# How Do You Apply Green's Formula with a Scaled Dimension?

• Somefantastik
In summary: This means that we can handle the scaling in one dimension by simply adding it to the other term on the boundary.In summary, when dealing with an integral involving a scaled second derivative term, we can apply integration by parts to expand the term into multiple components. Then, we can handle the scaling in one dimension by adding it to the other term on the boundary.
Somefantastik
for

$$\Delta w = \frac{ \partial^{2} w }{x_{1}^{2}} + \frac{ \partial^{2} w }{x_{2}^{2}}$$

and $$\nabla = \left(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial x_{2}}\right)$$

$$\int_{\Omega} \nabla v \nabla w d \Omega = \int_{\Gamma} v \frac{\partial w}{\partial n} d \Gamma - \int_{\Omega}v \Delta w d \Omega$$

where $$\Gamma$$ is the boundary of $$\Omega$$

so if I have

$$- \int_{\Omega}v \Delta w d \Omega$$

I can apply Green's thm to get

$$- \int_{\Omega}v \Delta w d \Omega = \int_{\Omega} \nabla v \nabla w d \Omega - \int_{\Gamma} v \frac{\partial w}{\partial n} d \Gamma$$

But what if I'm starting with

$$\int_{\Omega} v \left( \frac{\partial^{2} w }{\partial x_{1}^{2}} + k \frac{\partial^{2} w }{\partial x_{2}^{2}} \right) d \Omega$$

where k is some scalar? I'm thrown off by only one of the dimensions being scaled.

it is important to approach problems with a systematic and analytical mindset. In this case, we can break down the given equation into smaller components to better understand how to handle the scaling in one dimension.

First, let's consider the integral of the second derivative term with respect to x1. We can rewrite this term as:

\int_{\Omega} v \frac{\partial^{2} w }{\partial x_{1}^{2}} d \Omega

Using integration by parts, we can expand this term as:

\int_{\Omega} \frac{\partial v}{\partial x_{1}} \frac{\partial w}{\partial x_{1}} d \Omega - \int_{\Gamma} v \frac{\partial w}{\partial x_{1}} \frac{\partial n}{\partial x_{1}} d \Gamma

where n is the unit outward normal vector on the boundary, and we have assumed that v and w are smooth enough for these derivatives to exist.

Similarly, we can rewrite the second derivative term with respect to x2 as:

\int_{\Omega} k v \frac{\partial^{2} w }{\partial x_{2}^{2}} d \Omega

Using integration by parts again, we get:

\int_{\Omega} k \frac{\partial v}{\partial x_{2}} \frac{\partial w}{\partial x_{2}} d \Omega - \int_{\Gamma} k v \frac{\partial w}{\partial x_{2}} \frac{\partial n}{\partial x_{2}} d \Gamma

Now, putting these two terms together, we can rewrite the original equation as:

\int_{\Omega} \left( \frac{\partial v}{\partial x_{1}} \frac{\partial w}{\partial x_{1}} + k \frac{\partial v}{\partial x_{2}} \frac{\partial w}{\partial x_{2}} \right) d \Omega - \int_{\Gamma} \left( v \frac{\partial w}{\partial x_{1}} \frac{\partial n}{\partial x_{1}} + k v \frac{\partial w}{\partial x_{2}} \frac{\partial n}{\partial x_{2}} \right) d \Gamma

Notice that the scaling factor k now appears in both terms on the boundary

## 1. What is Green's Formula in 2D?

Green's Formula in 2D is a mathematical theorem that relates line integrals around a closed curve to double integrals over the region enclosed by the curve. It is also known as the Green's Theorem and is named after the mathematician George Green.

## 2. How is Green's Formula used in 2D?

Green's Formula is used to solve problems involving line integrals and double integrals in two dimensions. It helps in calculating the total flux, work, and circulation of a vector field over a region bounded by a curve.

## 3. What are the conditions for applying Green's Formula in 2D?

To apply Green's Formula in 2D, the region must be simply connected, meaning that there are no holes or gaps in the region. The curve must also be closed and piecewise smooth, meaning that it can be broken down into a finite number of smooth curves.

## 4. What are the advantages of using Green's Formula in 2D?

Green's Formula in 2D provides a useful tool for solving problems involving line integrals and double integrals. It reduces the complexity of the calculations and allows for the use of simpler integration techniques such as the Fundamental Theorem of Calculus.

## 5. What are the real-life applications of Green's Formula in 2D?

Green's Formula in 2D has many applications in different fields of science and engineering. It is used in fluid mechanics to calculate the flow of fluids in a region, in electromagnetism to calculate the electric and magnetic fields, and in economics to model consumer demand and producer supply curves.

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