Engineering Mathematics Formulas for GATE Exam

Engineering Mathematics Formulas for GATE Exam Notes | Study Engineering Mathematics - Civil Engineering (CE)

Document Description: Engineering Mathematics Formulas for GATE Exam for Civil Engineering (CE) 2022 is part of Engineering Mathematics preparation. The notes and questions for Engineering Mathematics Formulas for GATE Exam have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Engineering Mathematics Formulas for GATE Exam covers topics like and Engineering Mathematics Formulas for GATE Exam Example, for Civil Engineering (CE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Engineering Mathematics Formulas for GATE Exam.

Introduction of Engineering Mathematics Formulas for GATE Exam in English is available as part of our Engineering Mathematics for Civil Engineering (CE) & Engineering Mathematics Formulas for GATE Exam in Hindi for Engineering Mathematics course. Download more important topics related with notes, lectures and mock test series for Civil Engineering (CE) Exam by signing up for free. Civil Engineering (CE): Engineering Mathematics Formulas for GATE Exam Notes | Study Engineering Mathematics - Civil Engineering (CE)
``` 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
AB = 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
AB = 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 dened 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 dened by
jAj =
?
i, j,k,...

i jk...
A
1i
A
2 j
A
3k
. . .
where the number of the sufxes is equal to the order of the matrix.
22 matrices
If A =

a b
c d

then,
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
AB = 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 dened 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 dened by
jAj =
?
i, j,k,...

i jk...
A
1i
A
2 j
A
3k
. . .
where the number of the sufxes is equal to the order of the matrix.
22 matrices
If A =

a b
c d

then,
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
() = jAIj. If A is Hermitian then the eigenvalues

i
are real and the eigenvectors u
i
are mutually orthogonal. jAIj = 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 = UU
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 jOji
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
jiihiji
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
AB = 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 dened 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 dened by
jAj =
?
i, j,k,...

i jk...
A
1i
A
2 j
A
3k
. . .
where the number of the sufxes is equal to the order of the matrix.
22 matrices
If A =

a b
c d

then,
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
() = jAIj. If A is Hermitian then the eigenvalues

i
are real and the eigenvectors u
i
are mutually orthogonal. jAIj = 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 = UU
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 jOji
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
jiihiji
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
1
+
2
1
2
div(A
1
+ A
2
) div A
1
+ div A
2
1

2
)
1
2
+
2
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
1
(A
1
2
div(curl A) 0, curl(grad) 0
2
A
1
 A
2
) A
1
 (curl A
2
) + (A
1
2
+ A
2
 (curl A
1
) + (A
2
1
Vector Calculus
```

Engineering Mathematics

47 videos|79 docs|55 tests
 Use Code STAYHOME200 and get INR 200 additional OFF

Engineering Mathematics

47 videos|79 docs|55 tests

Top Courses for Civil Engineering (CE)

Track your progress, build streaks, highlight & save important lessons and more!

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;