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### Course: Multivariable calculus>Unit 5

Lesson 7: 3D divergence theorem

# 3D divergence theorem intuition

Intuition behind the Divergence Theorem in three dimensions. Created by Sal Khan.

## Want to join the conversation?

• Is there such a thing a 4D or even 5D?
• Sure, a lot of mathematics has been generalised into higher dimensions (often n dimensions, meaning you can have as many as you want/need).

We live in three spatial dimensions (space) and one temporal dimension (time): spacetime, so it's also 4D in a sense (except we can't really control our movement in the time dimension).
• So is divergence theorem the same as Gauss' theorem? Also, we have been taught in my multivariable class that Gauss' theorem only relates the Flux over a surface to the divergence over the volume it bounds and if you had for example a path in three dimensions you would apply Green's theorem and the line integral would be equivalent to the Curl of the vector field integrated over the surface it bounds - not the divergence - so I guess I don't understand how its being "generalized to more dimensions"?
• Gauss Theorem is just another name for the divergence theorem.

It relates the flux of a vector field through a surface to the divergence of vector field inside that volume. So the surface has to be closed! Otherwise the surface would not include a volume.

So you can rewrite a surface integral to a volume integral and the other way round.

On the other hand the Stokes Theorem relates a line integral along a closed loop through a vector field to the the surface integral of the curl of the vector field. Here the closed lopp has to beed the boundary of the surface over which you take to surface integral.

In other words you can turn a line integral to an surface integral and the other way round.

The Greens theorem is just a 2D version of the Stokes Theorem. Just remember Stokes theorem and set the z demension to zero and you can forget about Greens theorem :-)

So in general Stokes and Gauss are not related to each other. They are NOT the same thing in an other dimenson.
• i dont understand how the normal vector is inward facing and outward facing
• Imagine a piece of paper with a pencil stuck halfway through it. Both sides of the pencil are perpendicular to the paper. One side points to the inside, if you start from the center of the pencil. The other points to the outside. Since both sides are perpendicular to the paper, they both model a normal vector to the paper's surface.

Vectors have components that describe which way they face from the origin. So a vector with components (2, 5) would point up and to the right, into the first quadrant, while a vector with components (-2, -5), even though it has exactly the same length, would point down and to the left, into the third quadrant. If you imagine a vector that is normal to this surface, on the outside top right part of the surface like where Sal first drew the normal vector, then the first vector would be pointing out of the surface, and the second vector would be pointing into it. Both could be normal to the surface (perpendicular to the surface's tangent line).
• Is this considered Calculus 3?
• Yep. Multivariable calculus.

Pretty much every playlist after the practice AP Calc tutorials are Calc III,
• Such an intuitive video, thank you so much Sal! I have a question though, at , shouldn't it be a unit normal vector? Since if not, the magnitude of the resulting dot product would vary depending on the magnitude of the normal vector? (Don't we purely want the component of the vector field going in the perpendicular direction to the boundary, cos(theta) * |F||1| (|unit normal vector| = 1))? Thanks.
• yeah, i was thinking same. pro'lly a typo - since at he has used correct symbol i.e. hat or cap instead of arrow. Though still didnt mention the term "unit" but yeah - error, rather than intentional i'd assume.
• Can you calculate the field of g?
• is F.n dr as stated in this video the same as F.n dS?
I remember in past videos on 2-D Divergence Theorem it is stated as F.n dS.
• Thank you for clear explanation!!
And I have a question about application of div theorem.

When vector field is offered for 3 dimension(x,y,z),
I'd like to calculate 'out amount(out let)' in 2 dimension(x,y).

Namely, If given vector field is 3D, and given domain is 2D,
can I use the 2D or 3D div theorem??

I'd like to extend appreciate!!