Introduction to Mathematical Modeling and Computer Simulations

Vladimir Mityushev, Wojciech Nawalaniec, Natalia Rylko

• Publisher: Chapman and Hall/CRC
• Year: 2018
• ISBN:
  9781351998758
• Edition: 1
• Topic: Mathématiques
• Subtopic: Mathématiques générales

Introduction to Mathematical Modeling and Computer Simulations is written as a textbook for readers who want to understand the main principles of Modeling and Simulations in settings that are important for the applications, without using the profound mathematical tools required by most advanced texts. It can be particularly useful for applied mathematicians and engineers who are just beginning their careers. The goal of this book is to outline Mathematical Modeling using simple mathematical descriptions, making it accessible for first- and second-year students.Chapter1 and the Prefaceof this book is freely available as a downloadable Open Access PDF under a Creative Commons Attribution-Non Commercial-No Derivatives 4.0 license available at http://www.taylorfrancis.com/books/e/9781315277240

Part III

Advanced Applications

Chapter 7

Vector analysis

7.1 Euclidean space R³

7.1.1 Polar coordinates

7.1.2 Cylindrical coordinates

7.1.3 Spherical coordinates

7.2 Scalar, vector and mixed products

7.3 Rotation of bodies

7.4 Scalar, vector and mixed product in Mathematica

7.5 Tensors

7.6 Scalar and vector fields

7.6.1 Gradient

7.6.2 Divergence

7.6.3 Curl

7.6.4 Formulae for gradient, divergence and curl

7.7 Integral theorems

Exercises

This chapter can be considered as an introduction to vector analysis with applications in main to mechanics. Computer implementation of vector analysis is widely used.

7.1 Euclidean space ℝ³

Following classic mechanics (Newtonian mechanics) the Euclidean space ℝ³ is considered as a place of action and time t is an independent parameter. The notation x is used for the vector $\overrightarrow{OX}$ connecting O, the origin of ℝ³, to a point X. The same notation¹ x is used for the point X ∈ ℝ³. The standard orthogonal basis of ℝ³ consists of the vectors

$i_1=\left(1,0,0\right),i_2=\left(0,1,0\right)i_3=\left(0,0,1\right).$

(7.1)

Any vector = x ∈ ℝ³ can be presented uniquely in the form

$x=x_1{i}_1+x_2{i}_2+x_3{i}_3=\left(x_1,x_2{x}_3\right).$

(7.2)

The triple (x₁,x₂,x₃) denotes the Cartesian coordinate of the vector x.

Physical laws have to be stated in the space ℝ³. Frequently, it is easy to do using the fixed basis (i₁, i₂, i₃). However, physical laws must be invariant with respect to coordinates. A separate question exists in the form of physical laws independent on coordinates. This theoretical question is touched upon in Sec.7.5. In order to giv...