V ECTOR C ALCULUS AND T HE T HEORY OF F IELDS A COMPREHENSIVE GUIDE TO STUDY THE MATHEMATICAL THEORIES NEEDED FOR MANY PHYSICAL APPLICATIONS, ELECTROMAGNETISM , HYDRODYNAMICS, ETC. W RITTEN B Y A M ONAEM A LBUSTANJI Einstein’s Prophetic Legacy A MB 2022 A M ONAEM A LBUSTANJI Contents 1 A Brief Introduction To The Field Theory 1 1.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Tensor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The Fundamental theorem of Calculus . . . . . . . . . . . . . . . . 3 1.3 The Differential Vector Operator (⃗ ∇) . . . . . . . . . . . . . . . . . 3 1.3.1 Multiplication by a scalar; The Gradient . . . . . . . . . . 4 1.3.2 Multiplication by another vector; The Divergence . . . . . 5 1.3.3 Multiplication by another vector; The Curl . . . . . . . . . 5 1 Chapter 1 A Brief Introduction To The Field Theory 1.1 Fields The concept of the physical field is very important, and it makes the study of some phenomena easier, like electromagnetism, hydrodynamics, etc. We will find out that this concept, introduced by Faraday, is essential in the de velopment of the theory of vector calculus; you will probably appreciate it at the end of this course. Now let’s define the field, considering an infinite or finite region in space; if every point corresponds to a definite value of some physical property, then this region represents a field. But the physical quan tities can be classified, with necessity to the complete description, into three types, scalar quantities, vector and tensor quantities, since the physical field is not more than generalized representation of some physical quantity. So, the physical field can be classified in the same way. 1.1.1 Scalar Field Considering the same region, if the physical quantity represented by the field is a scalar quantity, then the field which represents this quantity is called a scalar field.  Examples of scalar fields: electric or magnetic or gravitational poten tial, mass or charge density. Figure (1.1)1 visualizes a scalar field representing temperature map for the moon, where the intensity of the field is represented by different hues of col ors. 1 Source: https://scx1.bcdn.net/csz/news/800a/2009/2newnasatempe.jpg. 1 CHAPTER 1. A BRIEF INTRODUCTION TO THE FIELD THEORY 2 1.1.2 Vector Field Again by considering the same region, if the physical quantity represented by the field is a vector quantity, then the field which represents this quantity is called a vector field  Examples of vector fields: an electric, magnetic, or gravitational field. Figure (1.2) 2 visualizes a vector field representing the motion of a snowflake, where the drawn vectors represent the velocity of the snowflakes. Figure 1.1 Figure 1.2 1.1.3 Tensor Field Now, if the physical quantity represented by the field is a tensor quantity, then the field that represents this quantity is called a tensor field. Fig ure(1.3)3 represents a stress forces acting on a particle in a homogeneous medium under a uniform linear stress. This illustration is no more than a representation of a 3rd rank tensor. Figure 1.3 2 Source: https://web.mit.edu/8.02t/www/802TEAL3D/visualizations/ coursenotes/modules/guide01.pdf 3 Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Cmec_stress_ ball_f02_t6.png CHAPTER 1. A BRIEF INTRODUCTION TO THE FIELD THEORY 3 1.2 The Fundamental theorem of Calculus The fundamental theorem of calculus states: Z β df dx = f (β) − f (α) (1.1) α dx where the essence behind this theorem lies in what Newton and Leibniz re alized, that differentiation and integration are inverse operations. Now you may wonder what exactly this theorem tell us?, let’s find out. The integrand on the LHS is the total differential of ( f (x)) that is df d f (x) = dx (1.2) dx notice that ( f ) is a single variable function, and if you integrate4 it, this will give you the change in ( f (x)) here from (α) to (β), and the RHS tells that you well get the same answer if you took the values at the ends and subtract them, you will understand this theorem in a better way when we generalize it in one of the preceding sections. The purpose of this short section is just to give you a glance about the deep meaning of this important theorem. 1.3 The Differential Vector Operator (⃗ ∇) You learned in high school how to differentiate a function of one variable, that is, in one dimension, which is why you didn’t care about the direction, or rather, nobody warned you about it before, but we live in three dimensions, so we must deal with problems in three dimensions. For example, the function that describes the temperature of your room is a function of three variables, and these variables are the dimensions of the point at which you measure the temperature T(x, y, z), and indeed, this is an example of a scalar field, as I told you in the previous section. Therefore, we must generalize our operators to include problems in three dimensions, but we must be careful because the direction has to be taken into account, and here I’m going to take care of differential operators, let’s write the threedimensional version of equation (1.2) ∂f ∂f ∂f d f (x, y, z) = dx + dy+ dz (1.3) ∂x ∂y ∂z now by getting around the problem to include the direction, we will rewrite equation (1.3) using the technique of the dot product ∂f ∂f ∂f µ ¶ df = x̂ + ŷ + ẑ . (dx x̂ + d y ŷ + dz ẑ) (1.4) ∂x ∂y ∂z notice that the first parenthesis on the RHS of the equation (1.4) is a differ ential operator, but it includes direction, and that’s what we aspired to have for generalization purposes. 4 Remember the concept of Riemann integral. CHAPTER 1. A BRIEF INTRODUCTION TO THE FIELD THEORY 4 Now let’s define this quantity as follows ∂ ∂ ∂ ∇= x̂ + ŷ + ẑ (1.5) ∂x ∂y ∂z where it is operated on the f (x) in equation (1.4), this operator is usually called the gradient5 or "Del". Also, this operator is more complicated than normal differentiation because it involves the direction. The vector property of this operator is treated as a normal vector, which follows the axioms of vector space. Since this operator is a differential operator, then we care about the multiplication operations between vectors, and the other operations are meaningless in this case, the multiplication operations are 1.3.1 Multiplication by a scalar; The Gradient And this is what we get in the equation (1.4): ∂f ∂f ∂f ∇f = x̂ + ŷ + ẑ (1.6) ∂x ∂y ∂z and this is the gradient of f (x, y, z) as I mentioned above. The question now is what is the geometrical interpretation of the gradient?  How can we imagine the gradient? Returning to the same example of the scalar field (1.1), or talking of any scalar field—let it represent pressure, temperature, or any other scalar quan tity—the direction of the largest change in the physical quantity function is in the direction of the gradient, by rewriting the equation (1.4) d f = ∇ f . ds (1.7) or ∇ f   ds cos θ d f = ∇ (1.8) where (ds) is the infinitesimal displacement vector ds = dx x̂ + d y ŷ + dz ẑ (1.9) the maximum change in f occurs at the maximum value of cos θ , i.e. when (θ = 0). So, d f = ∇ ∇ f   ds (1.10) for fixed (ds), (d f ), or the change in f is the greatest when we walk in the same direction as (∇ f ). 5 Remember that the differentiation in equation (1.5) does not operate on unit vectors; in Cartesian coordinates, this doesn’t matter because they are constants, but in spherical and cylindrical coordinates, the situation is totally different. CHAPTER 1. A BRIEF INTRODUCTION TO THE FIELD THEORY 5 1.3.2 Multiplication by another vector; The Divergence Using the dot product, that is ∂ ∂ ∂ µ ¶ ¡ ¢ ∇.E = x̂ + ŷ + ẑ . E x x̂ + E y ŷ + E z ẑ (1.11) ∂x ∂y ∂z ∂E x ∂E y ∂E z ∇.E = + + (1.12) ∂x ∂y ∂z we call this quantity the divergence of E(x, y, z)6 . The question now is what is the geometrical interpretation of the divergence?  How can we imagine the divergence? The divergence describes how much a vector diverges (expands) from a point in a vector field. Imagine yourself inside a hurricane—God forbid—and you Figure 1.4 Figure 1.5 are spinning inside it (figure(1.4)). There is something pushing you outward. The amount of this push represents the divergence. Look at the faucet and the drain (figure (1.5)). The faucet represents positive divergence, while the drain represents negative divergence. 1.3.3 Multiplication by another vector; The Curl Using the cross product, that is ∂ ∂ ∂ µ ¶ ¡ ¢ ∇ ×E= x̂ + ŷ + ẑ × E x x̂ + E y ŷ + E z ẑ (1.13) ∂x ∂y ∂z ∂E z ∂E y ∂E x ∂E z ∂E y ∂E x µ ¶ µ ¶ µ ¶ ∇ ×E= − x̂ + − ŷ + − ẑ (1.14) ∂y ∂z ∂z ∂x ∂x ∂y 6 Remember that the components of a vector field can be in three variables (x, y, z), i.e., E x = E x (x, y, z). CHAPTER 1. A BRIEF INTRODUCTION TO THE FIELD THEORY 6 we call this quantity the curl of E(x, y, z). The question now is what is the geometrical interpretation of the curl?  How can we imagine the curl? The curl describes how much a vector curls around a point in a vector field, or simply that it is the amount of circulation of a vector field. Again, considering Figure 1.6 the same example of the hurricane, the amount of rotation (spinning) you have in a hurricane represents the amount of curl (figure (1.6)).
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