Page 1
If i, j, k are orthonormal vectors and A = A
x
i + A
y
j + A
z
k thenjAj
2
= A
2
x
+ A
2
y
+ A
2
z
. [Orthonormal vectors
orthogonal unit vectors.]
Scalar product
A B =jAjjBj cos where is the angle between the vectors
= A
x
B
x
+ A
y
B
y
+ A
z
B
z
= [ A
x
A
y
A
z ]
2
4
B
x
B
y
B
z
3
5
Scalar multiplication is commutative: A B = B A.
Equation of a line
A point r (x, y, z) lies on a line passing through a point a and parallel to vector b if
r = a +b
with a real number.
Vector Algebra
Page 2
If i, j, k are orthonormal vectors and A = A
x
i + A
y
j + A
z
k thenjAj
2
= A
2
x
+ A
2
y
+ A
2
z
. [Orthonormal vectors
orthogonal unit vectors.]
Scalar product
A B =jAjjBj cos where is the angle between the vectors
= A
x
B
x
+ A
y
B
y
+ A
z
B
z
= [ A
x
A
y
A
z ]
2
4
B
x
B
y
B
z
3
5
Scalar multiplication is commutative: A B = B A.
Equation of a line
A point r (x, y, z) lies on a line passing through a point a and parallel to vector b if
r = a +b
with a real number.
Vector Algebra
Equation of a plane
A point r (x, y, z) is on a plane if either
(a) r
b
d =jdj, where d is the normal from the origin to the plane, or
(b)
x
X
+
y
Y
+
z
Z
= 1 where X, Y, Z are the intercepts on the axes.
Vector product
AB = njAjjBj sin, where is the angle between the vectors and n is a unit vector normal to the plane containing
A and B in the direction for which A, B, n form a right-handed set of axes.
A B in determinant form
i j k
A
x
A
y
A
z
B
x
B
y
B
z
A B in matrix form
2
4
0 A
z
A
y
A
z
0 A
x
A
y
A
x
0
3
5
2
4
B
x
B
y
B
z
3
5
Vector multiplication is not commutative: A B =B A.
Scalar triple product
A B C = A B C =
A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z
=A C B, etc.
Vector triple product
A (B C) = (A C)B (A B)C, (A B) C = (A C)B (B C)A
Non-orthogonal basis
A = A
1
e
1
+ A
2
e
2
+ A
3
e
3
A
1
=
0
A where
0
=
e
2
e
3
e
1
(e
2
e
3
)
Similarly for A
2
and A
3
.
Summation convention
a = a
i
e
i
implies summation over i = 1 . . . 3
a b = a
i
b
i
(a b)
i
="
i jk
a
j
b
k
where"
123
= 1; "
i jk
="
ik j
"
i jk
"
klm
=
il
jm
im
jl
Page 3
If i, j, k are orthonormal vectors and A = A
x
i + A
y
j + A
z
k thenjAj
2
= A
2
x
+ A
2
y
+ A
2
z
. [Orthonormal vectors
orthogonal unit vectors.]
Scalar product
A B =jAjjBj cos where is the angle between the vectors
= A
x
B
x
+ A
y
B
y
+ A
z
B
z
= [ A
x
A
y
A
z ]
2
4
B
x
B
y
B
z
3
5
Scalar multiplication is commutative: A B = B A.
Equation of a line
A point r (x, y, z) lies on a line passing through a point a and parallel to vector b if
r = a +b
with a real number.
Vector Algebra
Equation of a plane
A point r (x, y, z) is on a plane if either
(a) r
b
d =jdj, where d is the normal from the origin to the plane, or
(b)
x
X
+
y
Y
+
z
Z
= 1 where X, Y, Z are the intercepts on the axes.
Vector product
AB = njAjjBj sin, where is the angle between the vectors and n is a unit vector normal to the plane containing
A and B in the direction for which A, B, n form a right-handed set of axes.
A B in determinant form
i j k
A
x
A
y
A
z
B
x
B
y
B
z
A B in matrix form
2
4
0 A
z
A
y
A
z
0 A
x
A
y
A
x
0
3
5
2
4
B
x
B
y
B
z
3
5
Vector multiplication is not commutative: A B =B A.
Scalar triple product
A B C = A B C =
A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z
=A C B, etc.
Vector triple product
A (B C) = (A C)B (A B)C, (A B) C = (A C)B (B C)A
Non-orthogonal basis
A = A
1
e
1
+ A
2
e
2
+ A
3
e
3
A
1
=
0
A where
0
=
e
2
e
3
e
1
(e
2
e
3
)
Similarly for A
2
and A
3
.
Summation convention
a = a
i
e
i
implies summation over i = 1 . . . 3
a b = a
i
b
i
(a b)
i
="
i jk
a
j
b
k
where"
123
= 1; "
i jk
="
ik j
"
i jk
"
klm
=
il
jm
im
jl
Unit matrices
The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements
zero, i.e., (I)
i j
=
i j
. If A is a square matrix of order n, then AI = IA = A. Also I = I
1
.
I is sometimes written as I
n
if the order needs to be stated explicitly.
Products
If A is a (n l) matrix and B is a (l m) then the product AB is dened by
(AB)
i j
=
l
?
k=1
A
ik
B
k j
In general AB6= BA.
Transpose matrices
If A is a matrix, then transpose matrix A
T
is such that (A
T
)
i j
= (A)
ji
.
Inverse matrices
If A is a square matrix with non-zero determinant, then its inverse A
1
is such that AA
1
= A
1
A = I.
(A
1
)
i j
=
transpose of cofactor of A
i j
jAj
where the cofactor of A
i j
is (1)
i+ j
times the determinant of the matrix A with the j-th row and i-th column deleted.
Determinants
If A is a square matrix then the determinant of A,jAj ( det A) is dened by
jAj =
?
i, j,k,...
i jk...
A
1i
A
2 j
A
3k
. . .
where the number of the sufxes is equal to the order of the matrix.
22 matrices
If A =
a b
c d
then,
jAj = ad bc A
T
=
a c
b d
A
1
=
1
jAj
d b
c a
Product rules
(AB . . . N)
T
= N
T
. . . B
T
A
T
(AB . . . N)
1
= N
1
. . . B
1
A
1
(if individual inverses exist)
jAB . . . Nj =jAjjBj . . .jNj (if individual matrices are square)
Orthogonal matrices
An orthogonal matrix Q is a square matrix whose columns q
i
form a set of orthonormal vectors. For any orthogonal
matrix Q,
Q
1
= Q
T
, jQj =1, Q
T
is also orthogonal.
Matrix Algebra
Page 4
If i, j, k are orthonormal vectors and A = A
x
i + A
y
j + A
z
k thenjAj
2
= A
2
x
+ A
2
y
+ A
2
z
. [Orthonormal vectors
orthogonal unit vectors.]
Scalar product
A B =jAjjBj cos where is the angle between the vectors
= A
x
B
x
+ A
y
B
y
+ A
z
B
z
= [ A
x
A
y
A
z ]
2
4
B
x
B
y
B
z
3
5
Scalar multiplication is commutative: A B = B A.
Equation of a line
A point r (x, y, z) lies on a line passing through a point a and parallel to vector b if
r = a +b
with a real number.
Vector Algebra
Equation of a plane
A point r (x, y, z) is on a plane if either
(a) r
b
d =jdj, where d is the normal from the origin to the plane, or
(b)
x
X
+
y
Y
+
z
Z
= 1 where X, Y, Z are the intercepts on the axes.
Vector product
AB = njAjjBj sin, where is the angle between the vectors and n is a unit vector normal to the plane containing
A and B in the direction for which A, B, n form a right-handed set of axes.
A B in determinant form
i j k
A
x
A
y
A
z
B
x
B
y
B
z
A B in matrix form
2
4
0 A
z
A
y
A
z
0 A
x
A
y
A
x
0
3
5
2
4
B
x
B
y
B
z
3
5
Vector multiplication is not commutative: A B =B A.
Scalar triple product
A B C = A B C =
A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z
=A C B, etc.
Vector triple product
A (B C) = (A C)B (A B)C, (A B) C = (A C)B (B C)A
Non-orthogonal basis
A = A
1
e
1
+ A
2
e
2
+ A
3
e
3
A
1
=
0
A where
0
=
e
2
e
3
e
1
(e
2
e
3
)
Similarly for A
2
and A
3
.
Summation convention
a = a
i
e
i
implies summation over i = 1 . . . 3
a b = a
i
b
i
(a b)
i
="
i jk
a
j
b
k
where"
123
= 1; "
i jk
="
ik j
"
i jk
"
klm
=
il
jm
im
jl
Unit matrices
The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements
zero, i.e., (I)
i j
=
i j
. If A is a square matrix of order n, then AI = IA = A. Also I = I
1
.
I is sometimes written as I
n
if the order needs to be stated explicitly.
Products
If A is a (n l) matrix and B is a (l m) then the product AB is dened by
(AB)
i j
=
l
?
k=1
A
ik
B
k j
In general AB6= BA.
Transpose matrices
If A is a matrix, then transpose matrix A
T
is such that (A
T
)
i j
= (A)
ji
.
Inverse matrices
If A is a square matrix with non-zero determinant, then its inverse A
1
is such that AA
1
= A
1
A = I.
(A
1
)
i j
=
transpose of cofactor of A
i j
jAj
where the cofactor of A
i j
is (1)
i+ j
times the determinant of the matrix A with the j-th row and i-th column deleted.
Determinants
If A is a square matrix then the determinant of A,jAj ( det A) is dened by
jAj =
?
i, j,k,...
i jk...
A
1i
A
2 j
A
3k
. . .
where the number of the sufxes is equal to the order of the matrix.
22 matrices
If A =
a b
c d
then,
jAj = ad bc A
T
=
a c
b d
A
1
=
1
jAj
d b
c a
Product rules
(AB . . . N)
T
= N
T
. . . B
T
A
T
(AB . . . N)
1
= N
1
. . . B
1
A
1
(if individual inverses exist)
jAB . . . Nj =jAjjBj . . .jNj (if individual matrices are square)
Orthogonal matrices
An orthogonal matrix Q is a square matrix whose columns q
i
form a set of orthonormal vectors. For any orthogonal
matrix Q,
Q
1
= Q
T
, jQj =1, Q
T
is also orthogonal.
Matrix Algebra Solving sets of linear simultaneous equations
If A is square then Ax = b has a unique solution x = A
1
b if A
1
exists, i.e., ifjAj6= 0.
If A is square then Ax = 0 has a non-trivial solution if and only ifjAj = 0.
An over-constrained set of equations Ax = b is one in which A has m rows and n columns, where m (the number
of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the
errorjAx bj) is the solution of the n equations A
T
Ax = A
T
b. If the columns of A are orthonormal vectors then
x = A
T
b.
Hermitian matrices
The Hermitian conjugate of A is A
y
= (A
)
T
, where A
is a matrix each of whose components is the complex
conjugate of the corresponding components of A. If A = A
y
then A is called a Hermitian matrix.
Eigenvalues and eigenvectors
The n eigenvalues
i
and eigenvectors u
i
of an n n matrix A are the solutions of the equation Au = u. The
eigenvalues are the zeros of the polynomial of degree n, P
n
() = jAIj. If A is Hermitian then the eigenvalues
i
are real and the eigenvectors u
i
are mutually orthogonal. jAIj = 0 is called the characteristic equation of the
matrix A.
Tr A =
?
i
i
, alsojAj =
?
i
i
.
If S is a symmetric matrix, is the diagonal matrix whose diagonal elements are the eigenvalues of S, and U is the
matrix whose columns are the normalized eigenvectors of A, then
U
T
SU = and S = UU
T
.
If x is an approximation to an eigenvector of A then x
T
Ax=(x
T
x) (Rayleigh's quotient) is an approximation to the
corresponding eigenvalue.
Commutators
[A, B] AB BA
[A, B] =[B, A]
[A, B]
y
= [B
y
, A
y
]
[A + B, C] = [A, C] + [B, C]
[AB, C] = A[B, C] + [A, C]B
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
Hermitian algebra
b
y
= (b
1
, b
2
, . . .)
Matrix form Operator form Bra-ket form
Hermiticity b
A c = (A b)
c
Z
O =
Z
(O )
h jOji
Eigenvalues, real Au
i
=
(i)
u
i
O
i
=
(i)
i
Ojii =
i
jii
Orthogonality u
i
u
j
= 0
Z
i
j
= 0 hij ji = 0 (i6= j)
Completeness b =
?
i
u
i
(u
i
b) =
?
i
i
Z
i
=
?
i
jiihiji
Rayleigh?Ritz
Lowest eigenvalue
0
b
A b
b
b
0
Z
O
Z
h jOj i
h j i
Page 5
If i, j, k are orthonormal vectors and A = A
x
i + A
y
j + A
z
k thenjAj
2
= A
2
x
+ A
2
y
+ A
2
z
. [Orthonormal vectors
orthogonal unit vectors.]
Scalar product
A B =jAjjBj cos where is the angle between the vectors
= A
x
B
x
+ A
y
B
y
+ A
z
B
z
= [ A
x
A
y
A
z ]
2
4
B
x
B
y
B
z
3
5
Scalar multiplication is commutative: A B = B A.
Equation of a line
A point r (x, y, z) lies on a line passing through a point a and parallel to vector b if
r = a +b
with a real number.
Vector Algebra
Equation of a plane
A point r (x, y, z) is on a plane if either
(a) r
b
d =jdj, where d is the normal from the origin to the plane, or
(b)
x
X
+
y
Y
+
z
Z
= 1 where X, Y, Z are the intercepts on the axes.
Vector product
AB = njAjjBj sin, where is the angle between the vectors and n is a unit vector normal to the plane containing
A and B in the direction for which A, B, n form a right-handed set of axes.
A B in determinant form
i j k
A
x
A
y
A
z
B
x
B
y
B
z
A B in matrix form
2
4
0 A
z
A
y
A
z
0 A
x
A
y
A
x
0
3
5
2
4
B
x
B
y
B
z
3
5
Vector multiplication is not commutative: A B =B A.
Scalar triple product
A B C = A B C =
A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z
=A C B, etc.
Vector triple product
A (B C) = (A C)B (A B)C, (A B) C = (A C)B (B C)A
Non-orthogonal basis
A = A
1
e
1
+ A
2
e
2
+ A
3
e
3
A
1
=
0
A where
0
=
e
2
e
3
e
1
(e
2
e
3
)
Similarly for A
2
and A
3
.
Summation convention
a = a
i
e
i
implies summation over i = 1 . . . 3
a b = a
i
b
i
(a b)
i
="
i jk
a
j
b
k
where"
123
= 1; "
i jk
="
ik j
"
i jk
"
klm
=
il
jm
im
jl
Unit matrices
The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements
zero, i.e., (I)
i j
=
i j
. If A is a square matrix of order n, then AI = IA = A. Also I = I
1
.
I is sometimes written as I
n
if the order needs to be stated explicitly.
Products
If A is a (n l) matrix and B is a (l m) then the product AB is dened by
(AB)
i j
=
l
?
k=1
A
ik
B
k j
In general AB6= BA.
Transpose matrices
If A is a matrix, then transpose matrix A
T
is such that (A
T
)
i j
= (A)
ji
.
Inverse matrices
If A is a square matrix with non-zero determinant, then its inverse A
1
is such that AA
1
= A
1
A = I.
(A
1
)
i j
=
transpose of cofactor of A
i j
jAj
where the cofactor of A
i j
is (1)
i+ j
times the determinant of the matrix A with the j-th row and i-th column deleted.
Determinants
If A is a square matrix then the determinant of A,jAj ( det A) is dened by
jAj =
?
i, j,k,...
i jk...
A
1i
A
2 j
A
3k
. . .
where the number of the sufxes is equal to the order of the matrix.
22 matrices
If A =
a b
c d
then,
jAj = ad bc A
T
=
a c
b d
A
1
=
1
jAj
d b
c a
Product rules
(AB . . . N)
T
= N
T
. . . B
T
A
T
(AB . . . N)
1
= N
1
. . . B
1
A
1
(if individual inverses exist)
jAB . . . Nj =jAjjBj . . .jNj (if individual matrices are square)
Orthogonal matrices
An orthogonal matrix Q is a square matrix whose columns q
i
form a set of orthonormal vectors. For any orthogonal
matrix Q,
Q
1
= Q
T
, jQj =1, Q
T
is also orthogonal.
Matrix Algebra Solving sets of linear simultaneous equations
If A is square then Ax = b has a unique solution x = A
1
b if A
1
exists, i.e., ifjAj6= 0.
If A is square then Ax = 0 has a non-trivial solution if and only ifjAj = 0.
An over-constrained set of equations Ax = b is one in which A has m rows and n columns, where m (the number
of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the
errorjAx bj) is the solution of the n equations A
T
Ax = A
T
b. If the columns of A are orthonormal vectors then
x = A
T
b.
Hermitian matrices
The Hermitian conjugate of A is A
y
= (A
)
T
, where A
is a matrix each of whose components is the complex
conjugate of the corresponding components of A. If A = A
y
then A is called a Hermitian matrix.
Eigenvalues and eigenvectors
The n eigenvalues
i
and eigenvectors u
i
of an n n matrix A are the solutions of the equation Au = u. The
eigenvalues are the zeros of the polynomial of degree n, P
n
() = jAIj. If A is Hermitian then the eigenvalues
i
are real and the eigenvectors u
i
are mutually orthogonal. jAIj = 0 is called the characteristic equation of the
matrix A.
Tr A =
?
i
i
, alsojAj =
?
i
i
.
If S is a symmetric matrix, is the diagonal matrix whose diagonal elements are the eigenvalues of S, and U is the
matrix whose columns are the normalized eigenvectors of A, then
U
T
SU = and S = UU
T
.
If x is an approximation to an eigenvector of A then x
T
Ax=(x
T
x) (Rayleigh's quotient) is an approximation to the
corresponding eigenvalue.
Commutators
[A, B] AB BA
[A, B] =[B, A]
[A, B]
y
= [B
y
, A
y
]
[A + B, C] = [A, C] + [B, C]
[AB, C] = A[B, C] + [A, C]B
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0
Hermitian algebra
b
y
= (b
1
, b
2
, . . .)
Matrix form Operator form Bra-ket form
Hermiticity b
A c = (A b)
c
Z
O =
Z
(O )
h jOji
Eigenvalues, real Au
i
=
(i)
u
i
O
i
=
(i)
i
Ojii =
i
jii
Orthogonality u
i
u
j
= 0
Z
i
j
= 0 hij ji = 0 (i6= j)
Completeness b =
?
i
u
i
(u
i
b) =
?
i
i
Z
i
=
?
i
jiihiji
Rayleigh?Ritz
Lowest eigenvalue
0
b
A b
b
b
0
Z
O
Z
h jOj i
h j i
Pauli spin matrices
x
=
0 1
1 0
,
y
=
0 i
i 0
,
z
=
1 0
0 1
x
y
= i
z
,
y
z
= i
x
,
z
x
= i
y
,
x
x
=
y
y
=
z
z
= I
Notation
is a scalar function of a set of position coordinates. In Cartesian coordinates
= (x, y, z); in cylindrical polar coordinates = (,', z); in spherical
polar coordinates = (r,,'); in cases with radial symmetry = (r).
A is a vector function whose components are scalar functions of the position
coordinates: in Cartesian coordinates A = iA
x
+ jA
y
+ kA
z
, where A
x
, A
y
, A
z
are independent functions of x, y, z.
In Cartesian coordinatesr (`del') i
?
?x
+ j
?
?y
+ k
?
?z
2
6
6
6
6
6
6
6
4
?
?x
?
?y
?
?z
3
7
7
7
7
7
7
7
5
grad =r, div A =r A, curl A =r A
Identities
grad(
1
+
2
) grad
1
+ grad
2
div(A
1
+ A
2
) div A
1
+ div A
2
grad(
1
2
)
1
grad
2
+
2
grad
1
curl(A
1
+ A
2
) curl A
1
+ curl A
2
div(A) div A + (grad) A, curl(A) curl A + (grad) A
div(A
1
A
2
) A
2
curl A
1
A
1
curl A
2
curl(A
1
A
2
) A
1
div A
2
A
2
div A
1
+ (A
2
grad)A
1
(A
1
grad)A
2
div(curl A) 0, curl(grad) 0
curl(curl A) grad(div A) div(grad A) grad(div A)r
2
A
grad(A
1
A
2
) A
1
(curl A
2
) + (A
1
grad)A
2
+ A
2
(curl A
1
) + (A
2
grad)A
1
Vector Calculus
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