Chapter 12

Vector Analysis

The perceptible physical world is three dimensional (although additional hidden dimensions have been speculated in superstring theories and the like). The most general mathematical representations of physical laws should therefore be relations involving three dimensions. Such equations can be compactly expressed in terms of vectors. Vector analysis is particularly applicable in formulating the laws of electromagnetic theory.

12.1 Scalars and Vectors

A scalar is a quantity which is completely described by its magnitude—a numerical value and usually a unit. Mass and temperature are scalars, with values, for example, like 10 kg and 300 K. A vector has, in addition, a direction. Velocity and force are vector quantities. A vector is usually printed as a boldface symbol, like image, while a scalar is printed in normal weight, usually in italics, like a. (Vectors are commonly handwritten by placing an arrow over the symbol, like image or image.) A vector in three-dimensional space can be considered as a sum of three components. Figure 12.1 shows a vector image with its Cartesian components image, alternatively written image. The vector image is represented by the sum

image (12.1)

where i, j, k are unit vectors in the x, y, z directions, respectively. These are alternatively written image or image. The unit vectors have magnitude one and are directed along the positive x, y, z axes, respectively. They are pure numbers, so that the units of A are carried by the components image. A vector can also be represented in matrix notation by

image (12.2)

The magnitude of a vector A is written image or A. By Pythagoras’ theorem in three dimensions we have

image (12.3)

Newton’s second law, written as a vector equation

image (12.4)

is shorthand for the three component relations

image (12.5)

Also (12.4) implies the corresponding relation between the scalar magnitudes

image (12.6)


Figure 12.1 Vector A with Cartesian components image. Unit vectors i, j, k are also shown. The length or magnitude of A is given by image.

A significant mathematical property of vector relationships is their invariance under translation and rotation. For example, if F and a are transformed to image and image by translation and/or rotation, the analog of (12.4), namely

image (12.7)

along with all the corresponding component relationships, is also valid. Note that scalar quantities do not change under such transformations, so image.

A vector can be multiplied by a scalar, such that

image (12.8)

For image, this changes the magnitude of the vector while preserving its direction. For image, the direction of the vector is reversed. Two vectors can be added using

image (12.9)

As shown in Figure 12.2, a vector sum can be obtained either by a parallelogram construction or by a triangle construction—placing the two vectors head to tail. Vectors can be moved around at will, so long as their magnitudes and directions are maintained. This is called parallel transport. (Parallel transport is easy in Euclidean space but more complicated in curved spaces such as the surface of a sphere. Here the vector must be moved in such a way that it maintains its orientation along geodesics. Parallel transport around a closed path in a non-Euclidean space usually changes the direction of a vector.)


Figure 12.2 Vector sum: Parallelogram and triangle constructions.

The position or displacement vector r represents the three Cartesian coordinates of a point:

image (12.10)

A single symbol r can thus stand for the three coordinates image in the same sense that a complex number z represents the pair of numbers image and image. A unit vector in the direction of r can be written

image (12.11)

A differential element of displacement can likewise be defined by

image (12.12)

The notation ds is often used for a differential element of a curve in two- or three-dimensional space.

A function of x, y, and z can be compactly written

image (12.13)

If image is a scalar, this represents a scalar field. If there is, in addition, dependence on another variable, such as time, we could write image. If the three components of a vector image are functions of image, we have a vector field

image (12.14)

or A(r,t) if it is time dependent.

12.2 Scalar or Dot Product

Vectors can be multipled in two different ways to give scalar products and vector products. The scalar product, written image, also called the dot product or the inner product, is equal to a scalar. To see where the scalar product comes from, recall that work in mechanics equals force times displacement. If the force and displacement are not in the same direction, only the component of force along the displacement produces work. We can write image, where image is the angle between the vectors F and r.

We are thus led to define the scalar product of two vectors by

image (12.15)

When B = A, image and image, so the dot product reduces to the square of the magnitude of A:

image (12.16)

When A and B are orthogonal (perpendicular), image and image, so that

image (12.17)

The scalar products involving the three unit vectors are therefore given by

image (12.18)

These vectors thus constitute an orthonormal set with respect to scalar multiplication. To express the scalar product image in terms of the components of A and B, write

image (12.19)

Using (12.18) for the products of unit vectors, we find

image (12.20)

The scalar product has the same structure as the three-dimensional matrix product of a row vector with a column vector:

image (12.21)

12.3 Vector or Cross Product

Consider a point P rotating counterclockwise about a vertical axis through the origin, as shown in Figure 12.3. Let r be the vector from the origin to point P and v, the instantaneous linear velocity of the point as it moves around a circle of radius image. A rotation is conventionally represented by an axial vectorimage normal to the plane of motion, such that the trajectory of P winds about image in a counterclockwise sense, the same as the direction a right-handed screw advances as it is turned. This is remembered most easily by using the right-hand rule shown in Figure 12.4. A rotational velocity of image radians/s moves point P with a speed image around the circle. The velocity vector v thus has magnitude imageand instantaneous direction normal to both r and image. This motivates definition of the vector product, also known as the cross product or the outer product, such that

image (12.22)

The direction of v is determined by the counterclockwise rotation of the first vector image into the second r, shown also in Figure 12.4.


Figure 12.3 Rotation of point P about vertical axis, represented by the vector product relation image.


Figure 12.4 Right-hand rules. Left: Direction of axial vector image representing counterclockwise rotation. Right: Direction of vector product image.

As another way to arrive at the vector product, consider the parallelogram with adjacent sides formed by vectors A and B, as shown in Figure 12.5. The area of the parallelogram is equal to its base A times its altitude image. By definition, the vector product image has magnitude image and direction normal to the parallelogram. The operation of vector multiplication is anticommutative since

image (12.23)

This implies that the cross product of a vector with itself equals zero:

image (12.24)

By contrast, scalar multiplication is commutative with

image (12.25)


Figure 12.5 Vector product image. The parallelogram has area image.

The three unit vectors have the following vector products:

image (12.26)

Therefore, the cross product of two vectors in terms of their components can be determined from

image (12.27)

This can be compactly represented as a image determinant:

image (12.28)

The individual components are given by

image (12.29)

where “et cyc” stands for the other two relations obtained by cyclic permutations image. Another way to write a cross product is

image (12.30)

in terms of the Levi-Civita symbolimage, defined by

image (12.31)

The vector product occurs in the force law for a charged particle moving in an electromagnetic field. A particle with charge q moving with velocity v in an electric field E and a magnetic induction field B experiences a Lorentz force:

image (12.32)

The magnetic component of the force is perpendicular to both the velocity and the magnetic induction, which causes charged particles to deflect into curved paths. This underlies the principle of the cyclotron and other particle accelerators.

Inversion of a coordinate system means reversing the directions if the image axes, or equivalently replacing image by image. This amounts to turning a right-handed into a left-handed coordinate system. A polar vector is transformed into its negative by inversion, as shown in Figure 12.6. A circulation in three dimensions, which can be represented by an axial vector, remains, by contrast, unchanged under inversion. An axial vector is also called a pseudovector to highlight its different inversion symmetry. The vector product of two polar vectors gives an axial vector: symbolically, image. You can show also that image and image. In electromagnetic theory, the electric field E is a polar vector while the magnetic induction B is an axial vector. This is consistent with the fact that magnetic fields originate from circulating electric charges.

Problem 12.3.1

A charged particle in an electromagnetic field is described by the Lagrangian


where q is the electric charge and image, where A is the vector potential. As a generalization of the result of Problem 11.6.1, derive the corresponding Hamiltonian H(rp).


Figure 12.6 Inversion behavior of polar vector P and axial vector image. Axes and vectors before and after inversion are drawn.

12.4 Triple Products of Vectors

The triple scalar product, given by

image (12.33)

represents the volume of a parallelepiped formed by the three vectors A, B, and C, as shown in Figure 12.7. A image determinant changes sign when two rows are interchanged but preserves its value under a cyclic permutation, so that

image (12.34)

For three polar vectors, the triple scalar product changes sign upon inversion. Such a quantity is known as a pseudoscalar, in contrast to a scalar, which is invariant to inversion.


Figure 12.7 Triple scalar product as volume of parallelepiped: base area = BC sinimage, altitude = A cosimage, volume = ABC sinimage cosimage.

You might also encounter the triple vector productimage, which is a vector quantity. This can be evaluated using the Levi-Civita representation (12.30). The i component of the triple product can be written

image (12.35)

where we have introduced new dummy indices as needed. Focus on the sum over k of the product of the Levi-Civita symbols image. For nonzero contributions, i and j must be different from k, and likewise for image and m. We must have either that image or image. Therefore

image (12.36)

Since a sum containing a Kronecker delta reduces to a single term, we find

image (12.37)

Therefore, in full vector notation,

image (12.38)

which has the popular mnemonic “BAC(K) minus CAB.”

12.5 Vector Velocity and Acceleration

A particle moving in three dimensions can be represented by a time-dependent displacement vector:

image (12.39)

The velocity of the particle is the time derivative of image

image (12.40)

as represented in Figure 12.8. The Cartesian components of velocity are

image (12.41)

The magnitude of the velocity vector is the speed:

image (12.42)

The velocity vector will be parallel to the displacement only if the particle is moving in a straight line. In Figure 12.8, image can clearly be in a different direction. Thus, v(t) need not, in general, be parallel to r(t).


Figure 12.8 Velocity vector.

Acceleration is the time derivative of velocity. We find

image (12.43)

with Cartesian components

image (12.44)

12.6 Circular Motion

Figure 12.9 represents a particle moving in a circle of radius r in the x, y-plane. Assuming a constant angular velocity of image radians/s, the angular position of the particle is given by image. Cartesian coordinates can be defined with image, so that

image (12.45)

Differentiation with respect to image gives the velocity

image (12.46)

As we have seen, the axial vector representing angular velocity is given by image. Thus,

image (12.47)


image (12.48)

in agreement with Eq. (12.22).


Figure 12.9 Displacement vector r(t) for particle moving in a circular path with angular velocity image. The x- and y-components of r are shown.

To find the acceleration of a particle in uniform circular motion, note that image is a time-independent vector, so that

image (12.49)

In this case acceleration represents a change in the direction of the velocity vector, while the magnitude v remains constant. Using the right-hand rule, the acceleration is seen to be directed toward the center of the circle. This is known as centripetal acceleration, centripetal meaning “center seeking.” The magnitude of the centripetal acceleration is

image (12.50)

The force which causes centripetal acceleration is called centripetal force. By Newton’s second law

image (12.51)

For example, gravitational attraction to the Sun provides the centripetal force which keeps planets in nearly elliptical orbits. The effect known as centrifugal force is actually an artifice. It actually represents the tendency of a body to continue moving in a straight line, according to Newton’s first law. It might be naı¨vely perceived as a force away from the center. Actually, it is the centripetal force which keeps a body moving along a curved path, in resistance to this inertial tendency.

12.7 Angular Momentum

The linear momentumimage is a measure of inertia for a particle moving in a straight line. Accordingly, Newton’s second law can be elegantly written

image (12.52)

A result applying to angular motion can be obtained by taking the vector product of Newton’s law:

image (12.53)

The last equality follows from

image (12.54)

and the fact that image. Angular momentum—more precisely, orbital angular momentum—is defined by

image (12.55)

The analog of Eq. (12.52) for angular motion is

image (12.56)

where image is called the torque or turning force. The law of conservation of angular momentum implies that, in the absence of external torque, a system will continue in its rotational motion.

Consider a particle of mass m in angular motion with the angular velocity vector image. The angular momentum can be related to the angular velocity using Eqs. (12.48) and (12.38):

image (12.57)

where image is the moment of inertia tensor, represented by the image matrix

image (12.58)

Tensors are the next member of the hierarchy which begins with scalars and vectors. The dot product of a tensor with a vector gives another vector. Usually the moment of inertia is defined for a rigid body, a system of particles with fixed relative coordinates. We must then replace image in the matrix by the corresponding summation over all particles, image, and so forth. We will focus on the simple case of circular motion, with r being perpendicular to image as in Figure 12.9. This implies that image, so that the moment of inertia reduces to a scalar:

image (12.59)

Moment of inertia is a measure of a system’s resistance to change in its rotational motion, in the same way that mass is a measure of resistance to change in linear motion.

The kinetic energy for a mass in circular motion can be expressed in terms of the angular frequency. We find

image (12.60)

This can also be expressed

image (12.61)

Evidently, relations for linear motion can be transformed to their analogs for angular motion with the substitutions: image. Here is a handy table giving the linear and angular equivalents for some mechanical variables:


12.8 Gradient of a Scalar Field

A scalar field image can be represented graphically by a family of surfaces on which the field has a sequence of constant values (see Figure 12.10). If the scalar quantity is temperature, the surfaces are called isotherms. Analogously, barometric pressure is represented by isobars and electrical potential by equipotentials. (The less-familiar generic term for constant-value surfaces of a scalar field is isopleths.) When the position vector image is changed by an infinitesimal image, the change in image is given by the total differential [see Eq. (11.12)]

image (12.62)

This has the form of a scalar product of the differential displacement image with the gradient vector

image (12.63)

In particular

image (12.64)


Figure 12.10 Gradients image of a scalar field shown as red arrows. The gradient at every point image is normal to the surface of constant image in the direction of maximum increase of image.

When image happens to lie within one of the surfaces of constant image, then image, which implies that the gradient image at every point is normal to the surface image containing that point. This is shown in Figure 12.10, with arrows representing gradient vectors. Where constant image surfaces are closer together, the function must be changing rapidly and the magnitude of image is correspondingly larger. This is more easily seen in a two-dimensional contour map, Figure 12.11. The gradient at image points in the direction of maximum increase of image. The symbol image is called “del” or “nabla” and stands for the vector differential operator

image (12.65)

Sometimes image is written image.


Figure 12.11 Two-dimensional contour map showing, for example, the elevation around a hill. Where the contours are more closely spaced, the terrain is steeper and the gradient larger in magnitude. This is indicated by the thicker arrow.

The change in a scalar field image along a unit vector image is called the directional derivative, defined by

image (12.66)

A finite change in the scalar field image as r moves from image to image over a path C is given by the line integral

image (12.67)

Work in mechanics is given by a line integral over force: image. The work done by the system is the negative of this quantity. Thus, for a conservative system, force is given by the negative gradient of potential energy: image. An analogous relation holds in electrostatics between the electric field and the potential: image.

Problem 12.8.1

The force between two electrical charges image and image separated by a displacement vector r (in three dimensions) is given by Coulomb’s law, image. Derive the potential-energy function image.

12.9 Divergence of a Vector Field

One of the cornerstones of modern physics is the conservation of electric charge. If an element of volume contains a certain quantity of charge, that quantity can change only when charge flows through the boundary of the volume. The charge per unit volume measured in Coulombs per unit volume (C/m3), is designated the charge densityimage. The flow of charge across unit area per unit time is represented by the current densityJ(r, t). The current density at a point r equals the product of the density and the instantaneous velocity of charge at that point:

image (12.68)

This is dimensionally consistent since

image (12.69)

Note that J(rt) and v(rt) are vector fields, while image is a scalar field.

Consider the element of volume image in Cartesian coordinates, shown in Figure 12.12. Designating the coordinates at the center by image, the middle of the faces normal to the x-direction are image and image, and analogously for the faces normal to the other two directions. The electric charge contained in image can be approximated by image. The net charge leaving image per unit time is then equal to the integral over the current density normal to the surface image enclosing the element of volume. Such a surface integral can be written image, where image, an element of area with the direction of the outward normal from the surface. In the present case, the surface integral is the sum of contributions from the six faces of image, thus

image (12.70)

Divide each side by image and let the element of volume be shrunken to a point. In the limit image, the terms containing image reduce to a partial derivative, as follows:

image (12.71)

Adding the analogous image and image contributions, we find

image (12.72)

The right-hand side has the structure of a scalar product of image with image:

image (12.73)

called the divergence of J (also written div J).


Figure 12.12 Flux of a vector field J out of an element of volume image with surface image, for derivation of the divergence theorem.

The divergence of the current density at a point x,y,z represents the net outward flux of electric charge from that point. Since electric charge is conserved, the flow of charge from every point must be balanced by a reduction of the charge density image in the vicinity of that point. This leads to the equation of continuity

image (12.74)

In the steady-state case, in which there is no accumulation of charge at any point, the equation of continuity reduces to

image (12.75)

In defining the divergence of a vector field A(rt), we have transformed an integration over surface area into an integration over volume, as shown in Figure 12.12. When two such elements of volume are adjacent, the contribution from their interface cancels out, since flux into one element is exactly canceled by flux out of the adjacent element. By adding together an infinite number of infinitesimal elements, the result can be generalized to a finite volume of arbitrary shape. We arrive thereby at the divergence theorem (also known as Gauss’ theorem):

image (12.76)

A differential element of volume is conventionally written image. Sometimes the notation image is used for S, to indicate the surface area enclosing the volume V.

The divergence of the gradient of a scalar field occurs in several fundamental equations of electromagnetism, wave theory, and quantum mechanics. In Cartesian coordinates

image (12.77)

The operator

image (12.78)

is called the Laplacian or “del squared.” Sometimes image is abbreviated as image.

12.10 Curl of a Vector Field

As we have seen, a vector field can be imagined as some type of flow. We used this idea in defining the divergence. Another aspect of flow might be circulation, in the sense of a local angular velocity. Figure 12.13 gives a schematic pictorial representation of vector fields with outward flux (left) and with circulation (right). To be specific, consider a rectangular element in the x, y-plane as shown in Figure 12.14. The center of the rectangle at (xy), the four sides are at image and image. The area if the rectangle is image. The circulation of a vector field A(r) around the rectangle in the counterclockwise sense is approximated by

image (12.79)

Next we divide both sides by image and take the limits image. Since

image (12.80)


image (12.81)

we find

image (12.82)

It can be recognized that the difference of partial derivatives represents the z-component of a cross product involving the vector operator image:

image (12.83)

The vector field image is called the curl ofimage, also written curl A. For an element of area in an arbitrary orientation, we find the limit

image (12.84)

where image is directed normal to the element of area. An arbitrary surface, such as the one drawn in Figure 12.15, can be formed from an infinite number of infinitesimal elements of area such as Figure 12.14. The line integral of a vector field A(r) counterclockwise around a path closed C can be related to surface integral of image over the enclosed area by Stokes’ theorem:

image (12.85)

In two dimensions, Stokes’ theorem reduces to Green’s theorem (11.75):

image (12.86)


Figure 12.13 Schematic representations of vector fields with divergence (left) and curl (right).


Figure 12.14 Circulation of a vector field A(r) about an element of area image. In the limit image, this gives the z-component of the curl image.


Figure 12.15 Stokes’ theorem: image.

For arbitrary scalar and vector fields image and image the following identities involving divergence and curl can be readily derived:

image (12.87)


image (12.88)

An intriguing combination is image also known as “curl curl” (Curl Curl Beach is a suburban resort outside Sydney, Australia). This has the form of a triple vector product (12.38). We must be careful about the order of factors, however, since image is a differential operator. If the vector image is always kept on the right of the image’s, we obtain the correct result:

image (12.89)

Note that image is a vector quantity such that

image (12.90)

12.11 Maxwell’s Equations

These four fundamental equations of electromagnetic theory can be very elegantly expressed in terms of the vector operators, divergence and curl. The first Maxwell equation is a generalization of Coulomb’s law for the electric field of a point charge:

image (12.91)

where Q is the electric charge and image, the permittivity of free space. The unit vector image indicates that the field is directed radially outward from the charge (or radially inward for a negative charge). We write

image (12.92)

Since image represents an element of area with an outward-directed normal, this can be transformed into the integral relationship

image (12.93)

where image is the charge density within a sphere of volume V with surface area S. But by the divergence theorem,

image (12.94)

The corresponding differential form is then the first of Maxwell’s equations:

image (12.95)

Free magnetic poles, the magnetic analog of electric charges, have never been observed (although their existence has been postulated in some theories). This implies that the divergence of the magnetic induction B equals zero, which serves as the second Maxwell equation:

image (12.96)

According to Faraday’s law of electromagnetic induction, a time-varying magnetic field induces an electromotive force in a conducting loop through which the magnetic field threads, as shown in Figure 12.16. This is somewhat suggestive of a linear relationship between the time derivative of the magnetic induction and the curl of the electric field. Indeed, the appropriate equation is

image (12.97)


Figure 12.16 Faraday’s law of electromagnetic induction, leading to third Maxwell equation: image.

Oersted discovered that an electric current produces a magnetic field winding around it, as shown in Figure 12.17. The quantitative result, suggested by analogy with Eq. (12.97), is Ampère’s law:

image (12.98)

where image is the permeability of free space. Maxwell showed that Ampère’s law must be incomplete by the following argument. Taking the divergence of (12.98) we find

image (12.99)

since the divergence of a curl is identically zero [Eq. (12.88)]. This is consistent only for the special case of steady currents [Eq. (12.75)]. To make the right-hand side accord with the full equation of continuity, Eq. (12.74), we can write

image (12.100)

This is still equivalent to 0 = 0 but suggests by (12.95) that

image (12.101)

With Maxwell’s displacement current hypothesis, the generalization of Ampère’s law is

image (12.102)

the added term being known as the displacement current.


Figure 12.17 Magnetic field produced by electric current, described by Ampére’s law: image.

We have given the forms of Maxwell’s equations in free space. In material media, two auxilliary fields are defined: the electric displacement image and the magnetic field image. In terms of these, we can write compact forms for Maxwell’s equations in SI units:

image (12.103)

In the absence of charges and currents (image and image), Maxwell’s equations reduce to

image (12.104)

Taking the curl of the third equation, the time derivative of the fourth and eliminating the terms in B, we find

image (12.105)

Using the curl curl equation (12.89) and noting that image we obtain the wave equation for the electric field:

image (12.106)

An analogous derivation shows that the magnetic induction also satisfies the wave equation:

image (12.107)

Maxwell proposed that Eqs. (12.106) and (12.107) describe the propagation of electromagnetic waves at the speed of light

image (12.108)

Note that this conclusion would not have been possible without the displacement current hypothesis.

By virtue of the vector identities (12.87) and (12.88), the two vector fields E and B can be represented by electromagnetic potentials—one vector field and one scalar field. The second Maxwell equation (12.96) suggests that the magnetic induction can be written

image (12.109)

where A(rt) is called the vector potential. Substituting this into the third Maxwell equation (12.97), we can write

image (12.110)

This suggests that the quantity in parentheses can be represented as the divergence of a scalar function, conventionally written in the form

image (12.111)

where image is called the scalar potential. In the time-independent case, the latter reduces to the Coulomb or electrostatic potential, with image. The electric field and magnetic induction are uniquely determined by the scalar and vector potentials:

image (12.112)

The converse is not true, however. Consider the alternative choice of electromagnetic potentials

image (12.113)

where image is an arbitrary function. The modified potentials image and image can be verified to give the sameE and B as the original potentials. This property of electromagnetic fields is called gauge invariance. Extension of this principle to quantum theory leads to the concept of gauge fields, which provides the framework of the standard model for elementary particles and their interactions.

12.12 Covariant Electrodynamics

Electromagnetic theory can be very compactly expressed in Minkowski four-vector notation. Historically, it was this symmetry of Maxwell’s equations which led to the Special Theory of Relativity. Now that we know about partial derivatives, the gradient four-vector can be defined. This is a covariant operator

image (12.114)

The corresponding contravariant operator is

image (12.115)

The scalar product gives

image (12.116)

The D’Alembertian operatorimage is defined by

image (12.117)

a notation is suggested by analogy with the symbol image for the Laplacian operator image. The wave equations (12.106) and (12.107) can thus be compactly written

image (12.118)

For the Minkowski signature image, image. The alternative choice of signature image implies image. To confuse matters further, image is sometimes written in place of image.

The equation of continuity for electric charge (12.74) has the structure of a four-dimensional divergence

image (12.119)

in terms of the charge-current four-vector

image (12.120)

In Lorenz gauge, the function image in Eqs. (12.113) is chosen such that image. The scalar and vector potentials are then related by the Lorenz condition

image (12.121)

This can also be expressed as a four-dimensional divergence

image (12.122)

in terms of a four-vector potential

image (12.123)

Maxwell’s equations in Lorenz gauge can be expressed by a single Minkowski-space equation

image (12.124)

where the image component gives

image (12.125)

(Incidentally, the Lorenz gauge was proposed by the Danish physicist Ludvig Lorenz. It is often erroneously designated “Lorentz gauge,” after the more famous Dutch physicist Hendrik Lorentz. In fact, the condition does fulfill the property known as Lorentz invariance.)

The electric and magnetic fields obtained from the potentials using (12.112) can be represented by a single relation for the electromagnetic field tensor

image (12.126)

Maxwell’s equations, in terms of the electromagnetic field tensor, can be written:

image (12.127)

Problem 12.12.1

An electromagnetic field dual tensor can be defined by


where image is the four-dimensional Levi-Civita tensor. Verify that


This can be obtained from the image tensor by the formal substitutions image and image.

Problem 12.12.2

Show that the second of Maxwell’s equations in (12.127) can be written more compactly as


12.13 Curvilinear Coordinates

Vector equations can provide an elegant abstract formulation of physical laws, but, in order to solve problems, it is usually necessary to express these equations in a particular coordinate system. Thus far we have considered Cartesian coordinates almost exclusively. Another choice, for example, cylindrical or spherical coordinates, might prove more appropriate, depending on the symmetry of the problem.

We will consider the general class of orthogonal curvilinear coordinates, designated image, whose coordinate surfaces always mutually intersect at right angles. The Cartesian coordinates of a point in three-dimensional space can be expressed in terms of a set of curvilinear coordinates by relations of the form

image (12.128)

The differentials of the Cartesian coordinates are

image (12.129)

with analogous expressions for dy and dz. Thus a differential element of displacement image can be written

image (12.130)

where image are unit vectors with respect to the curvilinear coordinates image and image are scale factors. As shown in Figure 12.18, the elements of length in the three coordinate directions are equal to image, image, and image. The element of volume is evidently given by

image (12.131)

where the image are, in general, functions of image.


Figure 12.18 Volume element in curvilinear coordinates q1q2q3: image.

We can evidently identify

image (12.132)

so that the element of volume can be equated to a triple scalar product (cf Section 12.4):

image (12.133)

This can therefore be connected to the Jacobian determinant:

image (12.134)

The components of the gradient vector represent directional derivatives of a function. For example, the change in the function image along the image-direction is given by the ratio of image to the element of length image. Thus, the gradient in curvilinear coordinates can be written

image (12.135)

The divergence image represents the limiting value of the net outward flux of the vector quantity image per unit volume. Referring to Figure 12.19, the net flux of the component image in the image-direction is given by the difference between the outward contributions image on the two shaded faces. As the volume element approaches a point, this reduces to

image (12.136)

Adding the analogous contributions from the image- and image-directions and diving by the volume image, we obtain the general result for the divergence in curvilinear coordinates:

image (12.137)

The curl in curvilinear coordinates is given by

image (12.138)

The Laplacian is the divergence of the gradient, so that substitution of (12.135) into (12.137) gives

image (12.139)


Figure 12.19 Evaluation of divergence in curvilinear coordinates.

The three most common coordinate systems in physical applications are Cartesian, with image, cylindrical, with image, and spherical polar, with image. We frequently encounter the spherical polar volume element

image (12.140)

and the Laplacian operator

image (12.141)

Problem 12.13.1

Determine the forms of the scalar and vector products, image and image, in terms of the components of image and image in spherical polar coordinates.

12.14 Vector Identities

For ready reference, we list a selection of useful vector identities.


Problem 12.14.1

Derive some of the above vector identities. It is usually sufficient to demonstrate a single Cartesian component of the formula.

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