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$\,\,\,\textbf{row vector}$.

(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$\,\,\,\textbf{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$\,\,\,\textbf{unchanged}$vector$\bold{A}$now takes the$\,\,\,\textbf{changed}$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

Summary : The transformation from one orthogonal Cartesian coordinate system to another Cartesian system is described by an$\,\,\,\textbf{orthogonal}$matrix.

$\square$Reflections

For simplicity , consider first the$\,\,\,\textbf{inversion}$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$\,\,\,\textbf{axial vectors}$, or$\,\,\,\textbf{pseudovectors}$, while the other are called$\,\,\,\textbf{polar vectors}$, or just$\,\,\,\textbf{vectors}$. 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$\,\,\,\textbf{pseudoscalar}$. 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$\,\,\,\textbf{Euler angles}$. 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$\,\,\,\textbf{vector field}$. 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 and learn how to use them.

$\square$Gradient ,$\nabla$

The$\,\,\,\textbf{gradient}$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$\,\,\,\textbf{vector differential operator}$.

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$\,\,\,\textbf{divergence}$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$\,\,\,\textbf{equation of continuity}$, 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$\,\,\,\textbf{solenoidal}$.

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

This vector is called the$\,\,\,\textbf{curl}$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$\,\,\,\textbf{from the top down}$.

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 everywhere is termed$\,\,\,\textbf{irrotational}$.

$\S3.6$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$\,\,\,\textbf{Laplacian equation}$.

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

Example Laplacian of a Central Field Potential

Calculate$\laplacian\varphi(r)$.

$\square$Irrotational and Solenoidal Vector Fields

The curl of the divergence :