Questions on the fundamental theorems

[Reshma]

I have these two queries regarding the fundamental theorems:

1. The fundamental theorem for divergences is given by:

$$\int_v (\nabla.\vec v)d\tau = \oint_s \vec v.d\vec a$$

According to this theorem, the integral of the divergence of a vector over a volume is equal to its surface integral(i.e. the bounding surface).

Could someone interpret this geometrically taking the example of a fluid flow?

2. Fundamental theorem for curls:

$$\int_s {(\nabla\times \vec v).d\vec a = \oint_p\vec v.d\vec r$$

Here the intergral of a curl over a patch of area is equal to its primeter i.e. value of the function at the boundary.

How is this interpreted geometrically and how is the ambiguity for the direction of the line integral resolved?

 

[kanato]

1)

$$\oint_S \vec{v} \cdot d\vec{A}$$

Think about fluid flow, with v being a vector field representing the flow of some incompressible fluid, such as water. This integral is the net flow of fluid into or out through the surface integrated over.

$$\int_V \nabla \cdot \vec{v} d\tau$$

the divergence of a vector field is often though to be indicative of the sources/sinks for the field. This integral is the total amount of sources/sinks within the enclosed volume. The relationship between these two integrals tells you that if you have a net amount of fluid leaving through some closed surface, there must be some source for fluid within that surface.

You can see the source/sink relationship with the following description. In 2-D, draw a box, and imagine the box represents an area over which v is varying slowly. We can then write

$$\nabla \cdot \vec{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} = \frac{\Delta v_x}{\Delta x} + \frac{\Delta v_y}{\Delta y} = \frac{v_{right} - v_{left}}{\Delta x} + \frac{v_{top} - v_{bottom}}{\Delta y}$$

Now, suppose box is a square, so delta x and delta y are the same. Now think of v as representing the flow of something, so that v_right represents fluid flowing out to the right, and v_left represents fluid flowing in from the left. The delta v_x term is just the net amount of fluid flowing from the x direction. Similar for y.

Then if the divergence of v was zero, that would mean that any increase in v_x would be accompanied by an equivalent decrease in v_y, so that would mean that all the fluid that flowed into the box flowed out. If the divergence is positive, that means that there is some overall addition to v that is occurring, so that somehow more fluid flowed out of the box than flowed in. That could only happen if there was something inside that cell that was adding fluid.

The divergence theorem can be thought of as just looking at it as above, and taking the limit as delta x, delta y go to zero.

2)

I don't have as nice a geometric interpretation of this one. Once you get to studying magnetic fields, it might come to you after using it a while. A situation like this with line currents is also analogous to complex contour integration.

But as far as the ambiguity, there is actually two ambiguities: the direction of the line integral, and the direction of the surface normal in the area integral, so the problem can be resolved by picking a consistent convention for both of them. So the convention is to use the right hand rule. You might say that you pick a surface normal direction, and if that side of the surface the normal comes out of was the face of a clock, the integration would go counter-clockwise.

 

[Reshma]

Thank you so much for clarifying my doubts on the fundamental theorems.

 

[cepheid]

A somewhat dubious (i.e not very convincing or rigorous) geometrical interpretation of Stokes' theorem we were given (just to help us visualize it) is as follows: say you have closed curve bounding a surface. The integral of the curl of the vector field over the entire surface can be thought of as the sum of all the individual little eddies or "swirls" within that surface area. But the opposite swirls all internally cancel, so the "net curl" is just equal to the portions of those swirls at the edges of the surface, which amounts to adding up the components of the vector field parallel to the curve bounding the surface at every point (a diagram would be useful here, I know).

Notice something though...if a small portion of the vector field "swirls" counterclockwise, then in what direction is the curl of the field at that point? By convention (the right hand rule), it points out of the page. So the integral of the curl over the entire surface can still be thought of as the flux of those curl vectors through the surface, which is equal to the value of the vector field itself integrated over the entire boundary of the region (a common trend in these fundamental theorems). So moving from our "swirling fluids" type of analogy instead to magnetic fields, Ampere's law states that the line integral of of the magnetic field around a closed loop is equal to the total current enclosed by that loop, which is indeed equal to the "flux" of the curl of B through any surface bounded by that loop. I hope this helps.

 

[Reshma]

A somewhat dubious (i.e not very convincing or rigorous) geometrical interpretation of Stokes' theorem we were given (just to help us visualize it) is as follows: say you have closed curve bounding a surface. The integral of the curl of the vector field over the entire surface can be thought of as the sum of all the individual little eddies or "swirls" within that surface area. But the opposite swirls all internally cancel, so the "net curl" is just equal to the portions of those swirls at the edges of the surface, which amounts to adding up the components of the vector field parallel to the curve bounding the surface at every point (a diagram would be useful here, I know).

I've noticed a diagram as you have described in Griffith's book but his explanation wasn't satisfactory.

Notice something though...if a small portion of the vector field "swirls" counterclockwise, then in what direction is the curl of the field at that point? By convention (the right hand rule), it points out of the page. So the integral of the curl over the entire surface can still be thought of as the flux of those curl vectors through the surface, which is equal to the value of the vector field itself integrated over the entire boundary of the region (a common trend in these fundamental theorems). So moving from our "swirling fluids" type of analogy instead to magnetic fields, Ampere's law states that the line integral of of the magnetic field around a closed loop is equal to the total current enclosed by that loop, which is indeed equal to the "flux" of the curl of B through any surface bounded by that loop. I hope this helps.

Thank you so much for giving me additional insight on Stokes' theorem.

 

[aranyakam]

Thanks kanato for the nice explanation