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The aim of this book is to deliver a thorough introduction to the basic properties of Differential Geometry. The study of the basic applications of surfaces is involved here. The area of surfaces is very important in mathematics, so some problems involving it are discussed as well as related topics of the applications of Differential Geometry. The Tensor analysis has its origin in the developments of Differential Geometry. This study has derived some standard results by the help of Tensor analysis. Geodesics is an important characteristic of surfaces, so some Geodesics on surfaces has also been derived. Differential Geometry is related to Astronomy and Geographical problems. Therefore some practical examples are discussed in this book. A mathematical method of finding distance and direction between two places on the surfaces of the earth with respect to the geographical North Pole or South Pole is established.