EEE233 (SEM2-2012/13)
TUTORIAL 1: PARTIAL DIFFERENTIAL EQUATIONS
1. Solve the following equations
a) ∂2u∂x2=24x2(t-2), given that at x=0, u=e2tand ∂u∂x=4t. b) ∂2u∂x∂y=4eycos2x, given that at y=0, ∂u∂x=cosx and at x=π, u=y2. 2. A perfectly elastic string is stretched between two points 10 cm apart. Its centre point is displaced 2 cm from its position of rest at right angles to the original direction of the string and then released with zero velocity. Applying the equation

∂2u∂x2=1c2∙∂2u∂t2
with c2=1, determine the subsequent motion ux,t.

3. One end A of an insulated metal bar AB of length 2 m is kept at 0°C while the other end B is maintained at 50°C until a steady state of temperature along the bar is achieved. At t=0, the end B is suddenly reduced to 0°C and kept at that temperature. Using the heat conduction equation

∂2u∂x2=1c2∙∂u∂t , determine an expression for the temperature at any point in the bar distance x from A at any time t. 4. A square plate is bounded by the lines x=0, y=0, x=2, y=2. Apply the Laplace equation ∂2u∂x2+∂2u∂y2=0

to determine the potential distribution ux,y over the plate, subject to the following boundary conditions.
5. Show that the equation
∂2u∂x2-1c2∙∂2u∂t2=0
is satisfied by u=fx+ct+F(x-ct) where f and F are arbitrary functions. 6. If ∂2u∂x2=1c2∙∂2u∂t2 and c=3, determine the solution u=f(x,t) subject to the boundary conditions u0,t=0 and u2,t=0 for t≥0 ux,0=x(2-x) and ∂u∂tt=0=0 for 0≤x≤2. 7. The centre point of a perfectly elastic string stretched between two points A and B, 4 m apart , is deflected a distance 0.01 m from its position of rest perpendicular to AB and released initially with zero velocity. Apply the wave equation ∂2u∂x2=1c2∙∂2u∂t2 where c=10 to determine the...

...Chapter 4
Vector integrals and integral theorems
Last revised: 1 Nov 2010.
Syllabus covered:
1. Line, surface and volume integrals.
2. Vector and scalar forms of Divergence and Stokes’s theorems. Conservative ﬁelds: equivalence to curl-free
and existence of scalar potential. Green’s theorem in the plane.
Calculus I and II covered integrals in one, two and three dimensional Euclidean (ﬂat) space (i.e. R, R2
and R3 ). We are still working in R3 so there is no generalization to be applied to volume or triple integrals,
but we will generalise one dimensional integration from a straight line to an integral along a curve, and we
will generalise two-dimensional integration from a region in a plane to a curved surface.
We will also be working with integration of vectors, though in many cases we will be using a scalar
product so the ﬁnal quantity to be integrated becomes a scalar. In the cases with a scalar product:
f (x) dx generalizes to
C
F · dr on a curve C , called a line integral (section 4.1).
f (x, y) dx dy generalizes to
S
F · dS over a surface S , called a surface integral (section 4.2).
We will then have to study the generalizations of
b
a
df
dx = f (b) − f (a) ,
dx
(4.1)
called the ‘fundamental theorem of calculus’, which we use in the proofs. This theorem relates a onedimensional integral to a (pair of) zero-dimensional evaluations at the two endpoints x...

...ECE 352 VECTOR ANALYSIS
DEL OPERATOR
GROUP 3 Andaya, Rizalyn Ramos, Maria Issa P.
∇
Del is a symbol used in mathematics, in particular, in vectorcalculus, as a vector differential operator, usually represented by the nabla symbol ∇. Del may denote the gradient (locally steepest slope), the divergence of a vector field, or the curl (rotation) of a vector field. The symbol ∇ can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product of scalars, dot product, and cross product, respectively, of the del "operator" with the field.
A. GRADIENT If a scalar function , , is continuously differentiable with respect to its variables x, y, z through the region, then the gradient of , written grad , is defined as the vector grad = Using the vector differential operator ∇, ∇= We can write the gradient of as
+
+
+
+
grad = ∇ = ( = Example: If =
+ +
+ +
)
, determine the grad at the point P(1,3,2).
By definition: grad = ∇ =
+
+
All we have to do then is to find the partial derivatives at x=1, y=3, z=2 and insert their values.
Since = = =
= = = = 2(1) (3) (2)3 + (3)2(2)2 (1)2(2)3 + 2(1) (3) (2)2 3(1) (3) (2)2 + 2(1) (3)2(2) = 84 = 32 = 72
grad
= ∇ = 84 i + 32 j +72 k
B. DIVERGENCE If a...

...
1a. h=-4.9t^2+450
1b. h(t)=-4.9t^2+450
(h(2)-h(0))/(2-0)
((-4.9(〖2)〗^2+450)-(-4.9(0)^2+450))/2
=(430.4-450)/2
=-19.6
∴The average velocity for the first two seconds was 19.6 metres per second.
c. i)
i)
=
=-24.5
∴ The average velocity from is 24.5 metres/s.
ii)
= -14.7
iii)
= -12.25
∴ The average velocity from is 12.25 metres/s.
d) Instantaneous velocity at 1s:
=-9.8
∴ The instantaneous velocity at 1s is 9.8 metres/s.
2a)
=
=
=
=
=
b)
=
∴ The average rate of change from is -0.4g/s.
C)
∴The instantaneous rate at t = 2 seconds is -1.6g/s
3)
b)
=
=
=22
∴ from seconds the car moves at an average of 22m/s
c)
t=4
=
=16
∴ The instantaneous rate at 4s is 16m/s
4a) In order to determine the instantaneous rate of change of a function using the methods discussed in this lesson, we would use the formula where h will approach 0, and the closer it gets to 0 the more accurate our answer will be.
4b)
∴
=1
Therefore, = 1
5a)
Therefore the instantaneous rate at x=2 is 0.
5b)
Therefore at t=4 the instantaneous rate is 0 and the particle is at rest.
6a)
Rate of change is positive when:
Rate of change in negative when:
6b)
Rate of change is 0 when:
X=-1, x=1
6c)
Local Maximum: (-1,2)
Local Minimum: (1,-2)...

...Vector Analysis
Definition
A vector in n dimension is any set of n-components that
transforms in the same manner as a displacement when you
change coordinates
Displacement is the model for the behavior of all vectors
Roughly speaking: A vector is a quantity with both direction as well as
magnitude.
On the contrary, a scalar has no direction and remains unchanged when
one changes the coordinates.
Notation: Bold face A, in handwriting A . The magnitude of the vector is
denoted by A A A
Example:
Scalars: mass, charge, density, temperature
Vectors: velocity, acceleration, force, momentum
Vector Algebra
Vector Operations
(a) Addition of two vectors
Parallelogram law: To find A+B, place the tail of B at the head of A and
draw the vector from the tail of A to the head of B
B
A
A+B
From the definition, the addition of vectors is
(i) Commutative
A+B=B+A
(ii) Associative
(A+B)+C=A+(B+C)
(b) Negative of a vector
The negative of a vector is defined as the vector with the same magnitude
but opposite direction
A
-A
(c) Multiplication by a scalar
Multiplication by a positive real number a multiplies the magnitude by a
times while leaving the direction unchanged.
A
A
aA
aA
Multiplication by 0 gives the null vector 0 which...

...No 1. 2. 3. 4. 5. 6. 7. 8. Code: UCCM1153 Status: Credit Hours: 3 Semester and Year Taught:
Information on Every Subject Name of Subject: Introduction to Calculus and Applications
Pre-requisite (if applicable): None Mode of Delivery: Lecture and Tutorial Valuation: Course Work Final Examination 40% 60%
9. 10.
Teaching Staff: Objective(s) of Subject: • Review the notion of function and its basic properties. • Understand the concepts of derivatives. • Understand linear approximations. • Understand the relationship between integration and differentiation and continuity. Learning Outcomes: After completing this unit, students will be able to: 1. describe the basic ideas concerning functions, their graphs, and ways of transforming and combining them; 2. use the concepts of derivatives to solve problems involving rates of change and approximation of functions; 3. apply the differential calculus to solve optimization problems; 4. relate the integral to the derivative; 5. use the integral to solve problems concerning areas.
11.
12.
Subject Synopsis: This unit covers topics on Functions and Models, Limits and Derivatives, Differentiation Rules, Applications of Differentiation and Integrals.
13.
Subject Outline and Notional Hours: Topic Learning Outcomes 1 L 4 T 1.5 P SL 6.25 TLT 11.75
Topic 1: Functions and Models
• • • • • • Functions Models and curve fitting Transformations, combinations, composition and graphs of...

...Portfolio in Calculus
Submitted by:
Chloe Regina C. Paderanga
Submitted to:
Sir Ferdinand Corpuz
Journal for the Month of June
WHAT I LEARNED?
I learned many things this month. It was good that our teacher repeated the topics in basic math to strengthen our foundation. Even if we had a hard time, I don’t see any reason why we should complain because I understand that our wanted to master these topics to be able to move to a higher math. The topics tackled this month are namely:
Inequalities
Rational Inequalities
Circles
Distances
Slopes
Angles of Incidence
WHAT IS THE HARDEST TOPIC?
For me, the hardest topic to master was the inequalities, which I know I should master to be able to understand the next topics.
HOW DID I LEARN?
I reviewed my wrong answers in our summative tests because I don’t want to be left behind with the topics.
REFLECTION
When Sir Corpuz said that we are going to have a double program in Math, I was excited because we are not just advancing but are reviewing in the same time.
APPLICATION TO LIFE
A lot of advance technologies are product of such very simple concepts in math as long as it is utilized in a very good way. For example the distance formula, this is not just used in Math but also in Physics, Science and many other fields.
Journal for the Month of July
WHAT I LEARNED?
This month, I learned that there are also ‘other’ versions of circles. Namely:
Parabola
Ellipse...

...
Calculus in Medicine
Calculus in Medicine
Calculus is the mathematical study of changes (Definition). Calculus is also used as a method of calculation of highly systematic methods that treat problems through specialized notations such as those used in differential and integral calculus. Calculus is used on a variety of levels such as the field of banking, data analysis, and as I will explain, in the field of medicine. Medicine is defined as the science and/or practice of the prevention, diagnosis, and treatment of physical or mental illness (Definition). The term medicine can also mean a compound or a preparation applied in treatment or control of diseases, mostly in form of a drug that is usually taken orally (Definition). Calculus has been widely used in the medical field in order to better the outcomes of both the science of medicine as well as the use of medicine as treatment. (Luchko, Mainardi & Rogosin, 2011). There has been a strong movement towards the inclusion of additional mathematical training throughout the world for future researchers in biology and medicine. It can be hard to develop new courses as well as alter major requirements, but institutions should consider the importance of a clear understanding of the function of mathematics in science. However, scientists who have not had the level of mathematical training needed to work in...

...
Abstract
The experiment was about the resolution of vector quantities using different methods or techniques. Among those are the Parallelogram, Polygon, and the Analytical or Component methods. Using each method, it was found out that Component method is the most accurate as its approach is purely theoretical, that is, all other physical factors are neglected leaving only the appropriate ones to be calculated. In addition, properties of these quantities such as associativity and commutativity of the addition operation were also explored.
1. Introduction
Vectors play an important role in many aspects of our everyday lives or of one’s daily routine. It is a mathematical quantity that has both a magnitude and direction.
A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". The study of vectors had gone through a lot of revisions, starting from the 19th century where mathematicians used geometrical representations for complex numbers. Lots of changes and multiple varieties of altering were conducted to this study, which led to the discovery of the vector that we all know today. Operations on vectors are also made possible through time. Addition of vectors was clarified and can now be done in different ways. Vector addition in a graphical way can use the polygon method and the...