Download e-book for kindle: Differential geometry : a first course by D. Somasundaram

By D. Somasundaram

ISBN-10: 184265182X

ISBN-13: 9781842651827

Differential Geometry: a primary direction is an advent to the classical concept of area curves and surfaces provided on the Graduate and publish- Graduate classes in arithmetic. in line with Serret-Frenet formulae, the speculation of area curves is built and concluded with a close dialogue on basic life theorem. the idea of surfaces contains the 1st primary shape with neighborhood intrinsic houses, geodesics on surfaces, the second one basic shape with neighborhood non-intrinsic houses and the elemental equations of the skin conception with numerous functions.

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D. Somasundaram's Differential geometry : a first course PDF

Differential Geometry: a primary direction is an creation to the classical concept of area curves and surfaces provided on the Graduate and publish- Graduate classes in arithmetic. in keeping with Serret-Frenet formulae, the idea of area curves is constructed and concluded with a close dialogue on primary life theorem.

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Example text

2) ds Since p' * 0, we get p + a — {op') = 0 proving the necessity of the condition. ds 43 Theory of Space Curves Conversely let us assume that the condition is satisfied at every point of the curve. Multiplying the given condition throughout by p' and integrating with respect to s, we obtain p 2 + {op')2 = constant which shows that under the given condition the radius of the osculating sphere is constant at every point of the curve. The centre C of the osculating sphere is C = p + pn + op'b Differentiating C and simplifying as in the previous example, we have dC b which is zero by the given condition.

To prove the necessity of the condition, let us assume that the curve lies on a sphere. Then the sphere is the osculating sphere at every point of the curve so that the radius of the osculating sphere is constant. (2) ds Since p' * 0, we get p + a — {op') = 0 proving the necessity of the condition. ds 43 Theory of Space Curves Conversely let us assume that the condition is satisfied at every point of the curve. Multiplying the given condition throughout by p' and integrating with respect to s, we obtain p 2 + {op')2 = constant which shows that under the given condition the radius of the osculating sphere is constant at every point of the curve.

For different choices of c, we get different involutes of the system. Corollary 1. The distance between corresponding points of two involutes is constant. Let P be a fixed point on C. Let rx and r2 be the position vectors of the corresponding points on two involutes determined by c = cx and c = c2. Then rx = r + (c{ - s) t, r2 = r + (c2 - ^)t Then rx - r2 = (c{ - c2) t so that \rx - r2| = \cx - c2\ which is constant. Thus the length between two such corresponding points is constant. Corollary 2.

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Differential geometry : a first course by D. Somasundaram


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