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The Shapes of Things

Shawn W. Walker
Publication Date: 
Number of Pages: 
Advances in Design and Control
[Reviewed by
P. N. Ruane
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This book falls within the SIAM series ‘Advances in Design and Control’, and the intended readership includes those who specialise in continuum mechanics, fluid mechanics, and associated PDEs. This is compatible with the research interests of the author, which include PDEs for fluids and moving/free boundaries, geometric evolution problems and optimal PDE control of shape.

The overall purpose of the book is to present an overview of differential geometry insofar as it underpins the theory of shape differential calculus , which is the study of change with respect to an independent ‘shape variable’. The author suggests that readers should have previous exposure to the differential geometry of curves. So, contrary to my initial expectations, this is not an introductory text on differential geometry.

The notion of ‘shape variable’ relates to the theme of domain perturbation , which is useful for the understanding mathematical models containing PDEs, such as the surface version of the standard Laplace equation. The basic idea behind shape derivative is illustrated in the following example:

If \(f = f(r,\theta)\) is a smooth function defined on a disc \(\Omega\) of radius \(R\), and if, in terms of polar coordinates, \(\mathfrak{I} = \displaystyle\int_\Omega f = \displaystyle \int_0^{2\pi}\int_0^R f(r,\theta)r\,dr\,d\theta\), then, for optimization purposes, the sensitivity of \(\mathfrak{I}\)with respect to \(R\) is obtained from: \[ \frac{d}{dR}\mathfrak{I} = \int_0^{2\pi}\left(\frac{d}{dR}\int_0^R f(r,\theta)r\,dr\right)\,d\theta = \int_0^{2\pi}\int_0^R f'(r,\theta;R)r\,dr\,d\theta + \int_0^R f(R,\theta;R)R\,d\theta.\]

Replacing a disc of radius \(R\) by a more general domain \(\Omega\), and using Cartesian coordinates \(\mathbf{x}\), the above statement reduces to: \[ \frac{d}{dR}\mathfrak{I} = \int_\Omega f'(\mathbf{x};\Omega)\,d\mathbf{x} + \int_{\partial\Omega} f(\mathbf{x};\Omega)\,dS(\mathbf{x}). \]

Further generalisation arises by regarding \(R\) as a velocity vector field \(\mathbf{V}\) that drives points on \(\partial\Omega\) and therefore instantaneously perturbs the domain .

Overall, the ensuing framework of shape differential calculus is shown to yield PDEs relating to mean curvature flow and Willmore flow, which occur in various applications such as biology and fluid dynamics.

Peter Ruane is retired from the field of mathematics education, which involved the training of primary and secondary school teachers. His postgraduate study included of algebraic topology and differential geometry, with applications to superconductivity.