ENGR 0135
Chapter 5 2
Equivalent force-couple system
Department of Mechanical
Engineering
Topics
Couples
Definition of Couples
Characteristics of Couples
Equivalent force-couple system
Resultant
of non-concurrent force
system
Department of Mechanical
Today:
- Forces & moments in 3D
Book: Chapter 2.8-2.9, 3.4, 4.4-4.9 (3D)
Force in 3 dimensions
Force in 3 dimensions
Force in 3 dimensions
Projection of a force along an arbitrary line
Angle between two
vectors
Scalar product
The tension T in the cable CD
Math 201-203-RE - Calculus II
Integration by Parts
Page 1 of 5
Integration by Parts Formula
The integral is given in the form of a product of 2 expressions
u . dv
Therefore the formula is derived from the product rule formula as follows:
d(u . v)
du
dv
=
Math 102 Practice Exam 1 Solutions
Spring 2008
1. Determine whether or not the following integrals converge. If they converge, evaluate. If they diverge to , specify which one.
(a)
2 x
dx.
0 x2 1
x
2 1 is discontinuous
x
2
0
s1
1
2
s
x
dx = lim
21
x
s1
=
1
Method for Ordinary Differential Equations
This chapter will introduce the reader to the terminology and notation of differential equations.
Students will also be reminded of some of the elementary solution methods they are assumed to
have encountered i
Section 2.4 The Product and Quotient Rules
2010 Kiryl Tsishchanka
The Product and Quotient Rules
THE PRODUCT RULE: If f and g are both dierentiable functions, then
d
d
d
[f (x)g(x)] = g(x) [f (x)] + f (x) [g(x)]
dx
dx
dx
or
[f (x)g(x)] = f (x)g(x) + f (x)
Section 2.5 The Chain Rule
2010 Kiryl Tsishchanka
The Chain Rule
PROBLEM: Let f (x) = (1 + x)2 . Find f (x).
Solution 1: To nd the derivative of this function, we do algebra rst and then apply calculus
rules:
f (x) = [(1 + x)2 ] = (1 + 2x + x2 ) = 1 + 2(x
Section 2.3 Basic Dierentiation Formulas
2010 Kiryl Tsishchanka
Basic Dierentiation Formulas
DERIVATIVE OF A CONSTANT FUNCTION:
d
(c) = 0
dx
c = 0
or
Proof: Suppose f (x) = c, then
cc
0
f (x + h) f (x)
= lim
= lim = lim 0 = 0
h0
h0 h
h0
h0
h
h
f (x) = lim
4
Separable First-Order Equations
As we will see below, the notion of a differential equation being separable is a natural generalization of the notion of a rst-order differential equation being directly integrable. Whats more,
a fairly natural modication