The Divergence Theorem

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Divergence Theorem. In this video, I give an example of the divergence theorem, also known as the Gauss-Green theorem, which helps us simplify surface integrals tremendously. It's, in my opinion, the most important theorem in multivariable calculus. It is also extremely useful in physics, more specifically in electromagnetism (Gauss law) and heat diffusion. Enjoy!

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Dr P:: *Uploads a video on the The Divergence Theorem*
Me: *Flashbacks to the hardest Calc 3 question*

frozenmoon
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we use it to compute area of polygons and inertias of arbitrary shapes .thanks

tomoki-vo
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I love the way you explain!! It made so much sense! Thank you!!!!

wakeupthisisntreal
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You are honestly such a great teacher. Thanks for all of your hard work!!!

joshuabrandt
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Dr. Peyam I have a question concerining your calculations. When I was learning the Divergence Theorem (or Divergenzsatz/Satz von Gauß in German) there was something different. It stated that the Integral over the volume of div(F) was equal to the integral of the surface of F DOTTED WITH THE OUTER NORMAL VECTOR. So if I were doing your problem by myself I would desperately try to find the normal vector of this surface.

Maybe someone can explain to me why it isn't necessarry here.

ridnap
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U are so passionate about math!! happy 58 k to you, my friend :D

Kdd
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Also you genuinely seem like such a likeable cool person!! Thanks for being such a positive happy teacher :)

wakeupthisisntreal
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thank you for refreshing my memory on calculus....

weiboliu
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I know it as Ostrogradsky's theorem, also my favourite as well...for flux calculations.

japotillor
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very beutifully worked out. what does a negative surface area mean? should have some physical meaning? please clarify. thanks for the same.

meiwinspoi
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Thank you! Excellent video. I have a couple questions:

6:06 I recall that the jacobian "tacks on" the r. Could you please remind me why the order of integration is "set" as dz dr d(theta), and not, for instance dr d(theta) dz. Any other sources or videos to review as to why this is would be appreciated.

9:51 what is the physical meaning of a negative surface integral = negative triple integral if the region was in the positive z?

drandrewsanchez
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Next video suggestion: banach spaces 😀

Caleepo
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Can you provide a simplified approach to Reynolds transport theorem, for your next calculus related video?

jonahellison
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Ich schreib mal auf deutsch. Verstehst du auch. :-)
Deine videos sind echt super. Hast du eigentlich mathematik studiert oder was anderes? Wenn ja wo hast du studiert?

revelationSandJ
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How do we konw when to use divergence theorem and when to use stokes thrm

ahmedmghabat
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"Just a triple integral...."

MathAdam
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Wait, does it make sense for this integral to be negative?

stevenwilson
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I think you need to change your video picture it has dx instead of dv in which the right integral should be triple integral of div(F) dv

jayjayf
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Unfortunately, i can't subscribe two times :(

wduandy