integration by part in high dimension

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.

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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

where  is the outward unit surface normal to Γ,  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

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

For  where , one gets

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: