Divergence theorem example 1 | Divergence theorem | Multivariable Calculus | Khan Academy

Example of calculating the flux across a surface by using the Divergence Theorem

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Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.

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[aecovarru]

I love this guy, and I'm not afraid to say it

[jisyang]

You don't need to integrate. At 2:30, div F turns out to be an odd function of x, and the shape of volume integral is symmetric upon yz plane. So the whole integration should cancel out to zero.

[ujjawalbaranwal]

Love from India. ....I am ready for coming in life of mathematics. ....really amazing solution with clear 3d images ....

[henrygomez]

One day, entire University classes will be just a series of links to channels like this.
Feeling a little ambivalent about it.
Thanks for the time and effort in making this channel what it is.

[numericbin]

And in the end, you say "all for naught"
Nice explanation !

[juanholguin]

I dont understand a damn thing but be assure i will become a fan of all this videos, ill watch them every chance i get, im tired of most utube videos, so many negative comments and anger between people.

[Mra]

, really you are a great teacher, im studying advanced fluid mechanics and this is so helpful thank you....

[eketony]

Thanks for this video! I have gone ahead to download Khan academy app to help me whenever I want to make references.💓

[purefrlyfe]

haha, that's awesome, i saw the original function and almost poo'd myself, then the divergence turned out to be laughable xD

[JFFisTheDude]

Although x is bounded -1 to 1, the other variables are bounded in relation to another (z to x and y to z). So say x is -1, then z is 0, and if x is zero, then z can be any number 0 to 1. You can find that for each value of x there is a certain range of z's and a certain range of y's.

[jasonclake]

The F vector equation isn't defined at y =0 for the j component due to ln(y)

[yli]

Strictly speaking, the edges of this shape is not differentiable, so the divergence theorem does not directly apply. But the edges are of measure 0, so it doesn't affect the integration.

[wunboonail]

Nice example. int(a, -a) f(x).dx = 0 is an odd function

[johnnyr]

At 5:50 I'm confused as to why he didn't integrate the 2x with respect to z. I know you disregard it as a constant when you are differentiating with respect to z, but when integrating wouldn't you take it into consideration which would make it 2xz???

[MileeWilson]

He forgot to put the Normal vector in the Flux equation at the start right?

[leotang]

where is the n unit normal vector here?

[asexualnerds]

I don't understand why x is just bounded by -1 to 1. That is only true when z is 0. At any other time it is bounded by a parabolic function, just like z is. Am I wrong?

[natnaelberhanu-iw]

How come our bounds for x range from -1 to 1?

[bellanursyahputri]

how If I integrate against z first?
So I'll substitute z = 1-x ^ 2 with the curve equation. the final result is same. it is wrong or not?

[SpaghettiEnterprises]

Whenever I get 0 as an answer I just assume I've done something wrong and try again. It makes sense conceptually but as a student it is very annoying.