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Introduction
Vectors will be needed throughout this course. We will begin with definition of vectors and identities involving dot and cross products.The identity for triple cross product \(\mathbf A\times (\mathbf B\times\mathbf C)\) is most important to remember.
| A.1 A Quick Review of Vectors |
| Lesson Overview |
| §1 Vectors as Geometrical Objects |
| §2 An Example |
| §3 Vector Algebra Identities |
| EndNotes |
A.2 Abstract mathematical spaces generalized from set of all vectors
t is important to remember that the usual definition in school text books that a vector is a quantity with a magnitude and a direction should be supplemented with rule for multiplication by a number, parallelogram law for addition of vector. I say this because this set of properties and rules, then can be generalized to define an abstract mathematical object Linear Vector Space.
This study of Abstract Vector Spaces by themselves turns out to be very fruitful. It has a diverse variety of applications physics.
The set of vectors, as mentioned in the above paragraph, supplemented with dot product has led to the abstract
Vector Space with Inner Product. Further generalization to infinite dimensional case, Hilbert Space which forms core of mathematical foundation of quantum mechanics.
| § 2.1 Summation Convention |
| §2.2 Einstein Summation Convention |
| §2.3 Kronecker Delta and Levi-Civita Symbols |
| §2.4 Examples |
| §2.5 Use of \(\epsilon,\delta\) symbols in vector algebra |
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EndNotes |
A.4 What is the next item in the list of preparation needed before Physics applications? and Why ?
Besides the basic properties of vectors, summation convention, epsilon and delta symbols we need to introduce the coordinate system. Given a set of coordinate axes components of a vector can be defined. This is an important step needed to solve differential equations of motion which involve vector objects. Using components also enables to make a details numerical predictions of theory with observed data. For example for orbits of planets detailed comparison of theory with data requires use of coordinates.
We also need to learn about change in components when a coordinate system is rotated. This will be required to set up equation of motion in rotating frames and for rigid body dynamics.
