Introduction to Vector and Tensor Analysis

Introduction to Tensor Calculus

Preface These are general notes on tensor calculus originated from a collection of personal notes which I prepared some time ago for my own use and reference when I was studying the subject. I decided to put them in the public domain hoping they may be beneficial to some students in their effort to learn this subject. Most of these notes were prepared in the form of bullet points like tutorials...

Vector and Tensor Analysis in Euclidean Space

Introduction to Support Vector Machines

Support Vector Machines (SVM’s) are a relatively new learning method used for binary classification. The basic idea is to find a hyperplane which separates the d-dimensional data perfectly into its two classes. However, since example data is often not linearly separable, SVM’s introduce the notion of a “kernel induced feature space” which casts the data into a higher dimensional space where the...

Introduction to topological vector spaces

Infinite-dimensional spaces are ubiquitous inmany branches of mathematics, and their topologies are almost always interesting. Samples of spaces that one is sure to encounter are Cc(R), C (R), L(R), and their continuous linear duals. There are two good reasons for making oneself at least somewhat familiar with the subject. On the one hand, without some preparation, it is easy to make false stat...

Introduction to Tensor Calculus for General Relativity

There are three essential ideas underlying general relativity (GR). The first is that spacetime may be described as a curved, four-dimensional mathematical structure called a pseudo-Riemannian manifold. In brief, time and space together comprise a curved fourdimensional non-Euclidean geometry. Consequently, the practitioner of GR must be familiar with the fundamental geometrical properties of c...