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...

...Sciences
ECE / ICE / MexE Department
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...

...
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...

...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...

...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...

...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...

...
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...

...
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...