Engineering Maths Flashcards

Question 1

Q

Number & Algebra

Algebra
  Laws of indices
  Factor Theorem
    Remainder Theorem
  Partial Fraction (for Integrals and Laplace Transforms)

Logarithm
Laws

Taylor series -

Maclaurin’s Series -

Answer

A

Number & Algebra

Algebra
Laws of indices - 6 laws
Factor Theorem - if for x=a p(x)=0 then x-a is factor of p(x)
Remainder Theorem - if ax2+bx+c divided by x-p, remainder is ap2+bp+c
Partial Fraction (for Integrals and Laplace Transforms)

Logarithm (exponential growth or decay)
5 Laws (product rule, quotient rule, power rule, identity rule)

Taylor series - expanding a function about any point as a function of its derivatives.

Maclaurin’s Series - some mathematical fn. as power series. It’s a taylor series about 0.

Question 2

Q

Matrices & Determinants

Determinant -
If two rows or columns identical then Det
If two rows or columns interchanged then Det
If rows interchanged to columns Det
Elementary operations performed Det
Factor Theorem for Det

Application of Det.
Area of triangle -
check collinearity of points

	Solutions of simultaneous linear equations
		Cramers Rule - 
		unique 
		Infinite soln 
		Inconsistent

Solutions of simultaneous homogeneous linear equations
	Infinite soln 
	Trivial Soln.

Matrix
Row -
Column -

	Null - 
	Square - 
	Diagonal - 
	Scalar - 
	Unit or Identity - 
	Elementary -

	Symmetric - 
	Skew-symmetric - 
	Traingular(Echelon form) - 
	Transpose - 
	Orthogonal - 
	Conjugate -
	Atheta - 
	Unitary -

	Idempotent -
	Periodic - 
	Nilpotent - 
	Involuntary - 
	Singular Matrix - 
	Inverse - 
	Adjoint - 
	Cofactors - 
	Minor -

Rank
Normal form (canonical form)
Rank -

Consistency - 
	unique 
	Infinite soln. 
Inconsistent - 

Homogeneous Eqn. - 
	Infinite soln 
	Trivial Soln

Linearly Dependent

Eigen Values, Eigen Vectors

characteristic polynomial
characteristic equation
characteristic roots or eigen values
Cayley-Hamilton Theorem

sum of eigen values - 
product of eigen values - 
    eigen values of transpose of matrix - 
    eigen values of matrix kA, Ainverse, A^m

if distinct eigen values - 
if two or more eigen values equal -

Answer

A

Matrices & Determinants

Determinant -
If two rows or columns identical then Det=0
If two rows or columns interchanged then Det changes sign
If rows interchanged to columns Det= remains same
Elementary operations performed Det = remains same
Factor Theorem for Det put x=y if Det=0 x-y is factor

Application of Det.
Area of triangle - 1/2 of Det
check collinearity of points (cordinates of points given)

Solutions of simultaneous linear equations
	Cramers Rule - x=D1/D, y=D2/D, z=D3/D
	unique for D not equal to 0
	Infinite soln for D=0 and D1=D2=D3=0
	Inconsistent for D=0 but D1 or D2 or D3 not equal to 0

Solutions of simultaneous homogeneous linear equations
	Infinite soln for D=0
	Trivial Soln. for D not equal to 0

Matrix
Row - only one row and any number of columns {2 7 3 9]
Column - one column and any number of rows [1 2 3]T

Null - all the elements are zeros
Square - number of rows is equal to the number of columns
Diagonal - square matrix with all its non-diagonal elements are zero
Scalar - diagonal matrix with all the diagonal elements equal to a scalar
Unit or Identity - all the diagonal elements are unity and non-diagonal elements are zero
Elementary - obtained by performing elementary operations on Identity matrix

	Symmetric - AT = A
	Skew-symmetric - AT = -A
	Traingular(Echelon form) - elements below the leading diagonal are zero UT &amp; elements above the leading diagonal are zero LT
	Transpose - interchange the rows and the corresponding columns
	Orthogonal - A.AT=I 
	Conjugate - z (x+iy) &amp; z bar (x-iy)
	Atheta - Transpose of conjugate
	Unitary - AthetaA=I

	Idempotent - A2=A
	Periodic - Ak+1=A
	Nilpotent - Ak=0
	Involuntary - A2=I
	Singular Matrix - det(A)=0
	Inverse - AA-1=I
	Adjoint - transpose of matrix of cofactors
	Cofactors - (-1)(i+j)Minor
	Minor - det obtained when i row &amp; j column deleted.

Rank
Normal form (canonical form)
Rank - Number of non-zero row in upper triangular matrix

Consistency - Rank A = Rank C
	unique when Rank A = Rank C = n
	Infinite soln. when Rank A = Rank C = r < n
Inconsistent - Rank A not equal to Rank C

Homogeneous Eqn. - Eqn are always consistent
	Infinite soln Rank A = n
	Trivial Soln Rank A < n

Linearly Dependent
if summation lamdai Xi = 0 (not all lamdas be zero)
else linearly independent

Eigen Values, Eigen Vectors

characteristic polynomial
characteristic equation
characteristic roots or eigen values
Cayley-Hamilton Theorem - every matrix satisfies its own characteristics equation 

sum of eigen values - trace (sum of principal diagonal elements)
product of eigen values - determinant
    eigen values of transpose of matrix - same 
    eigen values of matrix kA - k times lamda
                                Ainverse - lamda inverse
                                        A^m - lamda^m

if distinct eigen values - eigen vectors are linearly independent
if two or more eigen values equal it may or maynot be possible to get linearly independent eigen vectors.

Question 3

Q

Vector
  vector addition -
  multiplication -
   scalar product -
   cross product -
  triple product -
  unit vector-
position vector - 
  ratio formula -

point function -
grad phi -
directional derivative -

divergence -
curl -

conservative field -
solenoidal field -
Irrotational field -

Line Integral -
Surface Integral-
Volume Integral -

greens theorem -
stokes theorem -
Gauss theorem -

Answer

A

Vector - quantity having both magnitude and direction such as force, velocity, acceleration, displacement etc.
vector addition (triangle law and parallelogram law)
multiplication
scalar product - work done
cross product - area of parallelogram, moment, angular velocity
triple product - volume of parallelopiped
unit vector-
position vector -
ratio formula -

point function - value at point depends on position of point (scalar point function or vector point function)
grad phi - del Phi (phi - scalar point function)
(vector normal to surface & max rate of change of phi)
directional derivative - del Phi . eta

divergence - del F (F - vector function) (solenoidal field)
curl - del x F (rotational or irrotational field)

conservative field (gradient of some fn)
solenoidal field (div. free)
Irrotational field (curl is 0) &amp;

Line Integral - work, circulation
Surface Integral -
Volume Integral -

greens theorem - L & S
stokes theorem - L & S
Gauss theorem - S & V

Question 4

Q

Limits -

Continuous -

Types of discontinuities -

Differentiability -

Lagrange’s Mean Value Theorem -

Rolles Mean Value Theorem -

Cauchy’s Mean Value Theorem -

Homogeneous function -

Euler Theorem on Homogenous function -

Composite function -

Total Derivative -

Maxima -
Minima -

Definite Integral -
Improper Integral -

Answer

A

Limits - if both RHL & LHL exists and are equal.

Continuous - if Lt x tending to a f(x) = f(a)

Types of discontinuities - Type I, Type II, Type III
Type I - LHL not equal to RHL
Type II - Either LHL or RHL or both don’t exist
Type III - LHL = RHL but not equal to f(a)

Differentiability - if both LHD, RHD exists & equal to f’(a)

Lagrange’s Mean Value Theorem - if f(a) != f(b) for f being continuous in [a,b] & differentiable in (a,b) then f’(c) = [(f(b)-f(a))/(b-a)]

Rolles Mean Value Theorem - if f(a)=f(b) for f being continuous in [a,b] & differentiable in (a,b) then f’(c)=0 for c in (a,b)

Cauchy’s Mean Value Theorem - f(x) & g(x) {continuous & differentiable} then f’(c)/g’(c)=[f(b)-f(a)/g(b)-g(a)]

Homogeneous function - f(kx,ky) = k^n f(x,y)
product of two homogenous function is homogeneous & if Nr & Dr are homogenous rational fn is also homogenous

Euler Theorem on Homogenous function -
for z algebraic
xdel z by del x + y del z by del y = nz
x^2 del^2 z/del x^2 + 2xy del^2 z/del xy + y^2 del^2 z/del y^2 = n(n-1)z

for z not algebraic - phi (z) algebraic
xdel z by del x + y del z by del y = n phi (z)/ phi’(z)

Composite function - function in a function z=f(x) & x(t)

Total Derivative - Dz/Dt = del z/del x .dx/dt + del z/ del y.dy/dt

Maxima - if continuous fn increases upto certain value & then decreases, max value is called Maxima
Minima - opposite of above

Definite Integral -
Improper Integral - one or both limits infinite or f goes to infinite

Question 5

Q

Differential Equations

Differential Equations - Types.
Ordinary DE -
Partial DE -

Order of DE -
Degree of DE -

Solution of DE -

A). DE of first order & first degree -

SOV
Homogeneous Eqn.
Linear Equation of First Order
Exact DE

B). LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH CONSTANT COEFFICIENTS

Cauchy’s Equation -
Euler’s Equation -
Initial Value Problem -
Boundary Value Problem -

Partial Differential Equations -
types & solutions

Integral Transforms -

Laplace Transform -

Fourier Transform -

Formulae
Theorems
Use of LT for solving Differential Equations (IVP)

Answer

A

Differential Equations

Differential Equations - Equation that involves differential co-efficient. DE are of 2 Types.
Ordinary DE - DE involving derivatives with respect to a single independent variable.
Partial DE - DE involving partial derivatives with respect to more than one independent variable.

Order of DE - order of the highest differential co-efficient present in the equation.
Degree of DE - degree of the highest derivative after removing the radical sign and fraction.

Solution of DE -

A). DE of first order & first degree -

SOV
Homogeneous Eqn.
Linear Equation of First Order
Exact DE

B). LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH CONSTANT COEFFICIENTS

complete solution = complementary function + particular integral

Cauchy’s Equation -
Euler’s Equation -
Initial Value Problem -
Boundary Value Problem -

Partial Differential Equations - Wave equation, 1D transient Heat flow, 2D steady state Heat conduction

1-D Heat flow - SOV Putting, y=X(x)T(t) & solving for X&T
Wave Equation - solve using SOV
2-D steady Heat conduction - solve using SOV, u=X(x)Y(y)

Integral Transforms - Laplace transform, Fourier Transform, etc

Laplace Transform - type of Integral transform for solving ODE with IVP
Fourier Transform - type of Integral transform for solving PDE with BVP

Formulae
Theorems
Use of LT for solving Differential Equations (IVP)

Question 6

Q

Numerical Methods

Numerical solutions of linear and non-linear algebraic equations

Roots/Solution of an equation

Roots/Solution of System of Linear Equations
Direct Solvers -
Iterative Solvers -

Solution of ODE
single step methods -
multi-step methods -

Integration by trapezoidal and Simpson’s rule -

Answer

A

Numerical Methods

Numerical solutions of linear and non-linear algebraic equations

Roots/Solution of an equation
Bisection Method
Newton-Raphson Method

Roots/Solution of System of Linear Equations
Direct Solvers - Gauss Elimination, LU Decomposition (Crout’s)
Iterative Solvers - Jacobi, Gauss Siedal,

Solution of ODE
	Single step (Taylor Series), Picard, Multi step method (Runge Kutta)

Integration by trapezoidal and Simpson’s rule -

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Engineering Maths Flashcards

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