CME261
Engineering Mathematics
October 17, 2014
Kaiwen Xia
Practice Problems for Partial Differentiation - Solution
1. Suppose there are two vectors : u 1,1,2 ; v 2,0,1 . is the angle between
u and v . Find:
c) sin
d) cos
a) u v b) u v
2. Find the vecto

CME261, Part V
Curve Fitting
Prof. Kaiwen Xia
Department of Civil Engineering
University of Toronto
Split Hopkinson Pressure Bar
Semi-circular Bend (SCB)
K IC
Pmax S
a
Y ( )
3/ 2
R
BR
Y is calculated with
finite element method.
Fracture Toughness Data
Nor

CME261, Part IV
Partial Differentiation
Prof. Kaiwen Xia
Department of Civil Engineering
University of Toronto
Introduction to Vectors
Understanding vector notation.
Being able to identify the following types of
vectors: zero, unit, and base.
Knowing h

CME261 Engineering Math I
Kaiwen Xia
September 29, 2011
Practice Problems for systems of equations
1. Solve the following system using Cramers rule:
10 x1
3x1
x1
2 x2
6 x2
x2
x3
2 x3
5 x3
27
61.5
21.5
Solution:
x1
x3
27
2 1
61.5 6 2
21.5 1
5

CME261
Engineering Mathematics
September 15, 2013
Kaiwen Xia
Practice Problems for Rootfinding: Solution
1. Find the root of f ( x) x3 3 x 1 on [0, 1] using bisection method.
1) Determine how many iterations are needed for absolute error is smaller than 5

CME261
Engineering Mathematics
Kaiwen Xia
Practice Problems for Partial Differentiation - Solution
1. Suppose there are two vectors : u 1,1,2 ; v 2,0,1 . is the angle between
u and v . Find:
c) sin
d) cos
a) u v b) u v
a) u v 1 (2) (1) 0 (2) 1 4
i
b) u

CME261
Kaiwen Xia
Engineering Mathematics
Practice Problems for Curve Fitting - Solution
A. Linear Regression
1. Use least squares regression to fit a straight line to the data given in Table
1. Along with the slope and intercept, compute the standard err

CME261
Engineering Mathematics
September 15, 2013
Kaiwen Xia
Practice Problems for Rootfinding
1. Find the root of f ( x) x3 3 x 1 on [0, 1] using bisection method.
1) Determine how many iterations are needed for absolute error is smaller than 510-2.
2) C

CME261F Engineering Math I
Practice Problems for systems of equations
1. Solve the following system using Cramers rule:
10 x1
3x1
x1
2 x2
6 x2
x2
x3
2 x3
5 x3
27
61.5
21.5
2. Solve the system in Q(1) using nave Gauss elimination.
3. Employ partia

CME 261
Engineering Math I
Introduction
Professor Kaiwen Xia
Department of Civil Engineering, University of Toronto
Presentation date Friday September 6, 2013
Outline
Who am I ?
What is Engineering Math ?
Why Engineering Math ?
What will we cover ?
En