ME 210 Applied Mathematics for Mechanical Engineers
Determinants, Minors, and Cofactors
Determinant:
It is a scalar quantity and defined only for square arrays.
Every square array A of size n has a unique determinant value D and it is shown as
a11 a12 a1n
ME 210 Applied Mathematics for Mechanical Engineers
Vector Calculus
The basic concepts of calculus such as convergence, continuity, and
differentiability can be introduced to vector analysis.
Convergence: An infinite sequence of vectors , v n n = 1, 2, is
ME 210 Applied Mathematics for Mechanical Engineers
Tangent and Arc Length of a Curve
The tangent to a curve C at a point A on it is defined as the limiting position of the
straight line L through A and B, as B approaches A along the curve as illustrated
ME 210 Applied Mathematics for Mechanical Engineers
Scalar and Vector Fields
Scalar: A geometrical or physical quantity that can completely be characterized by
a single number. For example: length of a bar, mass of an object,
electrical resistivity of a m
ME 210 Applied Mathematics for Mechanical Engineers
VECTORS AND VECTOR ANALYSES
Introduction
Three dimensional Space Coordinates & Transformations:
Cartesian Coordinates
Cylindrical Coordinates
Spherical Coordinates
Prof. Dr. Faruk Arn
Spring 2010
ME 210
ME 210 Applied Mathematics for Mechanical Engineers
Equations of Lines and Planes in 3-D Space
Straight Lines
Consider the straight line, L. passing through a
P
given point P1(x1, y1, z1,) and parallel to a given
P1
vector V = a i + b j + c k . For any po
ME 210 Applied Mathematics for Mechanical Engineers
Curvature and Torsion of a Curve, Normal, Binormal, TNB Frame
The velocity, v( t ) , of a point P moving along a space curve C is always tangent to
this curve traced by the tip of the position vector, r
ME 210 Applied Mathematics for Mechanical Engineers
System of Linear Algebraic Equations
Consider a system of m linear algebraic equations with n unknowns: x1, x2, . , xn
where aij and bi are constants.
a11 x1 + a12 x 2 + . + a1n x n = b1
a21 x1 + a22 x 2
ME 210 Applied Mathematics for Mechanical Engineers
Similar Matrices, Similarity Transformation and Diagonalization
Two square matrices [A] and [B] of the same size n are called similar if there exits a
non-singular [P] matrix of size n such that the matr
ME 210 Applied Mathematics for Mechanical Engineers
Inverse of a Square Matrix
Adj [A]
[A]-1 =
det A
Adjoint matrix of A
Determinant of A
Adj [A] = [Cij]T
Transpose of the Cofactor Matrix, [Cij]T
Cofactor Matrix: Matrix formed from cofactors of elements (
ME 210 Applied Mathematics for Mechanical Engineers
Gauss(ian) Elimination
This is a systematic method in which (the coefficient of the) 1st variable is eliminated
(made equal to zero) from the (m1) equations first, the 2nd variable is eliminated
from the
ME 210 Applied Mathematics for Mechanical Engineers
Eigenvalues and Eigenvectors
In a large number of physical and technical problems, the system of linear
equations of the type
a11 x1 + a12 x2 + + a1n xn = x1
a21 x1 + a22 x2 + + a2n xn = x2
.
an1 x1 + an
ME 210 Applied Mathematics for Mechanical Engineers
Cramers Rule
A useful application of determinants to the solution of linear algebraic equations
[A] [x] = [b]
is the Cramers rule, which can be used only when the coefficient matrix [A] is a
non-singular
ME 210 Applied Mathematics for Mechanical Engineers
10. If [C] = [A][B] where [A], [B], and [C] are all square matrices of the same
size, then det[C] = det[A] det[B]
Submatrices and Rank
Submatrix: Any matrix obtained by deleting some rows and/or columns
ME 210 Applied Mathematics for Mechanical Engineers
INTRODUCTION TO LINEAR ALGEBRA
Matrices and Vectors
Matrix: A rectangular array of scalars (numbers, variables, or functions,
real or complex).
a11 a12
a
21 a 22
a m1 a m 2
Columns
Element ai,j
ith row
ME 210 Applied Mathematics for Mechanical Engineers
Linear Dependence and Independence of Vectors
Given any set of m vectors [x1], [x2], , [xm] with the same number of components
in each, and any set of m scalars c1, c2, , cm,
if the linear combination of
ME 210 Applied Mathematics for Mechanical Engineers
Velocity and Acceleration along a Curve
If a body (represented by a point P) moves along a path C whose parametric
representation is given by its position vector, then the successive derivatives r ( t )