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

This video introduces the gradient operator from vector calculus, which takes a scalar field (like the temperature distribution in a room) and returns a vector field with the direction of fastest change in the temperature at every point. The gradient is a fundamental building block in vector calculus and it is also used more broadly in optimization and machine learning algorithms, for example in gradient descent and stochastic gradient descent. We also discuss the directional derivative.

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This video was produced at the University of Washington

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0:00 Introduction & Overview
5:02 Example: Temperature Gradient
7:30 The Directional Derivative
10:09 Example: Gravitational Potential Field

[Bruh-vpqf]

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

[pablocb]

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.

[xenofurmi]

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.

[TheGamshid]

Sir, You are the Best, truly intuitive. Much appreciation!

[matrub]

Correct me if i'm wrong, but i think the radius vector is pointed from the center of the Earth to the mass. This way the minus sign in the equation of force make sense.

[shubhamkamble]

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

[rodionraskolnikov]

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

[suttikoonkoonkor]

At 12:05, shouldn't vector r point outwards from the centre of the Earth? So the gravitational force has the direction towards the centre given the minus sign in the equation.

[manoj_]

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

[mariovrpereira]

so glad to be seeing your lecture. thank you

[twisthz]

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.

[KamrulHasan-dpjk]

Please make a video on double curl of a vector field on a cartesian point.

[mar-a-lagofbibug]

Its the small details added that fill in the missing knowledge gaps for beginners.
Why are the vector symbol over the grad removed? I needed to know that!!!
Thanks teacher.

[yahiabousseloub]

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

[sarasabag]

Really great lecture. Thank you very much!

[Tyokok]

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

[filu]

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)

[rm_sm]

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

[gyorgyo]

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

[PingSharp]

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