Introduction and Vectors Lecture 1

1 Introduction Introduction and Vectors Lecture 1 This is a course on classical Electromagnetism. It is the foundation for more advanced courses in modern physics. All physics of the modern era, from quantum mechanics to quarks, can trace their theoretical development to field theories which were originally introduced in electrodynamics. Any student of physics, with the expectation of continuing the study or practice of physics, will need to carefully assimilate this material. The text book is probably the best that is presently available. It is not easy, nor is this course easy, but you must spend whatever time necessary to learn the material. Work as many problems as possible, ask questions, study the text and notes. We begin with the mathematics of vector calculus and partial differential equations. Hopefully much of this mathematical foundation will be a review. We cover it at a more rapid pace than would occur in several semesters of a math course. 2 Vector Algebra A familiar definition of a vector is a mathematical quantity which has magnitude and a direction. Every vector is associated with three functions of 3 variables, one for each of 3 spatial dimensions. Collectively, these functions have special properties under coordinate transformations. Thus a vector connects the geometry (space) to a mathematical measurement of position. This immediately introduces the need to define the mathematics of how one combines - adds, multiplies, divides - vectors. We introduce these mathematical operations by defining them through the symmetry transformations of that define the geometry of a coordinate system. In the case of a Cartesian system the direction of a vector may be defined by the direction cosines, that is the 3 angles (γ 1, γ 2, γ 3 ) between the direction of the vector and the three coordinate axes, (x,y,z). Note that the direction cosines are not independent since ; γ 2 1 + γ 2 2 + γ 2 3 = 1 As you should know, only 2 angles are necessary to define a direction (note this is obvious in spherical coordinates). 1

A A A Y Y Y Y Y Z X X X X X θ Z X Z Z Y Z Translation Rotation Inversion Figure 1: Symmetry operations of translation, rotation and inversion of a coordinate system. The red system is transformed under the various operations into the black system 3 Transformations We define a scalar variable as a quantity which is independent of the coordinate system. That is, it maintains the same value and form under translation, rotation, and inversion of coordinates ( inversion is a change from a right-handed system to a left-handed system). Thus, a vector preserves its magnitude (a scalar value), even though it changes its projection on the coordinate axes under rotation and inversion. This is as shown in figure 1. Consider a 3-D vector given by the projections on the 3 coordinate axes, (x,y,z). A = A xˆx + A y ŷ + A z ẑ In the above, A i (x, y, z) are scalar functions depending on x, y, z, for each of the coordinate directions, î. The forms, ˆx, ŷ, ẑ, are vectors of unit magnitude which point along the (x,y,z) axes, respectively. Under a right-hand rotation about the z axis through an angle, θ, the vector, A, takes the form of A ; A x = A x cos(θ) + A y sin(θ) A y = A x sin(θ) + A y cos(θ) A mathematical form is then determined to be a vector if it s functional forms transform, as demonstrated above, under rotational and inversion of the coordinate system. A scalar form does not change under these transformations. Under inversion a vector changes sign such that ˆx = ˆx 2

ŷ = ŷ ẑ = ẑ Later we extend this definition to other mathematical forms. Suppose a vector given by A = A xˆx + A y ŷ + A z ẑ. Mathematically, rotation is equivalent to a multiplication in matrix algebra. Suppose a rotation about the Z axis as illustrated in figure 1. The operation of a rotation matrix, R, on the column vector, A, produces the transformed column vector A. A = R A = cos(θ) sin(θ) 0 sin(θ) cos(θ) 0 0 0 1 Rotation through an arbitrary angle can then be described as a rotation about three angles (Euler angles) as follows. First rotate an angle α about the z axis, R z (α). Then rotate an angle β about the new x axis, R x (β), and finally rotate by an angle γ, R z (γ) about the new z axis. This is accomplished by the multiplication of the 3 ordered rotation matrices as shown below. R(α, β, γ) = R z (γ) R x (β) R z (α) A x A y A z 3

R(α, β, γ) = cos(γ) sinγ 0 sin(γ) cosγ 0 0 0 1 1 0 0 0 cos(β) sin(β) 0 sin(β) cos(β) cos(α) sin(α) 0 sin(α) cos(α) 0 0 0 1 4 Vector Addition Vector addition and subtraction is defined by adding scalar amplitudes along each direction of the vector. Addition is distributive, associative, and communicative. Thus for the addition(subtraction) of the vectors A and B ; A ± B = (A x ± B x )ˆx + (A y + B y )ŷ + (A z ± B z )ẑ Obviously, the form ( A ± B) rotates as vector as define above. 5 The Scalar Product We now define a form of vector multiplication for the vectors, A and B; A B = A x B x + A y B y + A z B z The result of this operation is a number, independent of a coordinate transformation, so that it produces a scalar. This type of multiplication is called the scalar, or dot, product. The dot product of a vector with itself gives the square of the magnitude of the vector, obviously it is a scalar mathematical form. A A = A 2 x + A2 y + A2 z If we write the dot product in polar coordinates one finds that A B represents the magnitude of A times the magnitude of the projection of the vector B on A or the magnitude of B times the magnitude of the projection of the vector A on B, figure 2. A B = A B cos(θ) 4

A B A x B B A cos( ) θ A A sin( θ ) Scalar Product Vector Product A B A x B Figure 2: The scalar and vector products in polar coordinates Here θ is the angle between vector directions. The dot product is both distributive and communicative. 6 The Cross Product Another way to multiply two vectors is called the cross or vector product, figure 2. In polar coordinates the magnitude of the cross product takes the form of the magnitude of A times the perpendicular projection of the vector B on A or the perpendicular projection of the vector A on B. A B = A B sin(θ) The result has some of the properties of a vector having a direction perpendicular to the plane formed by the vectors, and pointing in a direction given by the right-hand-rule. In Cartesian coordinates the cross product of the unit vectors, (ˆx, ŷ, ẑ) is; ˆx ŷ = ẑ ; ŷ ẑ = ˆx ; ẑ ˆx = ŷ ; In Cartesian coordinates the cross product operation can be written as the determinant; A B ˆx ŷ ẑ = A x A y A z B x B y B z 5

The result of the cross product operation transforms like a vector under rotations, but does not change sign under inversion. Remember that a true vector changes sign under inversion. Thus the cross product forms an axial-vector or pseudo-vector. It represents the area enclosed by the parallelogram that has sides defined by the vectors A and B. The triple cross product is not distributive and not communicative. We can check that the form of the cross product transforms as a vector by substitution for A j in the cross product amplitudes. 7 Direct Product A final way to multiply vectors is to multiply each component by every component of the other vector. This produces a mathematical form called a tensor of 2nd rank. Thus the direct product of A = (A x, A y, A z ) with B = (B x, B y, B z ) produces 9 elements which may be written; C ij = A i B j These elements can be placed in matrix form, and the linear algebra developed for matrix manipulation can be used. We discuss tensors later. Here we note that tensors, which incorporate scalar and vector forms, are defined by their transformation properties. A scalar is a tensor of rank 0, a vector is a tensor of rank 1, and a matrix is a tensor of rank 2. 8 Partial Derivatives We propose a function in 3-space given by F = F(x, y, z) this forms a surface in 4-D space. We then define the slope of this surface projected on the various coordinate directions. Thus for example; F F(x + δ, y, z) F(x, y, z) = lim x δ 0 δ This is the partial derivative of F with respect to a change in the x direction. 9 Fields A field is used to connect a mathematical description of a process to a geometry. Therefore a field is one or more functions which are defined at each point in space. Generally these 6

functions are continuous, but in principle any set of functions is possible. In Cartesian coordinates a scalar function has the form f(x, y, z) and represents a number at the coordinate position (x, y, z). (A scalar maintains the same value and form under a coordinate transformation) f(x, y, z ) = f(x, y, z) Suppose there are 3 functions defined so that they give 3 numbers at each spatial position. We also require that these functions transform like a vector under coordinate rotations, translations, and inversions. Thus for example, given the form; [f(x, y, z), g(x, y, z), h(x, y, z)] and with a right handed rotation about the z axis; f(x, y z ) = f(x, y, z) cos(θ) + g(x, y, z) sin(θ) g(x, y z ) = f(x, y, z) sin(θ) + g(x, y, z) cos(θ) h(x, y z ) = h(x, y, z) The magnitude of a vector field is obtained from the dot product with itself. M(x, y, z) = f 2 + g 2 + h 2 1/2 and is a scalar. Examples of fields characterized by their coordinate transformation properties are given in Table 1 Table 1: Examples of Fields with various coordinate transformation properties Transformation Type Field Scalar Electric Potential Vector Electric Field Tensor (rank 2) Gravitational Field Pseudo-Vector Magnetic Field Pseudo-Scalar Pion Field (Nuclear force) 7

10 Gradient We now look at the first of several differential forms. We define the gradient operation on a scalar field, F = F(x, y, z) by; F = ˆx F x + ŷ F y + ẑ F z This operation forms a vector as may be shown by its transformation properties under rotation and reflection. We write the following using the above definition of a partial derivative and the chain rule of differentiation. df = [F(x + δx, y + δy, z + δz) F(x, y, z)] df = [F(x + δx, y + δy, z + δz) F(x, y + δy, z + δz)] [F(x, y + δy, z + δz) F(x, y, z + δz)] + [F(x, y, z + δz) F(x, y, z)] df = F dx + F dy + F x y z dz = F ds Here ds = dx ˆx + dy ŷ + dz ẑ. In Cartesian coordinates the gradient operator is defined by ; = ˆx x + ŷ y + ẑ z Then we write in polar coordinates that the length of the differential vector is equal to ds = dx 2 + dy 2 + dz 2 ; F s = F cos(θ) The derivative df is a maximum only when ds is in the direction of F, and has magnitude equal to F. The gradient operation has the properties; 1. Its components are the rates of change of the function, F, along the coordinate axes 2. Its magnitude is the maximum rate of change of the function 3. Its direction is in the direction of maximum change 4. It points to larger values of the function 11 General Curvilinear Coordinates It is necessary that you understand and can visualize, 3 dimensional coordinate systems in addition to the Cartesian system. Other coordinate systems are defined relative to a Carte- 8

Z θ r (x,y,z) Y X φ Figure 3: The Spherical coordinate system sian coordinates at a point(s) in space. It is useful here to discuss the spherical coordinate system as an example. In spherical coordinates, the location of a point in space is given by its radial distance, and the polar and azimuthal angles (three variables for the 3 dimensions). The relationship to a Cartesian set is shown in figure 3. The bounds on the spherical variables are; 0 r 0 θ π 0 φ 2π This definition gives a one-to-one map of points defined in an infinite 3-D Cartesian space into an infinite spherical space. We find the following relation between the coordinates; x = r sin(θ) cos(φ) y = r sin(θ) sin(φ) z = r cos(θ) Because there is a one-to-one map there is also an inverse; r = x 2 + y 2 + x 2 positive root taken 9

dφ dθ r dθ r sin( θ ) d φ dr Figure 4: Differential length elements in spherical coordinates θ = cos 1 [ z x2 + y 2 + z 2] φ = tan 1 [ y x ] Here we see that x 2 + y 2 + z 2 = r 2 as expected, that is, r r = r 2. Now we need to look at the differential lengths corresponding to the differential changes, dr; dθ; dφ. These are, dr, rdθ, rsin(θ)dφ, respectively. This is shown in figure 4. By observation, the volume element is obtained to first order in the differentials by multiplying the 3 perpendicular side lengths of the volume element together. (Note that this works because the lengths are mutually perpendicular) d (V olume) = dxdy dz = r 2 sin(θ) dr dθ dφ On the other hand the volume element in general may be obtained by a Jacobian transformation ; ( x, y, z) dxdy dz = J( ) dr dθ dφ ( r, θ, φ) In the case of spherical coordinates J can always be obtained from the determinate. For spherical coordinates the Jacobian is;; x y z r r r J = x y z sin(θ) cos(φ) sin(θ) sin(φ) cos(θ) θ θ θ = x y φ φ φ z r cos(θ) cos(φ) rcos(θ) sin(φ) r sin(θ) r sin(θ) sin(φ) r sin(θ) cos(φ) 0 10

The transformational equations x = x(r, θ, φ), y = y(r, θ, φ), and z = z(r, θ, φ) are needed of course. Now consider a general curvilinear set of coordinates in 3 dimensions, ζ i (x, y, z) i = 1, 2, 3. An infinitesimal line segment can be written in terms of the Cartesian set of coordinates (x, y, z) with length d s = dx ˆx + dy ŷ + dz ẑ. If each ζ i = constant and we have a one-toone map then the equations may be inverted to obtain (x, y, z) in terms of (ζ 1, ζ 2, ζ3). For an orthogonal set of surfaces, the line element is then; ds 2 = 3 h 2 i dζ2 i i The magnitude of an area element perpendicular to the i direction in an orthogonal system is; dσ i = jk h j dζ j h k dζ k However by differentiation in the general case; dx = x ζ 1 dζ 1 + x ζ 2 dζ 2 + x ζ 3 dζ 3 The (y, z) coordinates are obtained in the same way. Then these are combined to obtain; ds 2 = dx 2 + dy 2 + dz 2 = g 11 dζ 1 dζ 1 + g 12 dζ 1 dζ 2 + ds 2 = ij g ij dζ i dζ j with; g ij = x ζ i x ζ j + y ζ i y ζ j + z ζ i z ζ j These coefficients define the metric of the Riemann space. For an orthogonal system g ij = 0 for i j. In this case h 2 i = g ii. As previously observed for orthogonal coordinates; d s = h 1 dζ 1 â 1 + h 2 dζ 2 â 2 + h 3 dζ 3 â 3 ds 2 = i h 2 i dζ2 i The factor h i is the scale factor of the coordinate system. It is the multiplier that changes the differential coordinate variable into a length; (ie dθ rdθ for the spherical system). In an orthogonal system the magnitude of the surface area is dσ i = h j h k dζ j dζ k and the volume is dτ = h i h j h k dζ i dζ j dζ k. If â i are the unit vectors in the coordinate directions ζ i, respectively, then the gradient operator in an arbitrary, orthogonal set of coordinates is written; 11

F = (1/h 1 ) F ζ 1 â 1 + (1/h 2 ) F ζ 2 â 2 + (1/h 3 ) F ζ 3 â 3 This can be obtained from df = F d s as follows. df(ζ 1, ζ 2, ζ 3 ) = F d s = F(ζ 1 + δζ 1, ζ 2 + δζ2, ζ 3 + δζ 3 ) F(ζ 1, ζ 2 + δζ2, ζ 3 + δζ 3 ) + F(ζ 1, ζ 2 + δζ2, ζ 3 + δζ 3 ) F(ζ 1, ζ 2, ζ 3 + δζ 3 ) + F(ζ 1, ζ 2 +, ζ 3 + δζ 3 ) F(ζ 1, ζ 2, ζ 3 ) 12 Flux The definition of flux comes from an analogy to flow through a surface. Suppose N particles per unit volume moving with a velocity V. The number of particles that flow through an elemental area d σ in a time dt is then; dn = [(N V ) d σ] dt This is generalized to define the differential flux for any vector field F as; d (flux) = F d σ The dot product projects the velocity of the particles perpendicular to the surface. Then the number of particles passing through the surface element per unit time is a measure of the differential flux through the surface. d (flux) = (N V ) d σ The total flux through an area is ; flux = area (N V ) d σ 13 Divergence The divergence of a vector field, F, is defined as the flux out of a volume per unit volume, and written, Div F or in Cartesian coordinates F. We develop this in a generalized, 12

ζ 3 F2 F2 dζ h 3 3 h 2 dζ2 h dζ 1 1 ζ 2 ζ 1 Figure 5: A differential volume element which is used to obtain the flux orthogonal set of coordinates, but note that applying the operator in non-cartesian coordinates for the divergence is NOT correct. What we wish to find is the flux out of a volume τ per unit volume. Div F = lim dτ 0 F d σ dτ To proceed, consider the figure 5 which illustrates an elemental volume in a general coordinate frame defined by ζ 1 (x, y, z), ζ 2 (x, y, z), ζ 3 (x, y, z) with surfaces, ζ i (x, y, z) = constant Begin by finding the differential flux between the flux into and out of the volume, dτ, due to F 2. 2 = [F 2 h 1 h 3 ] (ζ1,ζ 2 +d(ζ 2 /2),ζ 3 ) dζ 1 dζ 3 [F 2 h 1 h 3 ] (ζ1,ζ 2 d(ζ 2 /2),ζ 3 ) dζ 1 dζ 3 Collecting terms and using the definition of partial differentials; 2 = (F 2h 1 h 3 ) ζ 2 dζ 1 dζ 2 dζ 3 Combine this with the other 3 directions and divide by the differential volume element. From the definition, this results in the divergence operation on the vector field, F. 13

Div F = 1 [ (h 2h 3 F 1 ) + (h 1h 3 F 2 ) + h 1 h 2 h 3 ζ 1 ζ 2 (h 1 h 2 F 3 ) ] ζ 3 Then as an example, use spherical coordinates with; h 1 = 1 h 2 = r h 3 = r sin(θ) The differential variables are; dr dθ dφ. Substitution gives the divergence; DivF(r, θ, φ) = (1/r 2 ) r (r2 F 1 ) + ( 1 r sin(θ) ) θ (sin(θ) F 2) + ( 1 r sin(θ) ) φ F 3 Note that the divergence of a vector field results in a scalar field. 14 The Circulation Next consider the line integral of a vector field, F, between spatial points a and b. This is defined as; I = b a F d l This is shown in figure 6. The projection of the vector F along the line is incrementally multiplied by the line length. Then define the circulation as the line integral around the boundary that encloses an area σ. This is illustrated in figure 7 and written mathematically as; Γ = L F d l The differential operation of Curl is the circulation per unit enclosed area. As with the divergence, do not directly use the operator on a vector field F unless in Cartesian coordinates. F CurlF d l = lim dσ 0 dσ In evaluating the circulation on the surface element shown in figure 8 we obtain; Γ 3 = F 2 h 2 dζ 2 (ζ1 +(ζ1/2),ζ 2,ζ 3 ) + F 1 h 1 dζ1 (ζ1,ζ2 +d(ζ 2 /2),ζ 3 ) 14

b F F F F F dl a Figure 6: The line integral from a to b of the vector F dl F Figure 7: The circulation is the line integral over a closed path. In general the enclosed area does not lie in a plane 15

ζ 3 dσ 3 h 1 dζ1 dζ h 2 2 ζ 2 ζ 1 Figure 8: The evaluation of the circulation over a differential area in order to obtain the curl of a vector function F F 2 h 2 dζ 2 (ζ1 d(ζ 1 /2),ζ2,ζ3) F 1 h 1 dζ 1 (ζ1,ζ 2 d(ζ2/2),ζ 3 ) Collecting terms, applying the partial derivative, and dividing by the enclosed differential area, one obtains of the 3 rd component of the curl curl F 3 = 1 h 1 h 2 [ (F 2h 2 ) ζ 1 (F 1h 1 ) ζ 2 ] When collecting all terms the curl may be placed in the form; curlf â 1 h 1 â 2 h 2 â 3 h 3 = 1 h 1 h 2 h 3 ζ 1 ζ2 ζ3 h 1 f 1 h 2 F 2 h 3 F 3 Applying the scale factors and differential variables for spherical coordinates one obtains; curlf = â 1 ( 1 r sin(θ) )[ θ (sin(θ) F 3) F 2 φ ] + â 2 [( 1 F 1 r sin(θ) θ (1/r) (rf 3) ] + â r 3 (1/r)[ (rf 2) r F 1 θ ] 15 Integral Theorems Recall that the definition of the divergence operation is the flux out of a volume per unit volume. The area is defined as the outward normal to a closed surface. Thus the differential element of the flux through a small volume, d τ, is; 16

d (flux) = (Div F) d τ Then summing all the volume elements contained within a finite volume gives the flux out of this volume. This is a statement of Gauss theorem. d τ (Div F) = F d σ Here d σ is and element of the surface area enclosing the volume, d τ Then the Curl operation is defined as the circulation per unit area, so that for an infinitesimal area; [(Curl F) d σ] enclosedarea = ( F d l) closedpath By combining the small, infinitesimal paths around the perimeters of a set of infinitesimal areas we have; (Curl F) d σ = F d l This is stokes theorem. Finally if we consider the line integral over a path from a to b we have for a scalar function using the gradient operation; path d F = path F d s = F(a) F(b) 16 The Laplacian Operator On a number of occasions we will use the combination of the Curl operating on the of a scalar (and later a vector) function. In Cartesian coordinates this takes the form 2. Again be careful not to apply the Cartesian form of this operator in other coordinate systems, or to a vector function. We will discuss later the differential equations that are produced by these various vector operators, and their physical implications. 17