Notes on Mathematical Methods for Physicists Chapter3

Notes on Mathematical Methods for Physicists $\S3$

Chapter3 Vector Analysis

$\S3.1$ Review of Basic Properties

(1)

(2)

(3) Vector polynomials (e.g. $\bold{A}-\bold{2B}+\bold{3C}$) are well-defined.

(4) Unit vectors $\hat{\bold{e}}_i$.

(5) Magnitude

(6) Dot product

(7) Orthogonal

(8) Projection

(9) Direction cosines

(10) Single-column matrix

(11) The addition and multiplication of vectors can also be replaced by single-column matrices.

(12) Transpose of a single-column matrix , called a row vector.

(13) Dot product replaced by multiplication of matrices

$\S3.2$ Vectors in $3-D$ Space

$\square$ Vectors or Cross Product

Definition :

The direction of $\bold{C}$ , i.e., that of $\hat{\bold{e}}_C$ , is perpendicular to the plane of $\bold{A}$ and $\bold{B}$ , such that $\bold{A}$ , $\bold{B}$ , and $\bold{C}$ form a right-handed system.

This can be equivalently represented as

Note that the cross product is a quantity specifically defined for $3-D$ space.

$\square$ Scalar Triple Product

Definition :

This quantity is equivalent to the volume of the space in the picture above.

There is

Example Reciprocal Lattice

Let $\bold{a}$ , $\bold{b}$ , and $\bold{c}$ (not necessarily mutually perpendicular) represent the vectors that define a crystal lattice. The displacements from one lattice point to another may then be written

In the band theory of solids , it is useful to introduce what is called a reciprocal lattice $\bold{a}'$ , $\bold{b}'$ , $\bold{c}'$ such that

and with

Then

$\square$ Vector Triple Product

The equation can be proved by Levi-Civita symbols and Kronecker delta. There is

$\S3.3$ Coordinate Transformations

$\square$ Rotations

Rotate the axis as the picture below

So the unchanged vector $\bold{A}$ now takes the changed form

which is equivalent to the matrix equation

It is easy to find that

is orthogonal.

$\square$ Orthogonal Transformations

It is no accident that the transformation describing a rotation in $\R^2$ was orthogonal. We can rewrite $\bold{S}$ as

Summerizing : The transformation from one orthogonal Cartesian coordinate system to another Cartesian system is described by an orthogonal matrix.

$\square$ Reflections

For simplicity , consider first the inversion operation , in which the sign of each coordinate is reversed. This will lead to

which clearly results in $\det S=-1$. The change in sign of the determinant corresponds to the change from a right-handed to a left-handed coordinate system (which cannot be accomplished by a rotation). Reflection about a plane also changes the sign of the determinant.

This formula for vector addition , multiplication by a scalar , and the dot product are unaffected by a reflection transformation of the coordinates , but this is not true of the cross product. To see this , look at one component :

The unchanged vectors are called axial vectors , or pseudovectors , while the other are called polar vectors , or just vectors. There is

It is easy to find that $\bold{A}\cdot(\bold{B}\cross\bold{C})$ changes sign in the transformation , so it is called a pseudoscalar. And the vector $\bold{A}\cross(\bold{B}\cross\bold{C})$ is a vector , so there is a general principle that a product with an odd number of pseudo quantities is "pseudo" , while those with even numbers of pseudo quantities are not.

$\square$ Successive Operations

Two transformations can be carried out successively. This rule is only true for rotations and/or reflections by applying their relevant orthogonal transformations.

Notes :

1.The operations take place in right-to-left order.

2.The product of two orthogonal matrices is orthogonal , so the combined operation is an orthogonal transformation.

$\S3.4$ Rotations in $\R^3$

Rotations in $\R^3$ is important in practice , so we discuss now in some detail the treatment of rotations in $\R^3$. There is

The argument we made to evaluate $\hat{\bold{e}}'_\mu\cdot\hat{\bold{e}}_\nu$ could as easily have been made with the roles of the two unit vectors reversed

In $\R^2$ , we find that all the elements of $S$ depended on a single variable , the rotation angle. In $\R^3$ , the number of independent variables needed to specify a general rotation is three. The usual parameters used to specify $\R^3$ rotations are the Euler angles. The three steps describing rotation of the coordinate axes are the following :

(1) The coordinates are rotated about the $\hat{\bold{e}}_3$ axis counterclockwise (as viewed from positive $\hat{\bold{e}}_3$) through an angle $\alpha$ in the range $0\leq\alpha<2\pi$ , into new axes denoted $\hat{\bold{e}}'_1$ , $\hat{\bold{e}}'_2$ , $\hat{\bold{e}}'_3$. (The polar direction is not changed ; the $\hat{\bold{e}}_3$ and $\hat{\bold{e}}'_3$ axes coincide.)

(2) The coordinates are rotated about the $\hat{\bold{e}}'_2$ axis counterclockwise (as viewed from positive $\hat{\bold{e}}'_2$) through an angle $\beta$ in the range $0\leq\beta\leq\pi$ , into new axes denoted $\hat{\bold{e}}''_1$ , $\hat{\bold{e}}''_2$ , $\hat{\bold{e}}''_3$. (This tilts the polar direction toward the $\hat{\bold{e}}'_1$ direction , but leaves $\hat{\bold{e}}'_2$ unchanged.)

(3) The coordinates are now rotated about the $\hat{\bold{e}}''_3$ axis counterclockwise (as viewed from positive $\hat{\bold{e}}''_3$) through an angle $\gamma$ in the range $0\leq\gamma<2\pi$ , into the final axes , denoted $\hat{\bold{e}}'''_1$ , $\hat{\bold{e}}'''_2$ , $\hat{\bold{e}}'''_3$. (This rotation leaves the polar direction , $\hat{\bold{e}}''_3$ , unchanged.)

The matrices that represent those rotations are

Note the order of $S$.

Note that those matrices are orthogonal with determinant $+1$.

$\S3.5$ Differential Vector Operators

We are going to discuss vector field. Physicists need to be able to characterize the rate at which the values of vectors (and also scalars) change with position , and this is most effectively done by introducing differential vector operator concepts. It turns out that there are a large number of relations between these differential operators , and it is our current objective to identify such relations andlearn how to use them.

$\square$ Gradient , $\nabla$

The gradient characterizes the change of a scalar field , here $\varphi$ , with position. There is

which is of the form corresponding to the dot product of

We have given the $3\cross1$ matrix of derivatives the name $\nabla\varphi$ (often referred to in speech as "del phi" or "grad phi") ; we give the differential of position its customary name $d\bold{r}$.

The $\nabla\varphi$ is a vector. To prove this , we must show that it transforms under rotation of the coordinate system according to $(\nabla\varphi)'=S(\nabla\varphi)$. We have

Having now established the legitimacy of the form $\nabla\varphi$ , we proceed to give $\nabla$ a life of its own. We therefore define

Note that $\nabla$ is a vector differential operator.

In physics , we have $\bold{F}(\bold{r})=-\nabla V(\bold{r})$.

There is a simple geometric interpretation of the gradient : $\nabla V$ is the direction of most rapid increase in $V$ , and its magnitude is associated with the speed of increasing.

Example Gradient of $r^n$

$\square$ Divergence , $\nabla\cdot$

The divergence of a vector is defined as the operation

Example Divergence of Central Force Field

Consider $\nabla\cdot f(r)\hat{\bold{r}}$. We write

Using

We can simplify the equation :

Of course , I have already learnt about the origin of divergence ---- the representative of flux in a differential form. If the physical problem being described is one in which fluid (molecules) are neither created or destroyed , we will also have an equation of continuity , of the form

If the divergence of a vector field is zero everywhere , its lines of force will entirely of closed loops ; such vector fields are termed solenoidal.

$\square$ Curl , $\nabla\cross$

This vector is called the curl of $\bold{V}$. Note that when the determinant is evaluated , it must be expanded in a way that causes the derivatives in the second row to be applied to the functions in the third row (and not to anything in the top row) ; we will encounter this situation repeatedly , and will identify the evaluation as being from the top down.

Example Curl of a Central Force Field

Consider $\nabla\cross[f(r)\hat{\bold{r}}]$. Writing

so the result is 0.

Noting that a vector whose curl is zero eveerywhere is termed irrotational.

$\S3.6$ Differential Vector Operators : Further Properties

$\square$ Successive Applications of $\nabla$

The possible results include the following :

All five of these expressions involve second derivatives , and all five appear in the second-order differential equations of mathematical physics , particularly in electromagnetic theory.

$\square$ Laplacian

Laplacian is the divergence of the gradient. We have

When $\varphi$ is the electrostatic potential , we have $\nabla\cdot\nabla\varphi=0$ in vacuum. This is called the Laplacian equation.

Often the Laplacian is written as $\nabla^2$ , or $\Delta$ in the older European literature.

Example Laplacian of a Central Fiel Potential

Calculate $\laplacian\varphi(r)$.

$\square$ Irrotational and Solenoidal Vecter Fields

The curl of the divergence :

The divergence of the curl :

We can thus see that the divergence is irrotational , and the curl is solenoidal. Therefore ,

$\square$ Maxwell's Equations

We who are learning physics all know this well.

$\square$ Vector Laplacian

There is

(This can be proved by Levi-Civita symbols.) The term $\nabla\cdot\nabla\bold{V}$ is called the vector Laplacian , written as $\laplacian\bold{V}$.

Example Electromagnetic Wave Equation

In vacuum , we can derive the Maxwell equation , and thus we have

Then the equation is decided only by $\bold{E}$. In vacuum , there is $\nabla\cdot\bold{E}=0$. The result is the vector electromagnetic wave equation for $\bold{E}$ ,

The components are three scalar wave equations.

$\square$ Miscellaneous Vector Identities

These are further examples of usages of $\nabla$.

Example Divergence and Curl of a Product

First , simplify $\nabla\cdot(f\,\bold{V})$ ,

Now , simplify $\nabla\cross(f\,\bold{V})$. Consider the $x$-component :

This is the $x$-component of $f(\nabla\cross\bold{V})+(\nabla f)\cross\bold{V}$ , so we have

Example Gradient of a Dot Product

We can prove the equation by applying $\nabla_A$ and $\nabla_B$ , which operate only on $\bold{A}$ or $\bold{B}$.

$\S3.7$ Vector Intergration

$\square$ Line Integrals

Possible forms :

These integrals are calculated over some path $C$.

$\square$ Surface Integrals

Possible forms :

Here , $d\bold{\sigma}$ is $\hat{\bold{n}}dA$ , where $\hat{\bold{n}}$ is a unit vector indicating the normal direction.

$\square$ Volume Integrals

$\S3.8$ Integral Theorems