Integration by parts, more than one variable, and green's identities etc.

In summary, integration by parts can be used to solve PDEs involving multiple variables, and it can be related to Green's theorem through the use of surface integrals.
  • #1
ericm1234
73
2
I was wondering if someone can show me or point me to a worked out example using integration by parts for more than one variable (as used in relation to pde's, for example). While I took pdes and calc 3, its been awhile and I don't know if I ever understood how to work out a concrete example regardless. When I look on wikipedia,say, they reference replacing certain parts using 'green's identities'; but I don't understand green's identities either. SO in conclusion I can not find a good concrete example for doing this intergration by parts involving more than one variable and relating it to all that green stuff.
 
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  • #2
Integration by parts can be used to solve partial differential equations (PDEs). To illustrate this, consider the PDE $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$Using integration by parts, we can rewrite this equation as:$\int \frac{\partial^2 u}{\partial x^2} \, dx + \int \frac{\partial^2 u}{\partial y^2} \, dy = 0$Now, let $v = \frac{\partial u}{\partial x}$ and $w = \frac{\partial u}{\partial y}$. By the product rule, we have$\frac{\partial^2 u}{\partial x^2} = \frac{\partial v}{\partial x}$and $\frac{\partial^2 u}{\partial y^2} = \frac{\partial w}{\partial y}$Using integration by parts, we can rewrite the equation as:$\int \frac{\partial v}{\partial x} \, dx + \int \frac{\partial w}{\partial y} \, dy = 0$By Green's theorem, we can rewrite the integrals on the left-hand side as surface integrals. Specifically, we have$\iint_S \left(\frac{\partial v}{\partial x} + \frac{\partial w}{\partial y}\right) \, dS = 0$where $S$ is the surface enclosed by the boundary of the domain. Finally, since $\frac{\partial v}{\partial x} + \frac{\partial w}{\partial y} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$, we can write the PDE in the form$\iint_S \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) \, dS = 0$which is a form of the PDE that can be solved using the techniques of boundary value problems.
 

1. What is integration by parts?

Integration by parts is a method used in calculus to integrate the product of two functions. It follows the formula: ∫u dv = uv - ∫v du, where u and v are the two functions being multiplied together and du and dv are their respective derivatives.

2. How is integration by parts used in multiple variables?

In multiple variable calculus, integration by parts is used to integrate functions with more than one variable. It follows a similar formula as in single variable calculus, but with partial derivatives and multiple variables.

3. What are some common applications of integration by parts?

Integration by parts is commonly used in various fields of science and engineering, such as physics, chemistry, and engineering. It can be used to solve problems involving areas, volumes, and other physical quantities.

4. What are Green's identities?

Green's identities are a set of equations in vector calculus that relate the surface integral of a vector function to the volume integral of its divergence and the line integral of its curl. They are used to solve boundary value problems in various fields of science and engineering.

5. How are Green's identities related to integration by parts?

Green's identities involve integration by parts and are used to simplify the process of solving certain types of integrals, especially those involving multiple variables. They provide a more general approach to integration by parts and can be used in a variety of applications.

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