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- Differential Geometry and its Applications
- Differential Geometry: Handwritten Notes
- An Introduction to Differential Geometry with Applications to Elasticity
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Differential Geometry and its Applications
It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , [a] which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts.
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss ' Theorema Egregium remarkable theorem that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space.
This implies that surfaces can be studied intrinsically , that is as stand alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate non-Euclidean geometries can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry also known as combinatorial geometry , etc. Often developed with the aim to model the physical world, geometry has applications to almost all sciences , and also to art , architecture , and other activities that are related to graphics.
For example, methods of algebraic geometry are fundamental for Wiles's proof of Fermat's Last Theorem , a problem that was stated in terms of elementary arithmetic , and remainded unsolved for several centuries. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum.
These geometric procedures anticipated the Oxford Calculators , including the mean speed theorem , by 14 centuries. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' theorem.
Around BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time,  introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.
Indian mathematicians also made many important contributions in geometry. The Satapatha Brahmana 3rd century BC contains rules for ritual geometric constructions that are similar to the Sulba Sutras. They contain lists of Pythagorean triples ,  which are particular cases of Diophantine equations. The Bakhshali manuscript also "employs a decimal place value system with a dot for zero. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" including cube roots, fractions, ratio and proportion, and barter and "practical mathematics" including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain.
Chapter 12 also included a formula for the area of a cyclic quadrilateral a generalization of Heron's formula , as well as a complete description of rational triangles i. In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.
In the early 17th century, there were two important developments in geometry. Two developments in geometry in the 19th century changed the way it had been studied previously. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.
The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements ,  one of the most influential books ever written. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry.
Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of ways, including Euclid's definition as 'that which has no part'  and through the use of algebra or nested sets. However, there has been some study of geometry without reference to points. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself".
For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation ,  but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. A plane is a flat, two-dimensional surface that extends infinitely far. For instance, planes can be studied as a topological surface without reference to distances or angles;  it can be studied as an affine space , where collinearity and ratios can be studied but not distances;  it can be studied as the complex plane using techniques of complex analysis ;  and so on.
Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. In differential geometry and calculus , the angles between plane curves or space curves or surfaces can be calculated using the derivative.
A curve is a 1-dimensional object that may be straight like a line or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves. In topology, a curve is defined by a function from an interval of the real numbers to another space.
A surface is a two-dimensional object, such as a sphere or paraboloid. In algebraic geometry, surfaces are described by polynomial equations. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory.
Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem.
Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral  or the Lebesgue integral. The concept of length or distance can be generalized, leading to the idea of metrics. Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity.
In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning a size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations.
Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.
Where the traditional geometry allowed dimensions 1 a line , 2 a plane and 3 our ambient world conceived of as three-dimensional space , mathematicians and physicists have used higher dimensions for nearly two centuries. For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension Hilbert spaces , for example and positive real numbers in fractal geometry.
The theme of symmetry in geometry is nearly as old as the science of geometry itself. Escher , and others. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is.
A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem.
Euclidean geometry is geometry in its classical sense. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. In particular, differential geometry is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved.
Euclidean geometry was not the only historical form of geometry studied. Spherical geometry has long been used by astronomers, astrologers, and navigators. Immanuel Kant argued that there is only one, absolute , geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori. Riemann's new idea of space proved crucial in Albert Einstein 's general relativity theory.
Riemannian geometry , which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry. Topology is the field concerned with the properties of continuous mappings ,  and can be considered a generalization of Euclidean geometry. The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry , in which transformations are homeomorphisms.
Subfields of topology include geometric topology , differential topology , algebraic topology and general topology. The field of algebraic geometry developed from the Cartesian geometry of co-ordinates. One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry.
In general, algebraic geometry studies geometry through the use of concepts in commutative algebra such as multivariate polynomials. Complex geometry studies the nature of geometric structures modelled on, or arising out of, the complex plane. Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces.
Contemporary treatment of complex geometry began with the work of Jean-Pierre Serre , who introduced the concept of sheaves to the subject, and illuminated the relations between complex geometry and algebraic geometry.
Special examples of spaces studied in complex geometry include Riemann surfaces, and Calabi-Yau manifolds , and these spaces find uses in string theory. In particular, worldsheets of strings are modelled by Riemann surfaces, and superstring theory predicts that the extra 6 dimensions of 10 dimensional spacetime may be modelled by Calabi-Yau manifolds.
Discrete geometry is a subject that has close connections with convex geometry. Examples include the study of sphere packings , triangulations , the Kneser-Poulsen conjecture, etc. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects.
Important problems historically have included the travelling salesman problem , minimum spanning trees , hidden-line removal , and linear programming.
Although being a young area of geometry, it has many applications in computer vision , image processing , computer-aided design , medical imaging , etc. Geometric group theory uses large-scale geometric techniques to study finitely generated groups. Geometric group theory often revolves around the Cayley graph , which is a geometric representation of a group. Other important topics include quasi-isometries , Gromov-hyperbolic groups , and right angled Artin groups.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics. Convex geometry dates back to antiquity.
The isoperimetric problem , a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato , Euclid , and later Kepler and Coxeter all studied convex polytopes and their properties.
Differential Geometry: Handwritten Notes
It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , [a] which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss ' Theorema Egregium remarkable theorem that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space. This implies that surfaces can be studied intrinsically , that is as stand alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate non-Euclidean geometries can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
This concise guide to the differential geometry of curves and surfaces can be estimates of the Riemannian curvature and diameter, the solution to Denote by R(P,γ) the radius of C(P,γ), and then write the radii of the circles.
An Introduction to Differential Geometry with Applications to Elasticity
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. For some years now, I, as well as a number of other contributors to this column, have on occasion expressed appreciation to Dover Publications for the service it provides to the mathematical community by re-issuing classic textbooks and making them available to a new generation at an affordable price. Of late, however, it seems to me based on anecdotal evidence garnered from a highly unscientific survey that not as many departments offer such a course. Yet, there must still be some market for books like this, because several have recently appeared, including a second edition of Differential Geometry of Curves and Surfaces by Banchoff and Lovett and another book with the same title by Kristopher Tapp. Most books with titles like this offer similar content.
The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.
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