Differential Geometry in Hindi and Urdu |Handout Notes
Differential geometry intro
Differential geometry, branch of mathematics that studies the geometry of curves, surfaces, and manifolds (the higher-dimensional analogs of surfaces). The area owes its name to its use of ideas and techniques from differential calculus, although the current subject regularly makes use of algebraic and in basic terms geometric techniques as an alternative. Despite the fact that fundamental definitions, notations, and analytic descriptions range widely, the following geometric questions succeed: how does one degree the curvature of a curve within a surface (intrinsic) as opposed to inside the encompassing area (extrinsic)? How can the curvature of a surface be measured? What is the shortest route within a surface among two factors on the floor? How is the shortest direction on a surface related to the idea of a immediately line?
Whilst curves had been studied for the reason that antiquity, the invention of calculus inside the seventeenth century opened up the have a look at of greater complicated plane curves—together with those produced via the French mathematician René Descartes (1596–1650) together with his “compass” (see records of geometry: cartesian geometry).
Specifically, quintessential calculus brought about fashionable answers of the historical issues of finding the arc length of aircraft curves and the vicinity of aircraft figures. This in turn opened the level to the research of curves and surfaces in area—an investigation that changed into the begin of differential geometry. A number of the fundamental thoughts of differential geometry can be illustrated by way of the strake, a spiraling strip regularly designed by using engineers to offer structural help to huge steel cylinders which includes smokestacks. A strake may be shaped through cutting an annular strip (the area between two concentric circles) from a flat sheet of metallic and then bending it right into a helix that spirals around the cylinder, as illustrated inside the discern. What ought to the radius r of the annulus be to provide the high-quality in shape?
Differential geometry components the solution to this trouble via defining a precise measurement for the curvature of a curve; then r may be adjusted until the curvature of the inner fringe of the annulus matches the curvature of the helix. A vital question remains: can the annular strip be bent, without stretching, so that it bureaucracy a strake across the cylinder? Specifically, because of this distance measured along the surface (intrinsic) are unchanged. Two surfaces are stated to be isometric if one can be bent (or converted) into the alternative without converting intrinsic distances. (for example, because a sheet of paper may be rolled right into a tube without stretching, the sheet and tube are “domestically” isometric—best locally due to the fact new, and possibly shorter, routes are created by means of connecting the two edges of the paper.) Accordingly, the second one query will become: are the annular strip and the strake isometric? To reply this and comparable questions, differential geometry developed the belief of the curvature of a surface.
In this course we are mainly interested in doing calculus on surfaces. For this we aim at doing the followings.
- Review of differential calculus.
- Develop tools to study curves and surfaces in space.
- Proper definition of surface. How to do calculus on surface.
- A detailed study of geometry of surface.