Objectives
This Mathematics course provides requisite and relevant background necessary to understand the other important engineering mathematics courses offered for Engineers and Scientists. Three important topics of applied mathematics, namely the Multiple integrals, Vector calculus, Laplace transforms which require knowledge of integration are introduced. Expected Outcome
At the end of this course the students are expected to learn (i)
how to evaluate multiple integrals in Cartesian, Cylindrical and Spherical geometries. (ii)
the powerful language of Vector calculus with physical understanding to deal with subjects such as Fluid Dynamics and Electromagnetic fields.
(iii)
to solve ordinary differential equations directly and also use transform methods where its possible
Unit 1
Mutivariable Calculus
9L+4P hours
Functions of two variableslimits and continuitypartial derivatives –total differential–Taylor’s expansion for two variables–maxima and minima–constrained maxima and minimaLagrange’s multiplier method Jacobians
Unit 2
Mutiple Integrals
9L+4P hours
Evaluation of double integrals–change of order of integration– change of variables between cartesian and polar coordinates evaluation of triple integralschange of variables between cartesian and cylindrical and spherical polar coordinatesbeta and gamma functions– interrelationevaluation of multiple integrals using gamma and beta functionserror functionproperties. Unit 3
Vector Calculus
9L+4LP hours
Scalar and vector valued functions  gradient–physical interpretationtotal derivative–directional derivativedivergence and curl –physical interpretationsStatement of vector identities  scalar and vector potentialsline, surface and volume integrals Statement of Green’s , Stoke’s and Gauss divergence theorems...
...FIRSTORDER
DIFFERENTIALEQUATIONS
OVERVIEW In Section 4.8 we introduced differentialequations of the form dy>dx = ƒ(x),
where ƒ is given and y is an unknown function of x. When ƒ is continuous over some interval, we found the general solution y(x) by integration, y = 1 ƒ(x) dx. In Section 6.5 we
solved separable differentialequations. Such equations arise when investigating exponential growth or decay, for example. In this chapter we study some other types of firstorder
differentialequations. They involve only first derivatives of the unknown function.
15.1
Solutions, Slope Fields, and Picard’s Theorem
We begin this section by defining general differentialequations involving first derivatives.
We then look at slope fields, which give a geometric picture of the solutions to such equations. Finally we present Picard’s Theorem, which gives conditions under which firstorder
differentialequations have exactly one solution.
General FirstOrder DifferentialEquations and Solutions
A firstorder differentialequation is an equation
dy
= ƒsx, yd
dx
(1)
in which ƒ(x, y) is a function of two variables defined on a region in the xyplane. The...
...CHAPTER 1
INTRODUCTION TO
DIFFERENTIALEQUATIONS1
Chapter INTRODUCTION TO
DIFFERENTIAL1EQUATIONS
Outline:
1.1
1.2
1.3
1.4
1.5
1.6
Basic Definition
Types of DifferentialEquations
Order of a DifferentialEquation
Degree of a DifferentialEquation
Types of Solutions to a DifferentialEquation
Elimination of Arbitrary Constant
Sir Isaac Newton (December 25, 1642 – March 20, 1727) was an English
physicist and mathematician who is widely regarded as one of the most influential
scientists of all time and as a key figure in the scientific revolution. His book
Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural
Philosophy"), first published in 1687, laid the foundations for most of classical
mechanics. Newton also made seminal contributions to optics and shares credit with
Gottfried Leibniz for the invention of the infinitesimal calculus.
In 1671, Newton wrote his thenunpublished The Method of Fluxions and
Infinite Series (published in 1736), in which he classified first order differentialequations, known to him as fluxional equations, into three classes. The first two
classes contain only ordinary derivatives of one or more dependent variables, with
respect to a single...
...Differentialequations1 Introduction
These notes are to be read together with Chapter 7 in the textbook (Calculus:
Concepts and contexts, by James Stewart). Separable differentialequations
are dealt with in the textbook and in these notes; the notes then continue
with first order linear differentialequations.
Differentialequations describe most, if not all, processes that we try to understand
in our technological era. They can be used to describe the way
planets revolve around the sun, the way rockets travel to outer space, the
way chemicals interact, the way electricity flows, even the way matter itself
exists. They can also be used to describe economic processes, such as the
way the net worth of a company changes. We will only be able to scratch
the surface of this very important subject.
Recall that a differentialequation (d.e. for short) is any equation that involves
at least one derivative (of an unknown function). Solving a d.e. means
finding (all) functions that satisfy the d.e.
Here are two examples:
1.
dy
dx
= cos x
2.
d2p
dt2
− 4
dp
dt
+ 3p = 0
In Example 1, the independent variable is x; one then tries to find a formula
describing how y depends on x. We call y the dependent variable. In
Example 2, the independent...
...CHAPTER 2
FIRST ORDER DIFFERENTIALEQUATIONS
2.1 Separable Variables
2.2 Exact Equations
2.2.1 Equations Reducible to Exact Form.
2.3 Linear Equations
4. Solutions by Substitutions
2.4.1 Homogenous Equations
2.4.2 Bernoulli’s Equation
2.5 Exercises
In this chapter we describe procedures for solving 4 types of differentialequations of first order, namely, the class of differentialequations of first order where variables x and y can be separated, the class of exact equations (equation (2.3) is to be satisfied by the coefficients of dx and dy, the class of linear differentialequations having a standard form (2.7) and the class of those differentialequations of first order which can be reduced to separable differentialequations or standard linear form by appropriate.
2.1 Separable Variables
Definition 2.1: A first order differentialequation of the form
[pic]
is called separable or to have separable variables.
Method or Procedure for solving separable differentialequations
(i) If h(y) = 1, then
[pic]
or dy = g(x) dx
Integrating both sides we get
[pic]...
...DIFFERENTIALEQUATIONS: A SIMPLIFIED APPROACH, 2nd Edition
DIFFERENTIALEQUATIONS PRIMER By: AUSTRIA, Gian Paulo A. ECE / 3, Mapúa Institute of Technology NOTE: THIS PRIMER IS SUBJECT TO COPYRIGHT. IT CANNOT BE REPRODUCED WITHOUT PRIOR PERMISSION FROM THE AUTHOR. DEFINITIONS / TERMINOLOGIES A differentialequation is an equation which involves derivatives and is mathematical models which can be used to approximate realworld problems. It is a specialized area of differentialcalculus but it involves a lot of integral calculus as well, so in general, differentialequations straddle the specific parts of basic calculus or it can be considered part of advanced calculus. There are two general types of differentialequations. An ordinary differentialequation involves only two variables, whereas a partial differentialequation involves more than two. A differentialequation can have many variables. The independent variable is the variable of concern from which the terms are derived on, whereas if the same variable appears in its derivative, then it is a dependent variable. Variables are different from parameters, which are constants with no derivatives.
The given...
...INTERNATIONAL UNIVERSITY
Chapter 2. MultivariablecalculusCalculus 2B for Business Administration
Lecturer: Nguyen Minh Quan, PhD
Dr. Nguyen Minh Quan (HCMIUVNU)
Chapter 2. Multivariablecalculus
Summer 2013
1 / 80
Contents
1
Functions of several variables
2
Partial derivatives
3
Maxima and minima. Optimization
4
Constrained Optimization
5
Total differentials and approximations
6
Double integrals
Dr. Nguyen Minh Quan (HCMIUVNU)
Chapter 2. Multivariablecalculus
Summer 2013
2 / 80
Introduction
Example
If the company produces two products, with x of one product at a cost of
$10 each, and y of another product at a cost of $15 each, then the total
cost to the firm is a function of two independent variables, x and y. By
generalizing notation, the total cost can be written.
C (x, y ) = 10x + 15y
Dr. Nguyen Minh Quan (HCMIUVNU)
Chapter 2. Multivariablecalculus
Summer 2013
3 / 80
Introduction
Example (Production cost)
A company is developing a new soft drink. The cost in dollars to produce a
batch of the drink is approximated by
C (x, y ) = 2200 + 27x 3 − 72xy + 8y 2 ,
where x is the number of kilograms of sugar per batch and y is the number
of grams of flavoring per batch. Find the amounts of sugar and...
...MATHEMATICAL METHODS
PARTIAL DIFFERENTIALEQUATIONS
I YEAR B.Tech
By
Mr. Y. Prabhaker Reddy
Asst. Professor of Mathematics
Guru Nanak Engineering College
Ibrahimpatnam, Hyderabad.
SYLLABUS OF MATHEMATICAL METHODS (as per JNTU Hyderabad)
Name of the Unit
UnitI
Solution of Linear
systems
UnitII
Eigen values and
Eigen vectors
Name of the Topic
Matrices and Linear system of equations: Elementary row transformations – Rank
– Echelon form, Normal form – Solution of Linear Systems – Direct Methods – LU
Decomposition from Gauss Elimination – Solution of Tridiagonal systems – Solution
of Linear Systems.
Eigen values, Eigen vectors – properties – Condition number of Matrix, Cayley –
Hamilton Theorem (without proof) – Inverse and powers of a matrix by Cayley –
Hamilton theorem – Diagonalization of matrix – Calculation of powers of matrix –
Model and spectral matrices.
Real Matrices, Symmetric, skew symmetric, Orthogonal, Linear Transformation 
UnitIII
Linear
Transformations
Orthogonal Transformation. Complex Matrices, Hermition and skew Hermition
matrices, Unitary Matrices  Eigen values and Eigen vectors of complex matrices and
their properties. Quadratic forms  Reduction of quadratic form to canonical form,
Rank, Positive, negative and semi definite, Index, signature, Sylvester law, Singular
value decomposition.
Solution of Algebraic and Transcendental Equations...
...Your file name must be like this:
1 LIST OF SYMBOLS
Symbol
Description
Unit
T
Temperature
K
ΔP
Pressure Drop
Pa
ρ
Density
kg/m3
µ
Kinematic Viscosity
N*s/m2
V
Bulk Velocity
m/s
D
Diameter
m
A
Area
m2
Flow Rate
m3/s
Re
Reynolds Number

f
Friction Factor

L
Length
m
2 CALCULATIONS
For the sample calculations, we looked at the first sample point of the flow in Pipe1, the smallest diameter smooth copper tube:
The first step in determining the properties of the flow is finding the density and kinematic viscosity of the water. At 296.51 K, water has the following properties1:
From this we can determine the bulk velocity of the stream using Equation1.
(Eqn. 1)
Where is the flowrate in m3/s and A is the crosssectional area of the pipe. To find the flowrate, we multiply the flowmeter reading by the constant
and convert from gallons to cubic meters as follows:
The cross sectional area of the 7.75mm pipe is
Plugging these values into Equation1, we obtain a bulk velocity .
With the bulk velocity value, we can find the Reynolds number of the flow using Equation 2.
(Eqn. 2)
Plugging in known values to Equation 2, we find:
The experimental friction factor of the pipe can be calculated as:
(Eqn. 3)
Using...