Mathematical Exercises for Physics Olympiad
T
Vectors
-⃗ = 3𝚤̂ − 5𝑘+.
1. Consider two vectors 𝐴⃗ = 𝚤̂ + 2𝚥̂ + 3𝑘+ and 𝐵
-⃗ and 𝐴⃗ × 𝐵
-⃗.
(a) Compute 𝐴⃗ ⋅ 𝐵
(b) What is the angle between two vectors?
2. What is the area of the triangle with vertices 𝑃 = (−1, 3), 𝑄 = (1, 5) and 𝑅 = (−3, 7)?
Differentiation
!"
Find the derivative !# of the following functions:
$
$
#
√#
1. 𝑦 = + √𝑥 −
!
2. 𝑦 = 𝑥 & (𝑥 ' + 1)"
3. 𝑦 = 𝑥 − √𝑥 ' + 1
4. 𝑦 = 𝑥 cos 𝑥
5. 𝑥 ' 𝑦 − 𝑥 + 𝑦 ' − 𝑦 = 0
6. 𝑦 = ln cos 𝑥
7. 𝑦 = √1 + cos& 3𝑥
8. 𝑦 = 3 tan 2𝑥
#
9. 𝑦 = 𝑒 # ()* #
10. 𝑦 = 𝑥 + 𝑒 &#
Curve Sketching
For each problem, determine the following if applicable:
(a) x and y intercepts
(b) vertical and horizontal asymptotes
(c) open intervals on which the function is increasing and decreasing
(d) open intervals where the function is concave up and concave down
(e) coordinates of all relative extrema
(f) coordinates of all points of inflection
(g) sketch the curve
1. 𝑦 = 𝑥 , − 18𝑥 ' + 56
2. 𝑦 =
# # -$
#"
Integration
1.
𝐼 = K(𝑥 & + 3𝑥 + 2)𝑑𝑥
2.
$
1
K
-$ M1 − 𝑦 '
3.
𝑑𝑦
.
𝐼 = K 𝑒 -# 𝑑𝑥
$
4.
/
'
𝐼 = K cos& 𝑥 𝑑𝑥
0
5.
'
1
𝑑𝑥
$ 𝑥
𝐼=K
6.
'
𝐼=K
√𝑥 − 1
$
7.
𝑑𝑥
/
1
𝐼 = K 4 sin' 3𝑥 𝑑𝑥
0
8.
K
1
𝑑𝑥
𝑥(𝑥 + 2)
9.
'
1
𝑑𝑥
&
-' 𝑥
𝐼=K
10.
𝐼 = K arcsin 𝑥 𝑑𝑥
11.
𝐼=K
𝑥'
𝑑𝑥
𝑥, − 1
12. Find the area enclosed by the curve 𝑦 = 𝑥(𝑥 ' − 1) and the 𝑥-axis from 𝑥 = 1 to 𝑥 = 3.
13.
𝐼=K
1
𝑑𝑥
1 + 𝑥'
Integration for fun:
You are not supposed to solve them in PhO. I put them here just for fun.
1. (For Fun)
/
'
K √tan 𝑥 𝑑𝑥
0
14.3 Rotation
407
2. One very useful integration in Physics
Now we will consider an x ′ −y ′ −z ′ coordinate system which is obtained by shifting
.
the old system by the vector r 0 .
#
The new origin of the coordinate system coincides with the center
ofK
the sphere
𝐼
=
𝑒 -# 𝑑𝑥
(see Fig. 14.9). The equation of the sphere in the x ′ −y ′ −z ′ coordinate system is
0
well known to be
R2 = x ′2 + y ′2 + z ′2
3.This
Feynman
equation is Trick:
obtained by applying the transformation rule to the previous equa.
tion.
sin 𝑥
Hence the equation of a sphere and other equations as well can
𝐼 =often
K be made 𝑑𝑥
𝑥
simpler by shifting the origin of the coordinate system.
0
14.3 Rotation
Coordinates:
1.14.3.1
Consider
theinposition
Rotation
a Plane vector 𝑟⃗ = 𝑥𝚤̂ + 𝑦𝚥̂ in the x-y system of coordinates. We can now rotate this
system through an angle 𝜙 into a new position, as shown in the figure below. The vector can be
2 in an x−y system of coordinates. We
Consider the position
represented
by 𝑟⃗vector
= 𝑥r2 𝚤̂=2 xi
+ +𝑦yj
𝚥̂′, where 𝚤̂′ and 𝚥̂′ are the unit vectors in the rotated coordinates.
can now rotate this system through an angle ! into a new position, as shown in
(a)
Find
out
𝑥′
and
𝑦′.
Express
the answers
𝑥, 𝑦 and 𝜃.
andterms
the unitof
vectors
Fig. 14.10. The new coordinate axes are denoted
by x ′ and y ′in
′
by iExpress
and j ′ , respectively.
(b)
the vectors 𝚤̂′ and 𝚥̂′ in terms of 𝚤̂, 𝚥̂ and 𝜃.
Fig. 14.10
2. Polar coordinate
and derivatives
In the x ′ −y ′ coordinate system the vector r is given by
(a) Consider a point 𝑟⃗ is the ′Cartesian
coordinate that can be represented by the polar coordinate
r = x i ′ + y ′j ′
𝑟⃗ = 𝑟𝑟̂
The problem now is to find the relationship between the original coordinates (x, y)
+ in terms of 𝚤̂, 𝚥̂ and 𝜃.
′ , y ′ ).unit vectors of the polar coordinate. Express 𝑟̂ and 𝜃
where
𝑟̂ and
𝜃+ are(xthe
and the new
coordinates
We start with the components (x, y) of r in the original system. These are separated into components in the direction of the new axes; we need to find these components.
Finally,
we will collect
corresponding
(b)
If the
trajectory
of the
particleterms.
is given by 𝑟⃗ = U𝑟(𝑡), 𝜃(𝑡)W. Find the velocity of the particle 𝑟⃗̇(𝑡) in
!3̂
!6̇
polar coordinates. Notice that !5 and !5 are non-zero.
3. Integration using spherical and cylindrical coordinates.
(a) Compute the volume of the hollow cylinder of height ℎ shown in the figure using cylindrical
coordinates.
(b) Calculate the volume of a sphere of radius 𝑅 using spherical coordinate.
Taylor’s series
1. Find the Taylor’s series (up to 2nd order) of the following functions:
8 $%
(a) 𝑓(𝑥) = sin' 𝑥
(b) 𝑓(𝑥) = $-#
2. Expand the following functions at 𝑥0 = 𝜋:
(a) 𝑔(𝑥) = sin 𝑥
(b) 𝑔(𝑥) = cos 𝑥
3. Calculate √37 = √36 + 1 up to 4 d.p.
Complex number
1. Determine and simplify
$
(a) '√' U16 + √2𝑖W
$9:
$-:
(b) $-: − $9:
$
(c) $9:
2. Using Euler’s formula, compute cos 𝛼 and sin 𝛼 and convert to the algebraic form:
(a) 𝑒 ://'
(b) 𝑒 ://&
3. Convert the complex number to the polar form 𝑧 = 𝑟(cos 𝛼 + 𝑖 sin 𝛼):
(a) 𝑧 = −1 + 𝑖
(b) 𝑧 = −(1 + 𝑖)
Differential equation:
1. Solve the following differential equation:
where 𝑦(𝑥 = 0) = 1.
𝑑𝑦
= 2𝑦𝑥
𝑑𝑥
2. Solve the following differential equation of 𝑥(𝑡),
𝑥̈ + 3𝑥̇ + 2𝑥 = 0
with initial conditions 𝑥(0) = 0 and 𝑥̇ (0) = 1.
3. Solve the following differential equation of 𝑥(𝑡),
𝑥̈ + 3𝑥̇ + 2𝑥 = 1 + 𝑡
with initial conditions 𝑥(0) = 0 and 𝑥̇ (0) = 1.