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## Multivariable calculus

### Course: Multivariable calculus>Unit 5

Lesson 4: 2D divergence theorem

# Conceptual clarification for 2D divergence theorem

Understanding the line integral as flux through a boundary. Created by Sal Khan.

## Want to join the conversation?

• Can you generalise the concept of mass density to that of say heat density (ie temperature) for any type of vector? Does Sal just use the idea of mass density as a means of visualising what he is trying to explain?
• The scalar density function can apply to any density for any type of vector, because the basic concept is the same: density is the amount of something (be it mass, energy, number of objects, etc.) per unit of space (area, volume, etc.). Sal just used mass as an example.
• When Sal introduces the density function at , he separates it from the scalar functions M and N. In the general theory of divergence, could density be considered implicit in M and N? That is to say, did he just separate density for clarification of the concept?
• What you said is correct. The product of density times velocity in this video is just one example of how the divergence theorem can be applied in physics.
• I though the dimession of Force is ML/T^2 where M is mass, L is length and T is time?
• F doesn't represent force in this example. It's a function that represents the product of the local density of mass and the local velocity.
• If anybody is interested here is a 3D vector simulator that helps see vector fields, it's a nice complement to the material from Khan Academy. http://www.falstad.com/vector3d/
• What does this F function mean? i thought it means force
• In this case, F is a vector field. So you can kind of visualize it as at each point (x, y, z), F(x, y, z) gives a different force vector.
• Can you do a video with examples of the divergence theorem?!
• Is the divergence stated above not meant to be the partial of the j component with respect to x minus the partial of the i component with respect to y? which would mean its meant to be:

div F = d(pN)/dx - d(pM)/dy

maybe sals simplying but i just dont see it?
• what you just said is green's theorem. in the divergence theorem. it shows that the integral of [normal (on the curve s) of the vector field] around the curve s is the integral of the divergence of the vector field inside the curve. i have a feeling i just made it more complicated.
• At and onwards, isn't Sal wrong in differentiating the i component with respect to x and not with respect to y?