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Taha Sochi - Introduction to Differential Geometry of Space Curves and Surfaces

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Taha Sochi Introduction to Differential Geometry of Space Curves and Surfaces
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    Introduction to Differential Geometry of Space Curves and Surfaces
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Introduction to Differential Geometry of Space Curves and Surfaces: summary, description and annotation

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This book is about differential geometry of space curves and surfaces. The formulation and presentation are largely based on a tensor calculus approach. It can be used as part of a course on tensor calculus as well as a textbook or a reference for an intermediate-level course on differential geometry of curves and surfaces. The book is furnished with extensive sets of exercises and many cross references, which are hyperlinked, to facilitate linking related concepts and sections. The book also contains a considerable number of 2D and 3D graphic illustrations to help the readers and users to visualize the ideas and understand the abstract concepts. We also provided an introductory chapter where the main concepts and techniques needed to understand the offered materials of differential geometry are outlined to make the book fairly self-contained and reduce the need for external references.

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Preface
The present book is about differential geometry of space curves andsurfaces. The formulation and presentation are largely based on atensor calculus approach, which is the dominant trend in the modernmathematical literature of this subject, rather than the geometricapproach which is usually found in some old style books. The book isprepared, to some extent, as part of tutorials about topics andapplications related to tensor calculus. It can therefore be used aspart of a course on tensor calculus as well as a textbook or areference for an intermediate-level course on differential geometry ofcurves and surfaces.
Apart from general background knowledge in anumber of mathematical branches such as calculus, geometry and algebra,an important requirement for the reader and user of this book isfamiliarity with the terminology, notation and concepts of tensorcalculus at reasonable level since many of the notations and conceptsof differential geometry in its modern style are based on tensorcalculus.
The book contains a mathematical background section in the firstchapter to outline some important pre-required mathematical issues.However, this section is restricted to materials related directly tothe contents of differential geometry of the book and hence the readerand user should not expect this mathematical background section to becomprehensive in any way. General mathematical knowledge, plus possibleconsultation of mathematical textbooks related to other disciplines ofmathematics when needed, should therefore be considered.
The book is furnished with an index in the end ofthe book as well as sets of exercises in the end of each chapter toprovide useful revisions and practice. To facilitate linking relatedconcepts and parts, and hence ensure better understanding of theprovided materials, cross referencing is used extensively throughoutthe book where these referrals are hyperlinked in the electronicversion of the book for the convenience of the ebook users. The bookalso contains a considerable number of graphic illustrations to helpthe readers and users to visualize the ideas and understand theabstract concepts.
The materials of differential geometry arestrongly interlinked and hence any text about the subject, like thepresent one, will face the problem of arranging the materials in anatural order to ensure gradual development of concepts. In this bookwe largely followed such a scheme. However, this is not always possibleand hence in some cases references are provided to materials in laterparts of the book for concepts needed in earlier parts. Nevertheless,in most cases brief definitions of the main concepts are provided inthe first chapter in anticipation of more detailed definitions andinvestigations in the subsequent chapters.
Regarding the preparation of the book, everythingis made by the author including all the graphic illustrations,indexing, typesetting, book cover, as well as overall design. In thisregard, I should acknowledge the use of LaTeX typesetting package andthe LaTeX based document preparation package LyX which facilitated thetypesetting and design of the book substantially.
Taha Sochi
London, March 2017
Table of Contents
Nomenclature
In the following table, we define some of thecommon symbols, notations and abbreviations which are used in the bookto provide easy access to the reader.
nabla differential operator
Laplacian operator
~isometric to
, subscriptpartial derivative with respect to the following index(es)
; subscriptcovariant derivative with respect to the following index(es)
1D, 2D, 3D, n Done-, two-, three-, n -dimensional
overdot (e.g. r )derivative with respect to general parameter t
prime (e.g. r )derivative with respect to natural parameter s
tabsolute derivative with respect to t
, ipartial derivative with respect to th and ith variables
adeterminant of surface covariant metric tensor
asurface covariant metric tensor
a11,a12,a22coefficients of surface covariant metric tensor
a11,a12,a22coefficients of surface contravariant metric tensor
a , a , asurface metric tensor or its components
bdeterminant of surface covariant curvature tensor
bsurface covariant curvature tensor
Bbinormal unit vector of space curve
b11,b12,b22coefficients of surface covariant curvature tensor
b , b , bsurface curvature tensor or its components
Ccurve
CB , CN , CTspherical indicatrices of curve C
Ce , Cievolute and involute curves
Cnof class n
c , c , ctensor of third fundamental form or its components
dDarboux vector
d1 , d2unit vectors in Darboux frame
detdeterminant of matrix
dslength of infinitesimal element of curve
dsB,dsN,dsTlength of line element in binormal, normal, tangent directions
darea of infinitesimal element of surface
e,f,gcoefficients of second fundamental form
E,F,Gcoefficients of first fundamental form
, , Vnumber of edges, faces and vertices of polyhedron
Ei , Ejcovariant and contravariant space basis vectors
E , Ecovariant and contravariant surface basis vectors
Eq./Eqs.Equation/Equations
ffunction
Fig./Figs.Figure/Figures
gtopological genus of closed surface
gij , gijspace metric tensor or its components
Hmean curvature
IS , IIS , IIISfirst, second and third fundamental forms
IS , IIStensors of first and second fundamental forms
iffif and only if
JJacobian of transformation between two coordinate systems
JJacobian matrix
KGaussian curvature
Kttotal curvature
Llength of curve
nnormal unit vector to surface
Nprincipal normal unit vector to curve
Ppoint
r , Rradius
Ricci curvature scalar
rposition vector
r , r1st and 2nd partial derivative of r with subscripted variables
R1 , R2principal radii of curvature
nn -dimensional space (usually Euclidean)
Rij , R ijRicci curvature tensor of 1st and 2nd kind for space
R , RRicci curvature tensor of 1st and 2nd kind for surface
RijklRiemann-Christoffel curvature tensor of 1st kind for space
RRiemann-Christoffel curvature tensor of 1st kind for surface
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