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 CalculusRead More

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