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 and defined on the domain , and we would like to calculate

After this transformation, often times the integrand 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 and are two smooth functions defined on a 3D domain with boundary , then the formula for integration by parts becomes,

where is the outward unit surface normal to , and is its -th component, and ranges from to for 3D domains. Summation of this equation for yields the vector form of this equation,

where is a vector defined as

and similarly,

If the function is a vector-valued function of space, then the formula becomes This equitation was used in the derivation of the weak form of 3D BVP.

where 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.

If it happens that , then the formula becomes the first Green’s identity,

or

where is called Laplace operator or Laplacian, and is also a scalar,

The Laplace operator is also written as , for example, suppose is a continuous function defined on a 3D domain,

For more information, see the Wikipedia page on integration by parts.

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