This document provides an introduction to partial differential equations (PDEs). It defines PDEs as equations involving an unknown function of two or more variables and certain of its partial derivatives. The document then classifies PDEs as linear, semilinear, quasilinear, or fully nonlinear based on how they depend on the derivatives of the unknown function. It lists many common and important PDEs as examples, including the heat equation, wave equation, Laplace's equation, and Euler's equations. Finally, it outlines strategies for studying PDEs, such as seeking explicit solutions, using functional analysis to prove existence of weak solutions, and developing theories to handle both linear and nonlinear PDEs.

1. INTRODUCTION of PARTIAL DIFFERENTIAL EQUATIONS
1. PARTIAL DIFFERENTIAL EQUATIONS
• A partial differential equations (PDE) is an equation involving an unknown
function of two or more variables and certain of its partial derivatives.
• Using the notation explained in Appendix A, we can write out symbolically a
typical PDE, as follows. Fix an integer 𝑘 ≥ 1 and let 𝑈 denote an open subset of
ℝ 𝑛
.
DEFINITION. An expression of the form
𝐹(𝐷 𝑘
𝑢(𝑥), 𝐷 𝑘−1
𝑢(𝑥), ⋯ , 𝐷𝑢(𝑥), 𝑢(𝑥), 𝑥) = 0 (𝑥𝜖𝑈) (1)
is called 𝑎 kth
-order partial differential equation, where
𝐹: ℝ 𝑛 𝑘
× ℝ 𝑛 𝑘−1
× ⋯ ℝ 𝑛
× ℝ × 𝑈 → ℝ
is given, and
𝑢: 𝑈 → ℝ
is the unknown.
• We solve the PDE if we find all 𝑢 verifying (1), possibly only among those
functions satisfying certain auxiliary boundary conditions on some part Γof
𝜕𝑈.
• By finding the solutions we mean, ideally, obtaining simple, explicit solutions
or failing that, deducing the existence and other properties of solutions.
DEFINITIONS.
i. The partial differential equation (1) is called linear if it has the form
� 𝑎 𝛼(𝑥)𝐷 𝛼
𝑢 = 𝑓(𝑥)
|𝛼|≤𝑘
for given functions 𝑎 𝛼(𝛼| ≤ 𝑘), f. This linear PDE is homogeneous if 𝑓 ≡ 0.
ii. The PDE (1) is semilinear if it has the form
� 𝑎 𝛼(𝑥)𝐷 𝛼
𝑢 + 𝑎0(𝐷 𝑘−1
𝑢, ⋯ , 𝐷𝑢, 𝑢, 𝑥) = 0
|𝛼|=𝑘
iii. The PDE (1) is quasilinear if it has the form
� 𝑎 𝛼(𝐷 𝑘−1
𝑢, ⋯ , 𝐷𝑢, 𝑢, 𝑥)𝐷 𝛼
𝑢 + 𝑎0(𝐷 𝑘−1
𝑢, ⋯ , 𝐷𝑢, 𝑢, 𝑥) = 0
|𝛼|=𝑘
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 1

2. iv. The PDE (1) is fully nonlinear if it depends nonlinearly upon the higest order
derivatives.
DEFINITION. An expression of the form
𝐹(𝐷 𝑘
𝒖(𝑥), 𝐷 𝑘−1
𝒖(𝑥), ⋯ , 𝐷𝒖(𝑥), 𝒖(𝑥), 𝑥) = 𝟎 (𝑥𝜖𝑈) (2)
is called a kth
-order system of partial differential equations, where
𝐹: ℝ 𝑚𝑛 𝑘
× ℝ 𝑚𝑛 𝑘−1
× ⋯ ℝ 𝑚𝑛
× ℝ 𝑚
× 𝑈 → ℝ 𝑚
is given, and
𝒖: 𝑈 → ℝ 𝑚
, 𝒖 = (𝑢1
, ⋯ , 𝑢 𝑚)
is the unknown.
• Here we are supposing that the system comprises the same number 𝑚 of scalar
equations as unknowns (𝑢1
, ⋯ , 𝑢 𝑚).
• This is the most common circumstance, although other system may have fewer or
more equations than unknowns.
2. EXAMPLES
• There is no general theory known concerning the solvability of all partial
differential equations.
• Such a theory is extremely unlikely to exist, given the rich variety of physical,
geometric, probabilistic phenomena which can be modeled by PDE.
• Instead, research focuses on various particular partial differential equations that
important for applications within and outside of mathematics, with the hope that
insight from the origins of these PDE can give clues as to their solutions.
• Following is a list of many specific partial differential equations of interest in
current research.
• This listing is intended merely to familiarize the reader with the names and forms
of various famous PDE.
• To display most clearly the mathematical structure of the equations, we have
mostly relevant physical constants to unity.
• We will later discuss the origin and interpretation of many of these PDE.
• For each 𝑥 ∈ 𝑈, where 𝑈 is an open subset of ℝ 𝑛
, and 𝑡 ≥ 0. Also
𝐷𝑢 = 𝐷𝑥 𝑢 = �𝑢 𝑥1
, 𝑢 𝑥2
, ⋯ , 𝑢 𝑥 𝑛
� denotes the gradient of 𝑢 with respect to the
spatial variable 𝑥 = (𝑥1, ⋯ , 𝑥 𝑛).
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 2

3. 2.1 Single partial differential equations
A. Linear equations
1. Laplace’s equation
∆𝑢 = � 𝑢 𝑥 𝑖 𝑥 𝑖
= 0
𝑛
𝑖=1
2. Helmholtz’s (or eigenvalue) equation
−∆𝑢 = 𝜆𝑢
3. Linear transport equation
𝑢 𝑡 + � 𝑏 𝑖
𝑢 𝑥 𝑖
= 0
𝑛
𝑖=1
4. Liouville’s equation
𝑢 𝑡 − ��𝑏 𝑖
𝑢� 𝑥 𝑖
= 0
𝑛
𝑖=1
5. Heat (or diffusion) equation
𝑢 𝑡 − Δ𝑢 = 0
6. Schrodinger’s equation
𝑖𝑢 𝑡 + Δ𝑢 = 0
7. Kolmogorov’s equation
𝑢 𝑡 − � 𝑎𝑖𝑗
𝑢 𝑥 𝑖 𝑥 𝑗
+
𝑛
𝑖,𝑗=1
� 𝑏 𝑖
𝑢 𝑥 𝑖
= 0
𝑛
𝑖=1
8. Fokker-Planck equation
𝑢 𝑡 − � �𝑎𝑖𝑗
𝑢� 𝑥 𝑖 𝑥 𝑗
−
𝑛
𝑖,𝑗=1
��𝑏 𝑖
𝑢� 𝑥 𝑖
= 0
𝑛
𝑖=1
9. Wave equation
𝑢 𝑡𝑡 − Δ𝑢 = 0
10. Telegraph equation
𝑢 𝑡𝑡 + 𝑑𝑢 𝑡 − 𝑢 𝑥𝑥 = 0
11. General wave equation
𝑢 𝑡𝑡 − � 𝑎𝑖𝑗
𝑢 𝑥 𝑖 𝑥 𝑗
+
𝑛
𝑖,𝑗=1
� 𝑏 𝑖
𝑢 𝑥 𝑖
= 0
𝑛
𝑖=1
12. Airy’s equation
𝑢 𝑡 + 𝑢 𝑥𝑥𝑥 = 0
13. Beam equation
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 3

4. 𝑢 𝑡 + 𝑢 𝑥𝑥𝑥𝑥 = 0
B. Nonlinear equations
1. Eikonal equation
|𝐷𝑢| = 1
2. Nonlinear Poisson equation
−∆𝑢 = 𝑓(𝑢)
3. p-Laplacian equation
𝑑𝑖𝑣(|𝐷𝑢| 𝑝−2
𝐷𝑢) = 0
4. Minimal surface equation
𝑑𝑖𝑣 �
𝐷𝑢
(1 + |𝐷𝑢|2)
1
2
� = 0
5. Monge-Ampere equation
𝑑𝑒𝑡(𝐷2
𝑢) = 𝑓
6. Hamilton-Jacobi equation
𝑢 𝑡 + 𝐻(𝐷𝑢, 𝑥) = 0
7. Scalar reaction-diffusion equation
𝑢 𝑡 + 𝑑𝑖𝑣 𝐹(𝑢) = 0
8. Inviscid Burger’s equation
𝑢 𝑡 + 𝑢𝑢 𝑥 = 0
9. Scalar reaction-diffusion equation
𝑢 𝑡 − ∆𝑢 = 𝑓(𝑢)
10. Porous medium equation
𝑢 𝑡 − ∆(𝑢 𝛾) = 0
11. Nonlinear wave equation
𝑢 𝑡𝑡 − ∆𝑢 = 𝑓(𝑢)
𝑢 𝑡𝑡 − 𝑑𝑖𝑣 𝑎(𝐷𝑢) = 0
12. Korteweg-de Vries (KdV) equation
𝑢 𝑡 + 𝑢𝑢 𝑥 + 𝑢 𝑥𝑥𝑥 = 0
2.2 System partial differential equations
A. Linear systems
1. Equilibrium equations of elasticity
𝜇∆𝒖 + (𝜆 + 𝜇)𝐷(𝑑𝑖𝑣 𝒖) = 0
2. Evolution equations of linear elasticity
𝒖 𝑡𝑡 − 𝜇∆𝒖 − (𝜆 + 𝜇)𝐷(𝑑𝑖𝑣 𝒖) = 0
3. Maxwell’s equations
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 4

5. �
𝑬 𝑡 = 𝑐𝑢𝑟𝑙 𝑩
𝑩 𝑡 = −𝑐𝑢𝑟𝑙 𝑬
𝑑𝑖𝑣 𝑩 = 𝑑𝑖𝑣 𝑬 = 𝟎
B. Nonlinear systems
1. System of conservation laws
𝒖 𝑡 + 𝑑𝑖𝑣 𝑭(𝒖) = 𝟎
2. Reaction-diffusion system
𝒖 𝑡 − ∆𝒖 = 𝑓(𝒖)
3. Euler’s equations for incrompessible
�
𝒖 𝑡 + 𝒖 ∙ 𝐷𝒖 = −𝐷𝑝
𝑑𝑖𝑣 𝒖 = 0
4. Navier-Stokes equations for incompressible, viscous flow
�
𝒖 𝑡 + 𝒖 ∙ 𝐷𝒖 − ∆𝒖 = −𝐷𝑝
𝑑𝑖𝑣 𝒖 = 0
3. STRATEGIES FOR STUDYING PDE
• Our goal is the discovery of ways to solve partial differential equations of various
sorts.
• The very question of what it means to “solve” a given PDE can be subtle.
3.1 Well-posed problems, classical solutions.
• The problem in fact has a solution;
• This solution is unique;
• The solution depends continuously on the data given in the problem.
3.2 Weak solutions and regularity.
• Can we achieve this?
• The answer is that certain specific partial differential equations (ex. Laplace’s
equation) can be solved in the classical sense, but many others, cannot. Consider
for instance the scalar conservation law 𝑢 𝑡 + 𝐹(𝑢) 𝑥 = 0.
• In general, as we shall see, the conservation law has no classical solutions, but is
well-posed if we allow for properly defined generalized or weak solutions.
• The idea is to define for a given PDE a reasonable wide notion of a weak solution,
with the expectation that since we are not asking too much by way of smoothness
of this weak solution.
• For other equations we can hope that our weak solution.
• This leads to the question of regularity of weak solutions.
• As we will see, it is often the case that the existence of weak solutions depends
upon rather simple estimates plus ideas of functional analysis.
• Whereas the regularity of weak solutions, when true, usually rest upon many
intricate calculus estimates.
3.3 Typical difficulties
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 5

6. 1) Nonlinear equations are more difficult than linear equations, and indeed, the more
the nonlinearity affects the higher derivatives, the more difficult the PDE is.
2) Higher-order PDE are more difficult than lower-order PDE.
3) System are harder than single equations.
4) Partial differential equations entailing many independent variables are harder than
entailing few independent variables.
5) For most partial differential equations it is not possible to write out explicit
formulas for solutions.
4. OVERVIEW
This textbook is divided into three major Parts
PART I: Representation Formulas for Solutions
• Identify those important PDE for which in certain circumstances explicit or more-
or-less explicit formulas can be had for solutions.
• Chapter 2 is a detailed study of four exactly solvable PDE: the linear transport
equation, Laplace’s equation, the heat equation, and the wave equation.
• Chapter 3 continues the theme of searching for explicit formulas, now for general
first-order nonlinear PDE.
• We stipulate that once the problem becomes “only” the equation of integrating a
system of ODE.
• We introduce also the Hopf-Lax formula for Hamilton-Jacobi equations and the
Lax-Oleinik formula for scalar conservation laws.
• Chapter 4 is a grab bag of techniques for explicitly solving various linear and
nonlinear PDE.
PART II: Teori PDE Linear
• We abandon the search for explicit formulas and instead rely on functional
analysis and relatively easy “energy” estimates to prove existence of weak
solutions to various linear PDE.
• Chapter 5 is an introduction to Sobolev spaces, the proper setting for the study of
many linear and nonlinear PDE via energy methods.
• This is hard chapter, the real worth of which is only later revealed, and requires
some basic knowledge of Lebesgue measure theory.
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 6

7. • Chapter 6: we vastly generalize our knowledge of Laplace’s equation to other
second-order elliptic equations.
• Chapter 7 expands the energy methods to a variety of linear PDE characterizing
evolutions in time.
PART III: Theory for Nonlinear PDE
• This section parallels for nonlinear PDE the development in Part II, but is far less
unified in its approach, as the various types of nonlinearity must be treated in
quite different ways.
• Chapter 8 commences the general study of nonlinear PDE with an extensive
discussion of the calculus of variations.
• Chapter 9, a gathering of assorted other techniques of use for nonlinear elliptic
and parabolic PDE.
• Chapter 10 is an introduction to the modern theory of Hamilton-Jacobi PDE, and
particular to the notion of “viscosity solutions”.
• We encounter also the connections with optimal control of ODE, through
dynamic programming.
• Chapter 11 is the discussion of conservation laws, now systems of conservation
laws.
5. PROBLEMS
1) Classify each of PDE in [2] as follows:
a) Is the PDE linear, semilinear, quassilinear, or fully nonlinear?
b) What is the order of the PDE?
2) Prove the Multinomial Theorem
(𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑛) 𝑁
= � �
|𝛼|
𝛼
� 𝑥 𝛼
|𝛼|=𝑁
Where �|𝛼|
𝛼
� =
|𝛼|!
𝛼!
, 𝛼! = 𝛼1! 𝛼2! ⋯ 𝛼 𝑛!, and 𝑥 𝛼
= 𝑥1
𝛼1
⋯ 𝑥 𝑛
𝛼 𝑛
. The sum is taken over all
multiindecs 𝛼 = (𝛼1, 𝛼2, ⋯ , 𝛼 𝑛) with |𝛼| = 𝑁.
Multinomial Theorem is a natural extension of binomial theorem and proof gives a
good exercise for using the Principle of Mathematical Induction.
Theorem: Let P(n) be the proposition:
(𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑛) 𝑁
= � �
|𝛼|
𝛼
� 𝑥 𝛼
|𝛼|=𝑁
Where �|𝛼|
𝛼
� =
|𝛼|!
𝛼!
, 𝛼! = 𝛼1! 𝛼2! ⋯ 𝛼 𝑛!, and 𝑥 𝛼
= 𝑥1
𝛼1
⋯ 𝑥 𝑛
𝛼 𝑛
. The sum is taken over all
multiindecs 𝛼 = (𝛼1, 𝛼2, ⋯ , 𝛼 𝑛) with |𝛼| = 𝑁. In another form
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 7

8. (𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑛) 𝑁
= ∑ �
𝑁!
𝛼1!𝛼2!⋯𝛼 𝑛!
𝑥1
𝛼1
⋯ 𝑥 𝑛
𝛼 𝑛
�𝛼1,𝛼2,⋯,𝛼 𝑛≥0
𝛼1+𝛼2+⋯+𝛼 𝑛=𝑁
,where 𝑛, 𝑁 ∈ ℕ
Proof:
• 𝑃(1) → (𝑥1) 𝑁
= ∑ �
𝑁!
𝛼1!
𝑥1
𝛼1
�𝛼1≥0
𝛼1=𝑁
= ∑ �
𝑁!
𝑁!
𝑥1
𝑁
� =𝛼1≥0
𝛼1=𝑁
∑ [𝑥1
𝑁] = 𝑥1
𝑁
𝛼1≥0
𝛼1=𝑁
∴ 𝑃(1)𝑖𝑠 𝑡𝑟𝑢𝑒
• 𝑃(2) → (𝑥1 + 𝑥2) 𝑁
= ∑ �
𝑁!
𝛼1!𝛼2!
𝑥1
𝛼1
𝑥2
𝛼2
�𝛼1,𝛼2≥0
𝛼1+𝛼2=𝑁
, By Binomial Theorem:
(𝑥1 + 𝑥2) 𝑁
= � 𝐶𝑟
𝑁
𝑥1
𝑁−𝑟
𝑥2
𝑟
𝑁
𝑟=0
= �
𝑁!
(𝑁 − 𝑟)! 𝑟!
𝑥1
𝑁−𝑟
𝑥2
𝑟
𝑁
𝑟=0
= ∑
𝑁!
𝛼1!𝛼2!
𝑥1
𝛼1
𝑥2
𝛼2𝑁
𝛼2=0 , where 𝛼1 = 𝑁 − 𝑟, 𝛼2 = 𝑟.
= ∑
𝑁!
𝛼1!𝛼2!
𝑥1
𝛼1
𝑥2
𝛼2𝑁
𝛼1,𝛼2≥0
𝛼1+𝛼2=𝑁
∴ 𝑃(2)𝑖𝑠 𝑡𝑟𝑢𝑒
Assume 𝑃(𝑘) is true for some 𝑘 ∈ ℕ, that is:
(𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑘) 𝑁
= � �
𝑁!
𝛼1! 𝛼2! ⋯ 𝛼 𝑘!
𝑥1
𝛼1
⋯ 𝑥 𝑘
𝛼 𝑘
�
𝛼1,𝛼2,⋯,𝛼 𝑘≥0
𝛼1+𝛼2+⋯+𝛼 𝑘=𝑁
⋯ (1)
for 𝑃(𝑘 + 1),
(𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑘 + 𝑥 𝑘+1) 𝑁
= [(𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑘) + 𝑥 𝑘+1] 𝑁
= � 𝐶𝑟
𝑁( 𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑘) 𝑁−𝑟
𝑥 𝑘+1
𝑟
𝑁
𝑟=0
= � 𝐶𝑟
𝑁
� �
(𝑁 − 𝑟)!
𝛼1! 𝛼2! ⋯ 𝛼 𝑘!
𝑥1
𝛼1
⋯ 𝑥 𝑘
𝛼 𝑘
�
𝛼1,𝛼2,⋯,𝛼 𝑘≥0
𝛼1+𝛼2+⋯+𝛼 𝑘=𝑁−𝑟
𝑥 𝑘+1
𝑟
𝑁
𝑟=0
= � � �
𝑁!
(𝑁 − 𝑟)! 𝑟!
(𝑁 − 𝑟)!
𝛼1! 𝛼2! ⋯ 𝛼 𝑘!
𝑥1
𝛼1
⋯ 𝑥 𝑘
𝛼 𝑘
𝑥 𝑘+1
𝑟
�
𝛼1,𝛼2,⋯,𝛼 𝑘≥0
𝛼1+𝛼2+⋯+𝛼 𝑘=𝑁−𝑟
𝑁
𝑟=0
= ∑ ∑ �
𝑁!
𝛼1!𝛼2!⋯𝛼 𝑘!𝛼 𝑘+1!
𝑥1
𝛼1
⋯ 𝑥 𝑘
𝛼 𝑘
𝑥 𝑘+1
𝑟
�𝛼1,𝛼2,⋯,𝛼 𝑘≥0
𝛼1+𝛼2+⋯+𝛼 𝑘=𝑁−𝛼 𝑘+1
𝑁
𝑟=0 where 𝛼 𝑘+1 = 𝑟.
= � � �
𝑁!
𝛼1! 𝛼2! ⋯ 𝛼 𝑘! 𝛼 𝑘+1!
𝑥1
𝛼1
⋯ 𝑥 𝑘
𝛼 𝑘
𝑥 𝑘+1
𝑟
�
𝛼1,𝛼2,⋯,𝛼 𝑘,𝛼 𝑘+1≥0
𝛼1+𝛼2+⋯+𝛼 𝑘+𝛼 𝑘+1=𝑁
𝑁
𝑟=0
∴ 𝑃(𝑘 + 1)𝑖𝑠 𝑡𝑟𝑢𝑒
By the Principle of Mathematical Induction 𝑃(𝑛) is true ∀𝑛 ∈ ℕ.
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 8

9. 3) Prove Leibniz’ formula
𝐷 𝛼(𝑢𝑣) = ∑ � 𝛼
𝛽
�𝛽≤𝛼 𝐷 𝛽
𝑢𝐷 𝛼−𝛽
𝑣,
Where 𝑢, 𝑣: ℝ 𝑛
→ ℝ are smooth, � 𝛼
𝛽
� =
𝛼!
(𝛼−𝛽)!𝛽!
, and 𝛽 ≤ 𝛼means 𝛽𝑖 ≤ 𝛼𝑖, (𝑖 = 1,2, … , 𝑛)
Proof:
𝐷 𝛼(𝑢𝑣) =
𝑑 𝛼
𝑑𝑥 𝛼
(𝑢𝑣) = � �
𝛼
𝛽
�
𝛼
𝛽=0
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝛼−𝛽
𝑑𝑥 𝛼−𝛽
𝑣, 𝛼 ∈ ℤ+
By the Principle of Mathematical Induction on 𝛼. Recall
𝑑 𝛼
𝑑𝑥 𝛼
denotes the 𝛼 𝑡ℎ
derivative. The
product rule say that
𝑑(𝑢𝑣)
𝑑𝑥
= 𝑢
𝑑𝑣
𝑑𝑥
+ 𝑣
𝑑𝑢
𝑑𝑥
• 𝑃(1) →
𝑑(𝑢𝑣)
𝑑𝑥
= ∑ �1
𝛽
�1
𝛽=0
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑1−𝛽
𝑑𝑥1−𝛽
𝑣 = �1
0
�
𝑑0
𝑑𝑥0
𝑢
𝑑1−0
𝑑𝑥1−0
𝑣 + �1
1
�
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑1−1
𝑑𝑥1−1
𝑣 = 𝑢
𝑑𝑣
𝑑𝑥
+ 𝑣
𝑑𝑢
𝑑𝑥
• Assume 𝑃(𝑘) is true for some 𝑘 ∈ ℤ+
, that is:
𝑑 𝑘
𝑑𝑥 𝑘
(𝑢𝑣) = � �
𝑘
𝛽
�
𝑘
𝛽=0
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘−𝛽
𝑑𝑥 𝑘−𝛽
𝑣, 𝑘 ∈ ℤ+
• We want to prove the formula:
𝑑 𝑘+1
𝑑𝑥 𝑘+1
(𝑢𝑣) = � �
𝑘 + 1
𝛽
�
𝑘+1
𝛽=0
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣, 𝑘 ∈ ℤ+
First we compute:
𝑑 𝑘+1
𝑑𝑥 𝑘+1
(𝑢𝑣) =
𝑑
𝑑𝑥
�
𝑑 𝑘
𝑑𝑥 𝑘
(𝑢𝑣)�
=
𝑑
𝑑𝑥
�� �
𝑘
𝛽
�
𝑘
𝛽=0
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘−𝛽
𝑑𝑥 𝑘−𝛽
𝑣�
= � �
𝑘
𝛽
�
𝑘
𝛽=0
𝑑
𝑑𝑥
�
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘−𝛽
𝑑𝑥 𝑘−𝛽
𝑣�
= � �
𝑘
𝛽
�
𝑘
𝛽=0
�
𝑑 𝛽+1
𝑑𝑥 𝛽+1
𝑢
𝑑 𝑘−𝛽
𝑑𝑥 𝑘−𝛽
𝑣 +
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣�
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 9

10. = � �
𝑘
𝛽
�
𝑘
𝛽=0
𝑑 𝛽+1
𝑑𝑥 𝛽+1
𝑢
𝑑 𝑘−𝛽
𝑑𝑥 𝑘−𝛽
𝑣 + � �
𝑘
𝛽
�
𝑘
𝛽=0
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣
𝑟𝑒𝑝𝑙𝑎𝑐𝑒 𝑟 + 1 𝑏𝑦 𝑟 𝑎𝑛𝑑 𝑠𝑜 𝑟 = 𝑟 − 1, 𝑎𝑛𝑑 𝑠𝑢𝑚 𝑓𝑟𝑜𝑚 1 𝑡𝑜 𝑘
� �
𝑘
𝛽
�
𝑘
𝛽=0
𝑑 𝛽+1
𝑑𝑥 𝛽+1
𝑢
𝑑 𝑘−𝛽
𝑑𝑥 𝑘−𝛽
𝑣 = � �
𝑘 + 1
𝛽 − 1
�
𝑘+1
𝛽=1
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣
We get
𝑑 𝑘+1
𝑑𝑥 𝑘+1
(𝑢𝑣) = � �
𝑘 + 1
𝛽 − 1
�
𝑘+1
𝛽=1
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣 + � �
𝑘
𝛽
�
𝑘
𝛽=0
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣
= � ��
𝑘 + 1
𝛽 − 1
� + �
𝑘
𝛽
��
𝑘
𝛽=1
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣 + �
𝑘
𝑘
�
𝑑 𝑘+1
𝑑𝑥 𝑘+1
𝑢 + �
𝑘
0
�
𝑑 𝑘+1
𝑑𝑥 𝑘+1
𝑣
Since � 𝑘+1
𝛽−1
� + � 𝑘
𝛽
� = � 𝑘+1
𝛽
� and since � 𝑘
𝑘
� = � 𝑘+1
𝑘+1
�, and � 𝑘
0
� = � 𝑘+1
0
�
We get:
𝑑 𝑘+1
𝑑𝑥 𝑘+1
(𝑢𝑣) = � �
𝑘 + 1
𝛽
�
𝑘
𝛽=1
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣 + �
𝑘 + 1
𝑘 + 1
�
𝑑 𝑘+1
𝑑𝑥 𝑘+1
𝑢 + �
𝑘 + 1
0
�
𝑑 𝑘+1
𝑑𝑥 𝑘+1
𝑣
= � �
𝑘 + 1
𝛽
�
𝑘
𝛽=1
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣 + � �
𝑘 + 1
𝑘 + 1
�
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣
𝑘+1
𝛽=𝑘+1
+ � �
𝑘 + 1
0
�
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣
0
𝛽=0
= � �
𝑘 + 1
𝛽
�
𝑘+1
𝛽=0
𝑑 𝛽
𝑑𝑥 𝛽
𝑢
𝑑 𝑘+1−𝛽
𝑑𝑥 𝑘+1−𝛽
𝑣
∴ 𝑃(𝑘 + 1)𝑖𝑠 𝑡𝑟𝑢𝑒
By the Principle of Mathematical Induction 𝑃(𝛼) is true ∀𝛼 ∈ ℤ+
.
4) Assume that 𝑓: ℝ 𝑛
→ ℝ is smooth. Proof
𝑓(𝑥) = �
1
𝛼!
|𝛼|≤𝑘
𝐷 𝛼
𝑓(0)𝑥 𝛼
+ 𝑂(|𝑥| 𝑘+1) 𝑎𝑠 𝑥 → 0
For each 𝑘 = 1,2, ⋯.
𝛼 = (𝛼1, 𝛼2, ⋯ , 𝛼 𝑛), 𝛼1, 𝛼2, ⋯ , 𝛼 𝑛 ≥ 0
𝑥 = (𝑥1, 𝑥2, ⋯ , 𝑥 𝑛),
|𝛼| = 𝛼1 + 𝛼2 + ⋯ + 𝛼 𝑛,
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 10

11. 𝛼! = 𝛼1! 𝛼2! ⋯ 𝛼 𝑛!,
𝐷 𝛼
=
𝜕|𝛼|
𝜕 𝑥1
𝛼1⋯𝜕 𝑥 𝑛
𝛼 𝑛,
|𝑥| = (𝑥1
2
+ 𝑥2
2
+ ⋯ + 𝑥 𝑛
2)
1
2,
𝑥 𝛼
= 𝑥1
𝛼1
⋯ 𝑥 𝑛
𝛼 𝑛
.
This is Taylor’s Formula in multiindex notation. (Hint: fix 𝑥 ∈ ℝ 𝑛
and consider the
function of one variable 𝑔(𝑡) = 𝑓(𝑡𝑥).)
Proof: from 1D Taylor’s formula we have:
𝑔(𝑡) = 𝑔(0) + 𝑔′(0)𝑡 +
1
2
𝑔′′(0)𝑡2
+ ⋯ +
1
𝑘!
𝑔 𝑛(0)𝑡 𝑘
+
1
(𝑘 + 1)!
𝑔 𝑛+1(𝜀)𝑡 𝑘+1
Here (𝜀) lies between 0 and 𝑡.
Now as 𝑔(𝑡) = 𝑓(𝑡𝑥). All we need to do is to write the derivatives of 𝑔 using derivatives
of 𝑓. We compute:
𝑔 𝑘(𝑠) = �
𝑑 𝑘
𝑑𝑡 𝑘
𝑓(𝑡𝑥)��
𝑡=𝑠
= ( 𝑥1 𝜕1 + 𝑥2 𝜕2 + ⋯ + 𝑥 𝑛 𝜕 𝑛) 𝑘
𝑓(𝑡𝑥)� 𝑡=𝑠
By multinomial theorem, we get:
𝑔 𝑘(𝑠) = � ��
𝑘!
𝛼1! 𝛼2! ⋯ 𝛼 𝑛!
� �𝑥1
𝛼1
⋯ 𝑥 𝑛
𝛼 𝑛
� �
𝜕 𝛼1+𝛼2+⋯+𝛼 𝑛
𝜕 𝑥1
𝛼1
⋯ 𝜕 𝑥 𝑛
𝛼 𝑛
� 𝑓(𝑠𝑥)�
𝛼1,𝛼2,⋯,𝛼 𝑛≥0
𝛼1+𝛼2+⋯+𝛼 𝑛=𝑘
= � ��
𝑘!
𝛼1! 𝛼2! ⋯ 𝛼 𝑛!
� (𝑥 𝛼) �
𝜕|𝛼|
𝜕 𝑥1
𝛼1
⋯ 𝜕 𝑥 𝑛
𝛼 𝑛
� 𝑓(𝑠𝑥)�
𝛼1,𝛼2,⋯,𝛼 𝑛≥0
𝛼1+𝛼2+⋯+𝛼 𝑛=𝑘
= � ��
𝑘!
𝛼!
� (𝑥 𝛼)(𝐷 𝛼)𝑓(𝑠𝑥)�
|𝛼|=𝑘
Mohamad Tafrikan (moh.tafrikan@gmail.com) Page 11