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Valentine Ignatov
Valentine Ignatov

Tensor Analysis on Manifolds (Dover Books on Mathematics): A Mathematical Tool for Physics and Geometry


Tensor Analysis on Manifolds (Dover Books on Mathematics) Book PDF




If you are looking for a comprehensive and accessible introduction to tensor analysis on manifolds, you might want to check out the book Tensor Analysis on Manifolds by Richard L. Bishop and Samuel I. Goldberg. This book is a classic in the field of differential geometry and provides a solid foundation for further studies in mathematics and physics. In this article, we will give you an overview of the main topics covered in the book, as well as some applications and examples of tensor analysis on manifolds. We will also provide you with a link to download the book in PDF format for free.




Tensor Analysis on Manifolds (Dover Books on Mathematics) book pdf



Introduction




Before we dive into the details of tensor analysis on manifolds, let us first review some basic concepts and definitions that will help us understand the subject better.


What is tensor analysis?




A tensor is a mathematical object that can be used to describe various physical phenomena, such as forces, stresses, strains, electric fields, magnetic fields, etc. A tensor can be thought of as a generalization of scalars, vectors, and matrices, which are special cases of tensors. A scalar is a tensor of rank zero, a vector is a tensor of rank one, and a matrix is a tensor of rank two. A tensor of rank n can be represented by an array of n indices, each ranging from 1 to some dimension d.


For example, a scalar can be written as a = a1, where the index 1 indicates that it has only one component. A vector can be written as b = bi, where i ranges from 1 to d. A matrix can be written as c = cij, where i and j range from 1 to d. A general tensor of rank n can be written as t = ti1i2...in, where each index ranges from 1 to d.


Tensors can be manipulated by various operations, such as addition, subtraction, multiplication, contraction, etc. For example, two tensors of the same rank can be added or subtracted component-wise: t + s = (ti1i2...in + si1i2...in). A tensor can be multiplied by a scalar: k * t = (k * ti1i2...in). A tensor can be contracted by summing over repeated indices: ti = t11 + t22 + ... + tdd. A tensor can also be multiplied by another tensor, resulting in a new tensor of higher or lower rank, depending on the number of indices involved.


What is a manifold?




A manifold is a mathematical space that locally resembles a Euclidean space of some dimension, but globally may have a more complicated shape or structure. For example, the surface of a sphere is a two-dimensional manifold, because every small patch of it looks like a flat plane, but the whole surface is curved and finite. A torus (donut-shaped surface) is another example of a two-dimensional manifold, which has a different topology than a sphere. A manifold can also have more than two dimensions, such as a hypersphere or a hyperboloid.


A manifold can be equipped with various structures that allow us to measure distances, angles, areas, volumes, etc. on it. For example, a metric tensor is a tensor field that assigns a positive-definite symmetric matrix to each point of the manifold, which defines the inner product and the norm of vectors at that point. A metric tensor can be used to compute the length of curves, the angle between vectors, the area of surfaces, the volume of regions, etc. on the manifold. A metric tensor also induces a notion of curvature on the manifold, which measures how much the manifold deviates from being flat.


Why study tensor analysis on manifolds?




Tensor analysis on manifolds is a powerful tool that can be used to study various aspects of geometry, topology, physics, and engineering. By using tensors, we can express quantities and equations that are invariant under coordinate transformations, which means that they do not depend on the choice of coordinates used to describe the manifold. This makes tensors suitable for describing natural phenomena that obey universal laws, such as gravity, electromagnetism, fluid dynamics, etc.


Tensor analysis on manifolds also allows us to generalize concepts and results from Euclidean spaces to more general spaces that may have curvature or other features. For example, we can define derivatives of functions and vector fields on manifolds using covariant derivatives, which take into account the connection and curvature of the manifold. We can also define integrals of functions and differential forms on manifolds using integration on chains, which generalize the notions of line integrals, surface integrals, and volume integrals.


Main Concepts and Theorems




In this section, we will briefly introduce some of the main concepts and theorems that are covered in the book Tensor Analysis on Manifolds. We will not go into too much detail or provide proofs here, but we will give some examples and intuition for each topic. For more details and rigor, we refer you to the book itself.


Tensors and tensor fields




Definition and examples




A tensor on a manifold M is an n-linear map from n copies of the tangent space TpM at some point p to the real numbers R. That is, a tensor T takes n vectors v, v, ..., v at p and produces a real number T(v, v, ..., v). The number n is called the rank or order of the tensor. A tensor field is a function that assigns a tensor to each point of the manifold.


For example, a scalar function f: M -> R is a tensor field of rank zero, because it takes zero vectors at each point and produces a real number f(p). A vector field X: M -> TM is a tensor field of rank one, because it takes one vector at each point and produces a real number X(p)(v) = , where is the inner product induced by the metric tensor. A bilinear form B: M -> T*M x T*M -> R is a tensor field of rank two, because it takes two vectors at each point and produces a real number B(p)(v,w) = B(p)(v)(w), where B(p)(v) is a linear map from TpM to R.


Operations and properties