# Computational Biomechanics

Dr. Kewei Li

## Integration by Parts

### Introduction

In Mathematics, integration by parts is often used to transform the indefinite integral of a product of functions into an indefinite integral for which a solution can be obtained more easily. For example, we have two functions $u(x)$ and $v(x)$ defined on the domain $\Omega = [a, b]$, and we would like to calculate

After this transformation, often times the integrand $u'(x) v(x)$ becomes easier to work with than the original integrand.

### Integration by Parts for Higher Dimensions

The formula for integration by parts can be extended to functions of several variables. Sometimes, it is referred as formula of integration by parts for higher dimensions. Suppose $u(x_1,x_2,x_3)$ and $v(x_1,x_2,x_3)$ are two smooth functions defined on a 3D domain $\Omega$ with boundary $\Gamma$, then the formula for integration by parts becomes,

where ${\mathbf{n}}$ is the outward unit surface normal to $\Gamma$, and $n_i$ is its $i$-th component, and $i$ ranges from $1$ to $3$ for 3D domains. Summation of this equation for $i=1,2,3$ yields the vector form of this equation,

where $\boldsymbol{\nabla} u$ is a vector defined as

and similarly,

If the function $v(x_1, x_2, x_3)$ is a vector-valued function $\mathbf{v}(x_1, x_2, x_3)$ of space, then the formula becomes This equitation was used in the derivation of the weak form of 3D BVP.

where $\boldsymbol{\nabla}\cdot\mathbf{v}$ is a scalar,

This formula can be derived by using the relationship

With that, we have,

The left-hand side of this equation can be simplified by using the divergence theorem (see also the Green’s theorem)

Thus, we obtain the final form of the equation,

which is identical to the above equation. $\qquad \blacksquare$

If it happens that $\mathbf{v}= \boldsymbol{\nabla} v$, then the formula becomes the first Green’s identity,

or

where $\boldsymbol{\nabla}^2$ is called Laplace operator or Laplacian, and $\boldsymbol{\nabla}^2 v$ is also a scalar,

The Laplace operator $\boldsymbol{\nabla}^2$ is also written as $\boldsymbol{\Delta}$, for example, suppose $f(x,y,z)$ is a continuous function defined on a 3D domain,