# Notes on Mathematical Methods for Physicists Chapter3

本文最后更新于 2024年8月8日 晚上

# Notes on Mathematical Methods for Physicists

## Chapter3 Vector Analysis

Review of Basic Properties

(1)

(2)

(3) Vector polynomials (e.g.

(4) Unit vectors

(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

(13) Dot product replaced by multiplication of matrices

Vectors in Space

Vectors or Cross Product

Definition :

The direction of

This can be equivalently represented as

Note that the cross product is a quantity specifically defined for

Scalar Triple Product

Definition :

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

There is

##### Example Reciprocal Lattice

Let

In the band theory of solids , it is useful to introduce what is called a

and with

Then

Vector Triple Product

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

Coordinate Transformations

Rotations

Rotate the axis as the picture below

So the

which is equivalent to the matrix equation

It is easy to find that

is orthogonal.

Orthogonal Transformations

It is no accident that the transformation describing a rotation in

Summary : The transformation from one orthogonal Cartesian coordinate system to another Cartesian system is described by an

Reflections

For simplicity , consider first the

which clearly results in

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

It is easy to find that

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.

Rotations in

Rotations in

The argument we made to evaluate

In

(1) The coordinates are rotated about the

(2) The coordinates are rotated about the

(3) The coordinates are now rotated about the

The matrices that represent those rotations are

Note the order of

Note that those matrices are orthogonal with determinant

Differential Vector Operators

We are going to discuss

Gradient ,

The

which is of the form corresponding to the dot product of

We have given the

The

Having now established the legitimacy of the form

Note that

In physics , we have

There is a simple geometric interpretation of the gradient :*increase* in

##### Example Gradient of

Divergence ,

The

##### Example Divergence of Central Force Field

Consider

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

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

Curl ,

This vector is called the

##### Example Curl of a Central Force Field

Consider

so the result is 0.

Noting that a vector whose curl is zero everywhere is termed

Differential Vector Operators : Further Properties

Successive Applications of

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.

Laplacian

Laplacian is the divergence of the gradient. We have

When

Often the Laplacian is written as

##### Example Laplacian of a Central Field Potential

Calculate

Irrotational and Solenoidal Vector Fields

The curl of the divergence :