ME641 Engineering Analysis I
Homework Solution 1
ME641 Homework Solution 1
Problem 1
Show that if x is a nonzero vector,
is a unit vector.
Solution
Assume that x is a nonzero vector,
Thus, forming uu,
ME 641 Matrices
Learning Objectives
The student will be able to:
Define matrices as abstract vector space
Classify matrices
Rectangular, square, triangular, diagonal, identity
Symmetric, orthogon
ME641 Engineering Analysis I
Homework Assignment 1
ME641 Homework Assignment 1
Problem 1
Show that if x is a nonzero vector,
is a unit vector.
Problem 2
Show that the oblique rank 2 projection of vect
ME641 Engineering Analysis I
Homework Solution 5
ME641 Homework Solution 5
Problem 1
Solve the system of linear differential equations
x (t ) = x (t ) + y(t ), y (t ) = 4 x (t ) + y(t ), x (t = 0.1) =
ME641 Engineering Analysis I
Homework Solution 4
ME641 Homework Solution 4
Problem 1
Classify the following differential equations by type, order and linearity. Determine the
independent and dependent
ME641 Engineering Analysis I
Homework Solution 2
ME641 Homework Solution 2
Problem 1
Consider the square matrix A given below.
8
0 4
0
A = 1 0
1 3 7
a) Using Laplace expansion, calculate det(A).
b)
ME641 Engineering Analysis I
Homework Solution 7
ME641 Homework Solution 7
Problem 1
Using Lagrange polynomials, interpolate the function f(x)=tanh(x) on the interval [-1.5, 1.5] with
5 equally spaced
ME641 Engineering Analysis I
Homework Solution 4
ME 641 Homework Solution 4
Problem 1
Classify the following differential equations by type, order and linearity. Determine the
independent and dependen
ME641 Engineering Analysis I
Homework Solution 7
ME 641 Homework Solution 7
Problem 1
Using Lagrange polynomials, interpolate the function f(x) = tan(x) on the interval [-1.5, 1.5] with
5 equally spac
ME641 Engineering Analysis I
Homework Solution 11
ME641 Homework Solution 11
Problem 1
Show that (r, ) is the solution to Laplaces equation in polar coordinates:
a
(r, ) = 0 + a n r n cos(n) + b n r
ME641 Engineering Analysis I
Homework Solution 6
ME 641 Homework Solution 6
Problem 1
The statement t a means that a t a and the real variable t is defined on the interval
[ a, a ] including the end p
ME641 Engineering Analysis I
Homework Solution 5
ME 641 Homework Solution 5
Problem 1
Solve the system of linear differential equations
x t x t yt , y t 4 x t yt , x t 0.1 yt 0.1 1, 0.1 t 1
a) analyti
ME641 Engineering Analysis I
Homework Solution 10
ME641 Homework Solution 10
Problem 1
Using double integrals, calculate the following:
a) area of region bounded by straight line y(x ) = x and curve y
ME641 Engineering Analysis I
Homework Solution 2
ME 641 Homework Solution 2
Problem 1
The vector space of second-order polynomials on the interval 1 x 1 is spanned by the
following vectors:
0 x 1 1 x
ME641 Engineering Analysis I
Homework Solution 1
ME641 Homework Solution 1
Problem 1
Show that if x is a nonzero vector, u
x
xx
is a unit vector.
Solution
Assume that x is a nonzero vector,
x
x
u
1
x
ME641 Engineering Analysis I
Homework Solution 8
ME 641 Homework Solution 8
Problem 1
Find the domain for the function f (x , y ) = 6 (2 x + 3y ) .
Solution
domain is defined by the inequality:
6 (2x
ME641 Engineering Analysis I
Homework Solution 3
ME 641 Homework Solution 3
Problem 1
t 2t
Consider the matrix A
that depends on the parameter t. Find the eigenvalues.
2t t
Solution
writing chara
ME641 Engineering Analysis I
Homework Solution 9
ME641 Homework Solution 9
Problem 1
Let the temperature of a surface be T(x , y, z ) = x 2 + xy + yz . At point (2, 1, 4), determine the
following:
a)
ME641 Engineering Analysis I
Homework Assignment 9
ME641 Homework Assignment 9
Problem 1
Let the temperature of a surface be T(x , y, z ) = x 2 + xy + yz . At point (2, 1, 4), determine the
following:
ME641 Engineering Analysis I
Homework Assignment 2
ME 641 Homework Assignment 2
Problem 1
The vector space of second-order polynomials on the interval 1 x 1 is spanned by the
following vectors:
0 x 1
ME641 Engineering Analysis I
Homework Assignment 3
ME 641 Homework Assignment 3
Problem 1
t 2t
Consider the matrix A
that depends on the parameter t. Find the eigenvalues.
2 t t
Problem 2
2 1 1
ME641 Engineering Analysis I
Homework Assignment 2
ME641 Homework Assignment 2
Problem 1
Consider the square matrix A given below.
8
0 4
0
A = 1 0
1 3 7
a) Using Laplace expansion, calculate det(A)
ME641 Engineering Analysis I
Homework Assignment 10
ME641 Homework Assignment 10
Problem 1
Using double integrals, calculate the following:
a) area of region bounded by straight line y(x ) = x and cur
ME641 Engineering Analysis I
Homework Assignment 11
ME641 Homework Assignment 11
Problem 1
Show that (r, ) is the solution to Laplaces equation in polar coordinates:
a
(r, ) = 0 + a n r n cos(n) + b
ME 641 Calculus of Variations
Learning Objectives
The student will be able to:
Define functional
Classify functionals
Linear
Bilinear
Symmetric
Quadratic
Positive definite
Continuous
Calculate fir
ME 641 Matrices
Learning Objectives
The student will be able to:
Define matrices as abstract vector space
Classify matrices
Rectangular, square, triangular, diagonal, identity
Symmetric, orthogon
ME 641 Advanced Calculus
Learning Objectives
The student will be able to:
Define functions, single-variable and multi-variable
functions
Calculate limits of single-variable and multi-variable
funct
Approximation of Functions
Learning Objectives
The student will be able to:
Interpolate discrete data points by curves using
Lagrange polynomials Ln(x)
Cubic splines s3(x)
Non-polynomial function
ME 641 Integral Calculus
Learning Objectives
The student is able to:
Find antiderivatives of elementary functions
Evaluate definite integrals using Fundamental Theorem
of Calculus
Evaluate single
ME 641 Eigenvalues and Eigenvectors
Prerequisites
The student is able to:
Calculate roots of polynomials
Factor polynomials
Sven K. Esche, September 12, 2012
ME 641 Eigenvalues and Eigenvectors
Po
ME 641 Advanced Differential Equations
Prerequisites
The student is able to:
Determine whether a function is continuous, piecewise
continuous or discontinuous
Solve ODEs with piecewise continuous f
Approximation of Functions
Prerequisites
The student is able to:
Interpolate sets of discrete data points by curves using
polynomials
Calculate coefficients for
First-order polynomials
Second ord
ME 641 Matrices
Prerequisites
The student is able to:
Perform basic matrix operations
Check for equality
Addition/subtraction
Scalar multiplication
Matrix multiplication
Transpose
State properties
ME 641 Calculus of Variations
Prerequisites
The student is be able to:
Calculate partial and total derivatives of multivariate
functions
Differentiate integrals
Evaluate integrals using integratio
ME 641 Systems of Differential Equations
Prerequisites
The student is able to:
Classify systems of ODEs
by order
by linearity
by homogeneity
Write systems of 1st order ODEs
in component form
i