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integration by part in high dimension

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let's look at a question first.

how can we derive (2.10) from (2.9)?

Give the definition of integration by part in high dimension from wiki first.

{[from https://en.wikipedia.org/wiki/Integration_by_parts]

Higher dimensions[edit]

The formula for integration by parts can be extended to functions of several variables. Instead of an interval one needs to integrate over an n-dimensional set. Also, one replaces the derivative with apartial derivative.

\int_\Omega \varphi\, \operatorname{div}\, \vec v \; \mathrm d V = \int_{\partial \Omega} \varphi\, \vec v \cdot \mathrm d \vec S - \int_\Omega \vec v\cdot \operatorname{grad}\, \varphi \; \mathrm dV

.

More specifically, suppose Ω is an open bounded subset of ℝⁿ with a piecewise smooth boundary Γ. If u and v are two continuously differentiable functions on the closure of Ω, then the formula for integration by parts is

\int_{\Omega} \frac{\partial u}{\partial x_i} v \,d\Omega = \int_{\Gamma} u v \, \hat\nu_i \,d\Gamma - \int_{\Omega} u \frac{\partial v}{\partial x_i} \, d\Omega,

where $\hat{\mathbf{\nu}}$ is the outward unit surface normal to Γ, $\hat\nu_i$ is its i-th component, and i ranges from 1 to n.

Replacing v in the above formula with v_i and summing over i gives the vector formula

\int_{\Omega} \nabla u \cdot \mathbf{v}\, d\Omega = \int_{\Gamma} u (\mathbf{v}\cdot \hat{\nu})\, d\Gamma - \int_\Omega u\, \nabla\cdot\mathbf{v}\, d\Omega,

where v is a vector-valued function with components v₁, ..., v_n.

Setting u equal to the constant function 1 in the above formula gives the divergence theorem

\int_{\Gamma} \mathbf{v} \cdot \hat{\nu}\, d\Gamma = \int_\Omega \nabla\cdot\mathbf{v}\, d\Omega.

For $\mathbf{v}=\nabla v$ where $v\in C^2(\bar{\Omega})$ , one gets

\int_{\Omega} \nabla u \cdot \nabla v\, d\Omega = \int_{\Gamma} u\, \nabla v\cdot\hat{\nu}\, d\Gamma - \int_\Omega u\, \nabla^2 v\, d\Omega,

which is the first Green's identity.

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Then we give the relationship between the gradient and directional derivative:

{[from math guidebook for graduate entrance examination]

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At the end, the whole derivation process will be shown:

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淘宝虚拟试衣在哪

淘宝虚拟试衣在哪

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