Rules Summary
Tutorial 1: Elementary of Vectors - Vector Product
3 ways to express the components of a vector:
1. Using the given coordinates:
𝑎 =𝑥 −𝑥
𝑎 =𝑦 −𝑦
𝑎 =𝑧 −𝑧
2. By its components:
𝑎⃗ = [𝑎 , 𝑎 , 𝑎 ]
3. By its unit vectors
𝑎⃗ = 𝑎 𝒊 + 𝑎 𝒋 + 𝑎 𝒌
Sum & difference of two vectors:
𝑎⃗ ± 𝑏⃗ = [𝑏 ± 𝑎 , 𝑏 ± 𝑎 , 𝑏 ± 𝑎 ]
The length of a vector (magnitude of a vector):
|𝑎⃗| =
𝑎 +𝑎 +𝑎
|𝑎⃗| =
𝑎⃗ ∙ 𝑎⃗
𝑖 = [1,0,0]
𝑗 = [0,1,0] 𝑘 = [0,0,1]
|𝑣⃗| = 𝑎⃗ × 𝑏⃗ = |𝑎⃗| 𝑏⃗ 𝑠𝑖𝑛𝜃
The posi on vector (𝒓⃗):
𝑟⃗ = [𝑥, 𝑦, 𝑧]
The unit vector 𝒖⃗ in the direc on of vector 𝒗⃗:
𝑣⃗
𝑢⃗ =
|𝑣⃗|
The inner dot product of two vectors:
𝑎⃗ ∙ 𝑏⃗ = 𝑎 𝑏 + 𝑎 𝑏 + 𝑎 𝑏
The cross product of two vectors:
𝑖
⃗
𝑎⃗ × 𝑏 = 𝑎
𝑏
𝑗
𝑎
𝑏
𝑘
𝑎 = (𝑎 𝑏 − 𝑎 𝑏 )𝒊 − (𝑎 𝑏 − 𝑎 𝑏 )𝒋 + (𝑎 𝑏 − 𝑎 𝑏 )𝒌
𝑏
The angle between two vectors:
𝑐𝑜𝑠𝜃 = |
⃗∙⃗
⃗| ⃗
0≤𝜃≤𝜋
𝑎⃗ ∙ 𝑏⃗ < 0 → 𝜃 is an acute angle.
𝑎⃗ ∙ 𝑏⃗ = 0 → 𝜃 is a right angle. (𝑎⃗ & 𝑏⃗) are orthogonal/perpendicular/normal to each other
𝑎⃗ ∙ 𝑏⃗ > 0 → 𝜃 is an obtuse angle.
𝑠𝑖𝑛𝜃 =
⃗×⃗
| ⃗| ⃗
Scalar triple product:
𝑎
⃗
⃗
𝑎⃗ 𝑏 𝑐⃗ = 𝑎⃗ ∙ 𝑏 × 𝑐⃗ = 𝑏
𝑐
𝑎
𝑏
𝑐
𝑎
𝑏
𝑐
The resultant of vector:
𝑅⃗ = 𝑎⃗ + 𝑏⃗ + 𝑐⃗
Find 𝒑⃗ such that the given vectors & 𝒑⃗ are in equilibrium:
𝑝⃗ = −𝑅⃗
-
A scalar quan ty is determined by its magnitude.
A vector quan ty has both magnitude and direc on.
o Two vectors are equal if they have the same length and same direc on.
o A zero vector 0⃗ has length equal to 0 and no direc on.
Tutorial 2: Differen a on of a Vector – Gradient & Direc onal Deriva ve – Divergence &
Curl of vector
Vector func on:
𝒗⃗(𝑡) = [𝑣 (𝑡), 𝑣 (𝑡), 𝑣 (𝑡)]
Deriva ve of a vector func on:
𝒗⃗′(𝑡) = [𝑣 ′(𝑡), 𝑣 ′(𝑡), 𝑣 ′(𝑡)]
Derive 𝑣 (𝑡) with respect to 𝑥, 𝑣 (𝑡) with respect to 𝑦, 𝑣 (𝑡) with respect to 𝑧.
Parametric representa on for the vector func on of a curve with parameter 𝒕:
𝑟⃗(𝑡) = [𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)]
The tangent of a curve:
𝒓⃗′(𝑡) = [𝑥′(𝑡), 𝑦′(𝑡), 𝑧′(𝑡)]
Unit tangent vector:
𝑢⃗ =
𝒓⃗ ( )
|𝒓⃗ ( )|
Gradient of a scalar func on:
𝑔𝑎𝑟𝑑 𝑓 = ∇𝑓 =
𝜕𝑓 𝜕𝑓 𝜕𝑓
, ,
𝜕𝑥 𝜕𝑦 𝜕𝑧
Direc onal deriva ve of a func on in the direc on of a par cular vector:
𝐷 𝑓=
𝜕𝑓
𝑎⃗
=
∙ ∇𝑓
|𝑎⃗|
𝜕𝑠
Divergence of a vector:
𝑑𝑖𝑣 𝒗⃗ =
𝜕𝑣
𝜕𝑣
𝜕𝑣
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
Curl of a vector:
𝑖
𝜕
𝑐𝑢𝑟𝑙 𝒗⃗ = ∇ × 𝒗⃗ =
𝜕𝑥
𝑣
-
𝑗
𝜕
𝜕𝑦
𝑣
𝑘
𝜕
𝜕
𝜕
𝜕
𝜕
𝜕
𝜕
= 𝑣
−𝑣
𝑖− 𝑣
−𝑣
𝑗+ 𝑣
−𝑣
𝑘
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝑣
𝑑𝑖𝑣 (𝑐𝑢𝑟𝑙) = 0
𝑐𝑢𝑟𝑙 (𝑔𝑟𝑎𝑑) = 0
Gradient is the rate of change of a scalar func on in the direc on of maximum change.
Tutorial 3: Polar Coordinates – Line, Surface, & Volume Integrals
Plane Polar
𝑟 =𝑥 +𝑦
𝑦
𝑡𝑎𝑛𝜃 =
𝑥
𝑥 = 𝑟𝑐𝑜𝑠𝜃
𝑦 = 𝑟𝑠𝑖𝑛𝜃
𝑑𝑆 = 𝑟𝑑𝑟𝑑𝜃
Cylindrical Polar
𝑟=
𝑥 +𝑦
𝑦
𝑡𝑎𝑛𝜃 =
𝑥
𝑧=𝑧
𝑥 = 𝑟𝑐𝑜𝑠𝜃
𝑦 = 𝑟𝑠𝑖𝑛𝜃
𝑧=𝑧
𝑑𝑉 = 𝑟𝑑𝑟𝑑𝜃𝑑𝑧
Line integrals
1.
2.
3.
4.
𝑟⃗(𝑡) = [𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)]
𝒓⃗′(𝑡) = [𝑥′(𝑡), 𝑦′(𝑡), 𝑧′(𝑡)]
𝐹 𝑟⃗(𝑡)
𝐹 𝑟⃗(𝑡) ∙ 𝒓⃗′(𝑡)
5. ∫ 𝐹 𝑟⃗(𝑡) ∙ 𝒓⃗′(𝑡)𝑑𝑡
Surface integrals
1.
2.
3.
4.
5.
6.
7.
𝑟⃗(𝑢, 𝑣) = [𝑥(𝑢, 𝑣), 𝑦(𝑢, 𝑣), 𝑧(𝑢, 𝑣)]
𝒓⃗′(𝑢) = [𝑥′(𝑢, 𝑣), 𝑦′(𝑢, 𝑣), 𝑧′(𝑢, 𝑣)]
𝒓⃗′(𝑣) = [𝑥′(𝑢, 𝑣), 𝑦′(𝑢, 𝑣), 𝑧′(𝑢, 𝑣)]
𝑁 = 𝒓⃗′(𝑢) × 𝒓⃗′(𝑣)
𝐹 𝑟⃗(𝑢, 𝑣)
𝐹 𝑟⃗(𝑢, 𝑣) ∙ 𝑁
∬ 𝐹 𝑟⃗(𝑢, 𝑣) ∙ 𝑁𝑑𝑢𝑑𝑣
Volume integral
𝑓(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦𝑑𝑧
Volume integral is the triple integral of a func on over the region 𝑇
Unit normal vector:
𝒏=
𝑵
|𝑵|
Area of parallelogram with sides 𝒓𝒖 and 𝒓𝒗 :
|𝑵| = |𝒓 × 𝒓 |
Spherical Polar
𝑥 = 𝑟𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜙
𝑦 = 𝑟𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜙
𝑧 = 𝑟𝑐𝑜𝑠𝜃
𝑑𝑆 = 𝑟 𝑠𝑖𝑛𝜃𝑑𝑟𝑑𝜃
𝑑𝑉 = 𝑟 𝑠𝑖𝑛𝜃𝑑𝑟𝑑𝜃𝑑𝜙
Tutorial 4: Green’s Theorem – Divergence Theorem of Gauss – Stokes’ Theorem
Green’s Theorem in the Plane:
𝑑𝑥𝑑𝑦 = ∮(𝐹 𝑑𝑥 + 𝐹 𝑑𝑦)
−
∬
Green’s Theorem given the curl of vector func on 𝑭:
∬(𝑐𝑢𝑟𝑙 𝑭) ∙ 𝒌𝑑𝑥𝑑𝑦 = ∮ 𝑭 ∙ 𝑑𝒓
Green’s theorem steps of solving:
1.
2.
−
3.
4. ∬
−
𝑑𝑥𝑑𝑦
Divergence Theorem of Gauss:
𝑑𝑖𝑣 𝑭 =
+
+
∭ (𝑑𝑖𝑣 ⋅ 𝐅) 𝑑𝑉 = ∯ 𝐅 ⋅ 𝐧 𝑑𝐀
Stokes’ Theorem (generaliza on of greens theorem):
∫ ∫ (𝑐𝑢𝑟𝑙 𝐅) ⋅ 𝐧 𝑑𝐀 = ∮ 𝐅 ⋅ 𝐫′(𝐬) 𝑑𝐬
Stokes’ theorem steps of solving:
1. 𝑐𝑢𝑟𝑙 𝑭
2. 𝑵 = ∇ 𝑧 − 𝑓(𝑥, 𝑦)
3. 𝑐𝑢𝑟𝑙 𝑭 ∙ 𝑵
Tutorial 5: Elementary of Complex Numbers - Geometric Representa on in the Complex
Plane - Polar Form of Complex Numbers -Power and Roots
-
A complex number 𝑧 is an ordered pair of real numbers 𝑥 and 𝑦: 𝑧 = (𝑥, 𝑦).
o Real part: 𝑥 = 𝑅𝑒 𝑧
o Imaginary part: 𝑦 = 𝐼𝑚 𝑧
Addi on & Subtrac on of two complex numbers:
𝑧 ± 𝑧 = (𝑥 , 𝑦 ) ± (𝑥 , 𝑦 ) = (𝑥 ± 𝑥 ) + 𝑖(𝑦 ± 𝑦 )
Mul plica on of two complex numbers:
𝑧 𝑧 = (𝑥 , 𝑦 )(𝑥 , 𝑦 ) = (𝑥 𝑥 − 𝑦 𝑦 ,
𝑥 𝑦 +𝑥 𝑦 )
Division of complex numbers:
𝑧=
-
=
×
=
+𝑖
Complex number can be represented on a complex plane with the real part on the x-axis and the
imaginary part on the y-axis.
The complex conjugate (𝑧̅) is the reflec on of complex number on the real axis.
o 𝑧 = 𝑥 + 𝑖𝑦
o 𝑧̅ = 𝑥 − 𝑖𝑦
Proper es of complex conjugate numbers:
1
𝑅𝑒 𝑧 = 𝑥 = (𝑧 + 𝑧̅)
2
𝐼𝑚 𝑧 = 𝑦 =
1
(𝑧 − 𝑧̅)
2𝑖
(𝑧 ± 𝑧 ) = 𝑧 ± 𝑧
(𝑧 𝑧 ) = 𝑧 𝑧
𝑧
𝑧
-
=
𝑧
𝑧
The complex number can be represented in polar form: 𝑧 = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)
o Real part: 𝑥 = 𝑟𝑐𝑜𝑠𝜃
o Imaginary part: 𝑦 = 𝑟𝑠𝑖𝑛𝜃
 Absolute value (modulus of z): 𝑟 = |𝒛| = 𝑥 + 𝑦 = √𝑧𝑧̅

𝜃 = tan
The principal value 𝑨𝒓𝒈 𝒛:
−𝜋 < 𝐴𝑟𝑔 𝑧 ≦ 𝜋
- Angle 𝜽 = 𝐚𝐫𝐠 𝒛:
arg 𝑧 = 𝐴𝑟𝑔 𝑧 ± 2𝑛𝜋
The triangle inequality:
|𝑧 + 𝑧 | ≦ |𝑧 | + |𝑧 |
-
Mul plica on of two complex numbers in polar form:
𝑧 𝑧 = 𝑟 𝑟 [cos(𝜃 + 𝜃 ) + 𝑖𝑠𝑖𝑛(𝜃 + 𝜃 )]
|𝑧 𝑧 | = |𝑧 ||𝑧 |
arg(𝑧 𝑧 ) = arg 𝑧 + arg 𝑧 up to mul ples of 2𝜋
Division of two complex numbers in polar form:
= [cos(𝜃 − 𝜃 ) + 𝑖𝑠𝑖𝑛(𝜃 − 𝜃 )]
|𝑧 |
𝑧
=
𝑧 ≠0
|𝑧 |
𝑧
arg = arg 𝑧 − arg 𝑧 up to mul ples of 2𝜋
A complex number with integer power:
𝑧 = 𝑟 (𝑐𝑜𝑠 𝑛𝜃 + 𝑖𝑠𝑖𝑛 𝑛𝜃)
De Moivre’s formula:
(cos 𝜃 + 𝑖 sin 𝜃) = cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃
The 𝒏th root of 𝒛:
𝑤 = √𝑧 = √𝑟 cos
+ isin
𝑘 = 0,1, … . , 𝑛 − 1
Tutorial 6: Elementary of Complex Func ons – Limit, Con nuity, & Deriva ves – Cauchy
Riemann Equa ons – Laplace’s Equa ons
A circle in complex plane with center 𝒂 and radius
𝝆
Two circles in the complex plane both with center
𝒂 and radius 𝝆𝟏 , 𝝆𝟐
|𝑧 − 𝑎| = 𝜌
The set of all points located on the circle itself
𝜌 < |𝑧 − 𝑎| < 𝜌
The set of all points located in the interior between
the two circles (open annulus - circular ring)
|𝑧 − 𝑎| < 𝜌
The set of all points located in the interior of the
circle (open circular disk)
𝜌 ≤ |𝑧 − 𝑎| ≤ 𝜌
The set of all points located on the two circles and
in the interior between them (closed annulus)
|𝑧 − 𝑎| > 𝜌
The set of all points located in the exterior of the
circle
|𝑧 − 𝑎| ≤ 𝜌
The set of all points located on the circle itself and
its interior (closed circular disk)
𝑦>0
The set of all points
located in the open
upper half plane
Half Planes
𝑦<0
𝑥>0
The set of all points
The set of all points
located in the open
located in the open
lower half plane
right half plane
𝑥<0
The set of all points
located in the open le
half plane
-
The boundary of 𝑆 is the set of all points on the boundary of the set.
Set 𝑆 is open if every point of 𝑆 is interior and no points is boundary (open circular disk and open
half plane)
Set 𝑆 is Close if it includes every interior and boundary points (clothes circular desk and closed
annulus)
Set 𝑆 is connected if we can connect between any two points in 𝑆.
Set 𝑆 is called a domain if it's open and connected.
The set of all points in the complex plane that do not belong to the set 𝑆 is called the
complement of set 𝑆.
Set 𝑆 is region if it includes at least all points in the domain.
-
A func on is set to have the limit 𝑙 as 𝑧 approaches a point 𝑧 : lim 𝑓(𝑧) = 𝑙
-
For every posi ve real 𝜖 we can find a posi ve real 𝛿 such that for all 𝑧 ≠ 𝑧 in the disk
|𝑧 − 𝑧 | < 𝛿.
A func on is set to be con nuous at 𝑧 = 𝑧 if 𝑓(𝑧 ) is defined and lim 𝑓(𝑧) = 𝑓(𝑧 ).
-
-
→
→
𝑓(𝑧) Is set to be con nuous in the domain if it is con nuous at each point of this domain.
The deriva ve of a complex func on 𝑓 at point 𝑧 is wri en 𝑓′(𝑧 ) and is defined by
𝑓(𝑧 + ∆𝑧) − 𝑓(𝑧 )
𝑓 (𝑧 ) = lim
∆ →
∆𝑧
Differen a on rules
(𝑐𝑓) = 𝑐𝑓
(𝑓 + 𝑔) = 𝑓 + 𝑔
(𝑓𝑔) = 𝑓 𝑔 + 𝑔 𝑓
𝑓
𝑓 𝑔−𝑔 𝑓
=
𝑔
𝑔
(𝑧 ) = 𝑛𝑧
-
-
A func on is said to be analy c in a domain 𝐷 if 𝑓(𝑧) is defined and differen able at all points of
the domain.
The func on 𝑓(𝑧) is set to be analy c at point 𝑧 = 𝑧 in 𝐷 if 𝑓(𝑧) is analy c in a neighborhood
of 𝑧 .
Cauchy – Riemann Equa ons:
-
𝑓 is analy c in a domain 𝐷 if and only if the first par al deriva ves of 𝑢 and 𝑣 set aside the two
Cauchy – Riemann Equa ons.
𝑢 =𝑣
𝑢 = −𝑣
-
Cauchy – Riemann Equa ons in polar forms:
𝑢 = 𝑣
𝑣 =− 𝑢
Laplace’s Equa on:
If 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦) is analy c in a domain 𝐷 then 𝑢 & 𝑣 sa sfy Laplace’s Equa on:
∇ 𝑢=𝑢
+𝑢
=0
∇ 𝑣=𝑣
+𝑣
=0
Solu ons of Laplace’s Equa on having con nuous second order par al deriva ves are called harmonic
func ons, where 𝑣 set to be a harmonic conjugate func on of 𝑢