MHB How do I calculate and use the different formulas of Green's Theorem?

[evinda]

Hello! (Wave)

I have a question..

There are three formulas of the Green Theorem:

• $$\oint_S (Mdx+Ndy)=\iint_R \left( \frac{\partial{N}}{\partial{x}}-\frac{\partial{M}}{\partial{y}} \right) dxdy$$
  
• $$\oint \overrightarrow{F} \cdot d \overrightarrow{R}=\iint_R \nabla \times \overrightarrow{F} \cdot d \overrightarrow A$$
  
• $$\oint_S \overrightarrow{G} \cdot \hat{n} \cdot d \sigma=\iint_R \nabla \cdot \overrightarrow{G} dxdy $$

But...Is there a difference between $F \text{ and } G$?

Also...at which case do I use each formula?? (Thinking)

 

[I like Serena]

Hello! (Wave)

I have a question..

There are three formulas of the Green Theorem:

• $$\oint_S (Mdx+Ndy)=\iint_R \left( \frac{\partial{N}}{\partial{x}}-\frac{\partial{M}}{\partial{y}} \right) dxdy$$
  
• $$\oint \overrightarrow{F} \cdot d \overrightarrow{R}=\iint_R \nabla \times \overrightarrow{F} \cdot d \overrightarrow A$$
  
• $$\oint_S \overrightarrow{G} \cdot \hat{n} \cdot d \sigma=\iint_R \nabla \cdot \overrightarrow{G} dxdy $$

But...Is there a difference between $F \text{ and } G$?

Also...at which case do I use each formula?? (Thinking)

Hey! (Blush)

They are all identical.
The first form is the same as the second form - merely written out in its components.
The third form is the same, but applied to a perpendicular vector.

When to use them?
Whatever is closest to the problem that you are trying to solve.
If you're dealing with a path integral (e.g. work), you'd use the first or second form.
If you're dealing with a flux, you'd use the third form.

And if you're working in 3 dimensions, you'd use either Stokes' theorem or Gauss's theorem, which correspond to these forms. (Inlove)

 

[evinda]

Hey! (Blush)

They are all identical.
The first form is the same as the second form - merely written out in its components.
The third form is the same, but applied to a perpendicular vector.

When to use them?
Whatever is closest to the problem that you are trying to solve.
If you're dealing with a path integral (e.g. work), you'd use the first or second form.
If you're dealing with a flux, you'd use the third form.

A ok...I understand!

And if you're working in 3 dimensions, you'd use either Stokes' theorem or Gauss's theorem, which correspond to these forms. (Inlove)

So,do we use the Stokes' theorem and the Gauss's theorem also,in order to find the work and the flux,respectively, with the only difference that we have 3 dimensions?

 

[I like Serena]

So,do we use the Stokes' theorem and the Gauss's theorem also,in order to find the work and the flux,respectively, with the only difference that we have 3 dimensions?

Yep! (Nod)

More specifically Stokes' theorem is about the work along a closed curve in 3 dimensions, while Green's theorem is about a closed curve in 2 dimensions.

And Gauss's theorem is about the flux of a closed surface in 3 dimensions, while Green's theorem is about the flux of a closed curve in 2 dimensions. (Nerd)

 

[evinda]

Yep! (Nod)

More specifically Stokes' theorem is about the work along a closed curve in 3 dimensions, while Green's theorem is about a closed curve in 2 dimensions.

And Gauss's theorem is about the flux of a closed surface in 3 dimensions, while Green's theorem is about the flux of a closed curve in 2 dimensions. (Nerd)

Nice!Thank you very much!

 

[evinda]

Yep! (Nod)

More specifically Stokes' theorem is about the work along a closed curve in 3 dimensions, while Green's theorem is about a closed curve in 2 dimensions.

And Gauss's theorem is about the flux of a closed surface in 3 dimensions, while Green's theorem is about the flux of a closed curve in 2 dimensions. (Nerd)

And something else... At the formula:

$$\oint_S \overrightarrow{G} \cdot \hat{n} \cdot d \sigma=\iint_R \nabla \cdot \overrightarrow{G} dxdy $$

how can I calculate $\hat{n}$ and how $d \sigma$ ? Because I haven't found anything in my notes... (Worried)(Worried)(Worried)

 

[I like Serena]

And something else... At the formula:

$$\oint_S \overrightarrow{G} \cdot \hat{n} \cdot d \sigma=\iint_R \nabla \cdot \overrightarrow{G} dxdy $$

how can I calculate $\hat{n}$ and how $d \sigma$ ? Because I haven't found anything in my notes... (Worried)(Worried)(Worried)

It works almost the same as it would for $\oint_S \overrightarrow{F} \cdot \overrightarrow{dR}$. (Nod)

Let's take a look at an example.
Suppose we integrate along the unit circle.

Then:
$$\overrightarrow R = \hat \imath \cos \theta + \hat \jmath \sin \theta$$
$$\overrightarrow{dR} = (-\hat \imath \sin\theta + \hat \jmath \cos \theta) d\theta$$
This is an infinitesimal vector tangential to the unit circle.
What we get from this, is that:
$$\oint_S \overrightarrow{F} \cdot \overrightarrow{dR} = \int_0^{2\pi} \overrightarrow{F} \cdot (-\hat \imath \sin\theta + \hat \jmath \cos \theta) d\theta$$
(Mmm)To address your question, the normalized vector perpendicular to $\overrightarrow{dR}$ is:
$$\hat n = \hat \imath \cos \theta + \hat \jmath \sin \theta$$
This is a radial vector that is perpendicular to the unit circle.

The corresponding infinitesimal vector perpendicular to the curve is:
$$\hat n d\sigma = \hat n d\theta = (\hat \imath \cos \theta + \hat \jmath \sin \theta)d\theta$$
So:
$$\oint_S \overrightarrow{G} \cdot \hat n d\sigma = \int_0^{2\pi} \overrightarrow{G} \cdot (\hat \imath \cos \theta + \hat \jmath \sin \theta)d\theta$$
(Wait)