In this book I present diﬀerential geometry and related mathematical topics with
the help of examples from physics. It is well known that there is something
strikingly mathematical about the physical universe as it is conceived of in
the physical sciences. The convergence of physics with mathematics, especially
diﬀerential geometry, topology and global analysis is even more pronounced in
the newer quantum theories such as gauge ﬁeld theory and string theory. The
amount of mathematical sophistication required for a good understanding of
modern physics is astounding.
These notes were developed as a supplement to a course on Differential Geometry at the advanced undergraduate, first years graduate level, which by author has taught for several years. These many axcellent good text in Differential Geometry bye very few have an early introduction to differential forms and their applications to Physisc
Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine Department of Mathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries: Distance, Open Sets, Parametric Surfaces and Smooth Functions 2. Smooth Manifolds and Scalar Fields 3. Tangent Vectors and the Tangent Space 4. Contravariant and Covariant Vector Fields 5. Tensor Fields 6. Riemannian Manifolds 7. Locally Minkowskian Manifolds: An Introduction to Relativity 8.
Bản đồ không gian tiếp xúc này được sử dụng cho các đối tượng biến dạng, chẳng hạn như các ký tự. Bản đồ không gian Tangent ghi vector bình thường so với bề mặt của đối tượng. Trong không gian tiếp xúc, có nghĩa là lên khỏi bề mặt của đối tượng.
Chương 4. Các thuật toán Vertex và Pixel Shader
light vector. Vector hướng ánh sáng trong không gian tiếp tuyến (tangent space), vector này được tính trong Vertex Shader và được truyền vào Pixel Shader để sử dụng. view vector. Vector tính từ mắt đến điểm nhìn (tọa độ trong không gian tiếp tuyến), vector này được tính trong Vertex Shader và truyền vào Pixel Shader để sử dụng. specular constant. Hằng phản chiếu, giá trị càng lớn thì vùng phản chiếu càng nhỏ. specular lookup.
Institute for Theoretical Physics University of California Santa Barbara, CA 93106 firstname.lastname@example.org December 1997
Abstract These notes represent approximately one semester’s worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein’s equations, and three applications: gravitational radiation, black holes, and cosmology. Individual chapters, and potentially updated versions, can be found at http://itp.ucsb.edu/~carroll/notes/.
Table of Contents
This is a quick set of note on basic differential topogoly. It get sketchier as it goes on. The last few section are only introduce the terminology and the some of concepts. These note qre written faster than I can read and may make no sense in spots.