on Arrakis?

Thank you

Thank you, Sir
I am eagerly waiting for the coming lectures.

Coming to learn new knowledge！

Always wonderful to learn from you, i tried the example of gravitational potential, quite a neat example. Thanks for the video :)

Dear Professor Brunton, I heard mathematicians treat the output of the gradient operator as a row vector (for reasons supposedly to do with differential geometry), while engineers and statisticians typically treat it as a column vector. I'm curious if there is a preference for either convention in the optimization / machine learning literature, as well as your own preference.

Thank you Steve. Enjoy your sabatical 💗

Sir, please can u explain augmented Lagrangian (penalty function)

love this series

I have a question about calculating a gradient of gravitational potetnital:
V = -(GmM)/r
I can decompose r into a form where I keep each axis separate using pythagorean theorem:
r = (x^2 + y^2 + z^2)^(1/2)
now V equals:
V = -(GmM)*(x^2 + y^2 + z^2)^(-1/2)
Now I can calculate a derivative in each direction (x, y, z):
dV/dx = -(GmM)*(-1/2)*(x^2 + y^2 + z^2)^(-3/2)*2x = x(GmM)*(x^2 + y^2 + z^2)^(-3/2) = x(GmM)/(r^3)
y and z can be calculated correspondingly. Now, what's the next step? How do I go from this vector:
∇V=[dV/dx, dV/dy, dV/dz]
to:
F = -r(mMG)/(r^3)

Another nice graphic application of Grad is that it provides the normal direction of the surface T(x,y)=constant. This idea can be very useful for the geometrical understanding of our math problems. For any surface I can write as f(x,y)=constant, I can obtain the normal of this surface at any surface point just obtaining the Grad vector (whatever the complexity of this surface is). This directly applies to plasticity theory with respect to the yield surface and the normality rule.

Really great lecture. Thank you very much!

Worst Earth ever.

Hi. So F is minus the directional derivative of V ? Excellent video.

so glad to be seeing your lecture. thank you

Hi Steve, do you have a video for "Field"? like what is field and what is scalar or vector field? Thank you very much!

How are you so good at writting backwards?

Best series I found for catching up on uni maths I forgot. Goes into enough detail whilst being short enough to fit in at work

Is the concept of grad similar to maxima and minima?

Great explanation!

May I ask where is the I,j,k in nebla (delta ) why you don't put them in the column vector like [d/dx i, d/dy j, d/dz k]

If there are two sources of heat and the bee is unlucky, it could just stay at a non optimal point

Great video! Thanx! 😊

Thank you!

Great explanations for easy intuitive understanding of PDEs. I however was wondering whether we can add time in these equations, and how?

thanks for your great explaination

Think you sir,,,,,,love from Bangladesh 🇧🇩🇧🇩

Some hints to write V = -mMG/r into cartesian coord (x,y,z) ? I have no clue where to start thx

UC Santa Cruz - Physics 1987

Is he writingevery thing mirrored???

I have an exam on this on monday. ≈50% of students fail the course, where I study. Thank you for the great explanations

Great!

I didn’t understand how the gravitational force is equal to mMG/r^3 , Isn’t suppoed to be 1/r^2

thank you sir

This is what is good about the internet for education. A good succinct lecture really gets the information across. If you want to learn we are in a golden age. I am very excited for it all.

F = -GMm/r2 (NOT r cube)

Some ants do fly.

Thank you so much

Great lecture! one thing, shouldn't the field have only "one M"? i.e., it should be the gravitational force per unit of mass

Teachings efficient and therapeutic!
Do some more on ODE Sir

Sir, you are the best I have ever known you have done this difficult subject as easy as possible for me and for my friends so you deserve a great thank 🎉🎉❤❤