# Mathematics II Course Note

*Made by Mike_Zhang*

*Notice | 提示*

个人笔记，仅供参考PERSONAL COURSE NOTE, FOR REFERENCE ONLYPersonal course note of

AMA2112 Mathematics II, The Hong Kong Polytechnic University, Sem2, 2022/23.

本文章为香港理工大学2022/23学年第二学期AMA2112数学II(AMA2112 Mathematics II)个人的课程笔记。Mainly focus on

Multiple Integrals,Vector Calculus,Fourier Series, andPartial Differential Equation.

主要内容包括多重积分，向量微积分，傅立叶级数，以及偏微分方程。

Unfold **Study Note** Topics | 展开**学习笔记**主题 >

# 1 Multiple Integrals

## 1.1 Double Integral

or

Double Integral is the volume above the xy plane and below the surface $z=f(x,y)$, $(x,y)\in R$

### 1.1.1 Fubini’s Theorem

**Rectangle Type**

In range $R=[a,b]\times [c,d]$,

**[Example]**

**Type I region (vertically simple region)**

In range $R={(x,y)|a\leq x\leq b, g_1(x)\leq y\leq g_2(x)}$, where x is in constant range

**[Example]**

**Type II region (horizontally simple region)**

In range $R={(x,y)|c\leq y\leq d, h_1(y)\leq x\leq h_2(y)}$, where y is in constant range

**[Example]**

**Steps to calculate integrals on Type I and II regions**

- Check in Type I or Type II;
- Apply the Fubini’s theorem.

**Steps to calculate integrals not formulated explicitly but given geometrically.**

- Formulate D as a Type I or Type II region first
- Find out the intersection points of two curves
- Check in which Type;

- Apply the Fubini’s theorem.

**[Example]**

**[Example]**

**A region is both Type I and Type II, then we can choose either case of Fubini’s theorem to make the calculation easier by switch the order of an iterated integral.**

*Switch the order of an iterated integral*

**[Example]**

### 1.1.2 Properties of Double Integrals

**Area of a region**:

**[Example]**

**Linearity**:

**Domain additivity**:

If $D$ is the **union** of two **non-overlapping** region $D_1$ and $D_2$, then

- If the integral on
**D is difficult to calculate**, we may**divide D into non-overlapping subregion s**, calculate the integral on each subregion, and then D2 take a sum. - If D is
**neither Type I nor Type II**, we may consider**dividing D into none-overlapping Type I or Type II subregions**.

**Domination**:

If $f(x,y) \ge 0$ on $D$, then

If $f(x,y) \ge g(x,y)$ on $D$, then

### 1.1.3 Change of Variables

$x=x(u,v), y=y(u,v)$, then

where the $J$ is the Jacobian:

or simply:

- Sometimes, it is easy to get the partial derivatives of $u$ and $v$ with respect to $x$ and $y$

Steps:

- Change variables to $u$ and $v$, transformation maps $R$ to $R^\prime$;
- Compute the Jacobian, may using the easy way to get the partial derivatives of $u$ and $v$ with respect to $x$ and $y$;
- Rewrite and submit the $u$ and $v$,
**DO NOT forget to multiply the Jacobian**.

**[Example]**

### 1.1.4 Polar Coordinates

Rewrite the Cartesian coordinates $(x,y)$ as polar coordinates $(r,\theta)$, where

- $r$ is the distance from the origin to the point $(x,y)$
- $\theta$ is the angle between the positive $x$-axis and the line segment from the origin to the point $(x,y)$

The J is:

**[Example]**

**[Example]**

## 1.2 Triple Integrals

### 1.2.1 Fubini’s Theorem - Box Case

$D={(x,y,z)\in R^3|a\le x\le b, c\le y\le d, e\le z\le f}$, then

**[Example]**

### 1.2.2 Fubini’s Theorem - Shadow Case

$D={(x,y,z) | (x,y) \in R, g(x,y)\le z\le h(x,y)}$, then

- First integrate $f$ with respect to $z$, whose result is a function of $(x, y)$.
- Then integrate the result with respect to $(x, y)$ (
**a double integral**)

**[Example]**

### 1.2.3 Properties of triple integrals

**Volume of a region**:

**Linearity**:

**Domain additivity**:

If $R$ is the union of two non-overlapping regions $R_1$ and $R_2$,

**Domination**:

If $f(x,y,z)\ge 0$ on $R$, then

If $f(x,y,z)\ge g(x,y,z)$ on $R$, then

### 1.2.4 Change of variables

$x=x(u,v,w), y=y(u,v,w), z=z(u,v,w)$, then

Where J is the Jacobian of the transformation:

If $u=u(x,y,z), v=v(x,y,z), w=w(x,y,z)$, then

Steps:

- Choose a coordinates, rewrite the region $R\prime$;
- Compute the Jacobian;
- Rewrite the triple integral in terms of the new coordinates;

### 1.2.5 Cylindrical coordinates

For a point $P$ in the $xyz$ space,

- $r$ and $\theta$ be the polar coordinates for the projection of P on the xy plane,

$r$: distance from the $z$-axis;

$\theta$: angle between the $z$-axis and the line from the origin to $P$;

$z$: height of $P$ above the $xy$-plane;

The Jacobian is: $r$;

Steps:

- Find the formulation of $R$ in cylindrical coordinates ($r$, $\theta$, $z$)to get $R\prime$
- Rewrite:

- Calculate the above integral;

**[Example]**

### 1.2.6 Spherical coordinates

For a point $P$ in the $xyz$ space,

$\rho$: distance from the origin $O$ to $P$, $\rho \ge 0$;

$\phi$: the angle between $OP$ and the **positive** $z$ axis, $0\le \phi \le \pi$;

$\theta$: the $\theta$ in the cylindrical coordinates of $P$

The J is $\rho^2\sin \phi$;

Steps:

- Find the formulation of $R$ in cylindrical coordinates ($\rho$, $\phi$, $\theta$) to get $R\prime$
- Rewrite:

- Calculate the above integral;

**[Example]**

## 1.3 Applications of Double Integrals

Consider physical matter (particles) in a region $R$ with a density function $\rho(x, y)$

**Mass**:

First moment about the **x-axis**:

*pay attention to the order of x and y*

First moment about the **y-axis**:

*pay attention to the order of x and y*

**Center of mass/gravity: $( \bar{x}, \bar{y})$**

*pay attention to the order of x and y*

**Centroid**:

- Submit $\rho(x, y) = 1$, then
- apply the above formula to get the centroid of the region $R$;
- its center of mass is also called its centroid.

Moment of inertia (**second moment**) about a line L:

- where $d(x,y)$ is the distance from the point $(x,y)$ to the line L.
**moment of inertia about the x-axis**:*pay attention to the order of x and y*

**moment of inertia about the y-axis**:*pay attention to the order of x and y*

Step for finding the **Center of mass**:

- Get the mass $M$;
- Get the first moment $M_x$ and $M_y$;
- Get the center of mass $(\bar{x}=\frac{M_y}{M}, \bar{y}=\frac{M_x}{M})$;
*pay attention to the order of x and y*

**[Example]**

Step for finding the **Centroid**:

- Set the density function $\rho(x, y) = 1$;
- Repeat the above steps;

**[Example]**

## 1.4 Applications of Triple Integrals

Consider physical matter (particles) in a region $R$ with a density function $\rho(x, y, z)$

**Mass**:

First moment about the **yz-plane**:

*pay attention to the order of x, y, z*

First moment about the **zx-plane**:

*pay attention to the order of x, y, z*

First moment about the **xy-plane**:

*pay attention to the order of x, y, z*

**Center of mass/gravity: $( \bar{x}, \bar{y}, \bar{z})$**

*pay attention to the order of x, y, z*

**Centroid**:

- Submit $\rho(x, y, z) = 1$, then
- apply the above formula to get the centroid of the region $R$;
- its center of mass is also called its centroid.

**Moment of inertia (second moment)** about a line L:

- where $d(x,y,z)$ is the distance from the point $(x,y,z)$ to the line L.

**moment of inertia** about the **x-axis**:

**moment of inertia** about the **y-axis**:

**moment of inertia** about the **z-axis**:

**[Example]**

**[Example]**

# 2 Vector Calculus

## 2.1 Scalar Fields

The value of $F$ is a scalar, a number;

The function $T$ is a scalar function;

### 2.1.1 Gradient of Scalar Field

for a scalar field $f(x,y,z)$, the gradient is a vector field $\triangledown f$:

- The gradient of a scalar field is a
, which has a direction and a magnitude;*vector field* - It points in the direction where $f$
**has the greatest rate of increase**, and its magnitude is this rate.

Where $\triangledown$ is the ** del operator**, not a function:

For a scalar field $f(x,y)$, the gradient is a vector field $\triangledown f$:

## 2.2 Vector Fields

- The vector field $\bold{F}$ is a
of three variables, which has three directions in space, the function value is a vector;*function* - $F_1$, $F_2$, $F_3$ are
, function value is a number;*scalar fields* - $\hat{i}$, $\hat{j}$, $\hat{k}$ are
;*unit vectors*

The gradient of a scalar field is a ** vector field**.

### 2.2.1 Divergence of Vector Field

For a vector field $\bold{F}(x,y,z)=F_1(x,y,z)\hat{i}+F_2(x,y,z)\hat{j}+F_3(x,y,z)\hat{k}$, the divergence is a scalar field $\triangledown \cdot \bold{F}$ (dot product):

- The divergence of a vector field is a
, which is just a*scalar field***number**;

**[Example]**

### 2.2.2 Curl of Vector Field

For a vector field$\bold{F}(x,y,z)=F_1(x,y,z)\hat{i}+F_2(x,y,z)\hat{j}+F_3(x,y,z)\hat{k}$, the curl is a vector field $\triangledown \times \bold{F}$ (cross product):

- The curl of a vector field is a
, which is a*vector field***vector**;

**[Example]**

Summary:

### 2.2.3 Properties of Divergence and Curl

**Linearity**:

**Curl of a gradient is zero** (gradient fields are curl free):

**Div of curl is zero** (curl fields are divergence free):

## 2.3 Laplacian of Scalar Field

For a scalar field $f(x,y,z)$, the Laplacian of $f$ is $\Delta f$:

- The Laplacian of a scalar field is a
, which is just a*scalar field***number**;

Where the $\Delta$ is the ** Laplacian operator**:

**[Example]**

## 2.4 Conservative Vector Field

For a vector field $\bold{F}(x,y,z)=F_1(x,y,z)\hat{i}+F_2(x,y,z)\hat{j}+F_3(x,y,z)\hat{k}$, it is ** conservative** if itself is

**exactly**the

**gradient**of a scalar field $f$:

- Where the $\bold{F}$ is called a
**conservative vector field**; - $f$ or $f+C$ is the
**potential function**of $\bold{F}$.

- If an only if $\bold{F}$ is conservative.

**[Example]**

### 2.4.1 Find Potential Function of Conservative Vector Field

The $\bold{F}$ must be conservative first;

### 2.4.2 2D Conservative Vector Fields

Let $\bold{F}=F_1(x,y)\hat{i}+F_2(x,y)\hat{j}$ in R2 then it is conservative if and only if:

**[Example]**

## 2.5 Curves

### 2.5.1 Curves in R2

for $t\in [a,b]$:

it represents a curve in xy plane, where $\bold{r(t)}$ is a parametriation of the curve.

**[Example]**

### 2.5.2 Closed Curves and Simple Curves

**Closed** Curves:

- for curve $\bold{r(t)}$ in R2, if $\bold{r(a)}=\bold{r(b)}$ then it is a closed curve.

**Simple** Curves:

- for curve $\bold{r(t)}$ in R2, if $\bold{r(t_1)}\neq \bold{r(t_2)}$ for $a\le t_1\lt t_2 \le b$ unless $t_1=a$ and $t_2=b$ then it is a simple curve.

### 2.5.3 Orientation of a Parametrized Curve

- we can specify the
**orientation of the curve**by specifying the**direction in which $t$ changes**(from $a$ to $b$ or from $b$ to $a$).

### 2.5.4 Curves in R3

for $t\in [a,b]$:

it represents a curve in xyz space, where $\bold{r(t)}$ is a parametriation of the curve.

## 2.6 Line Integral of a Scalar Field

### 2.6.1 Line Integral of the First Kind

For $C$, a curve in R3, $\bold{r(t)}=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$, $t\in [a,b]$

For function: $f(x,y,z)$

Then the **Line Integral of the First Kind of $f$ on $C$** is:

where:

- $s$ is the
**arclength**of $C$; - $\bold{r}^{\prime}(t)=x^{\prime}(t)\hat{i}+y^{\prime}(t)\hat{j}+z^{\prime}(t)\hat{k}$;
- $||\bold{r}^{\prime}(t)||=\sqrt{x^{\prime}(t)^2+y^{\prime}(t)^2+z^{\prime}(t)^2}$

**Steps**:

- Get the parametrization of $C$;
- Get $||\bold{r}^{\prime}(t)||$;
- Rewrite the integral.

**[Example]**

### 2.6.2 Arc Length

For a $C$ with parametrization $\bold{r}(t)=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$, $t\in [a,b]$

The **arc length** of $C$ is:

In 3D:

In 2D:

**[Example]**

### 2.6.3 Line Integral of the Second Kind

For $C$, a curve in R3, $\bold{r(t)}=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$, $t\in [a,b]$

For function: $f(x,y,z)$

Then the **Line Integrals of the Second Kind of $f$ on $C$** are:

Note:

*Line integrals of the second kind*are**independent of the parametrization of C**, as long as the**orientation keeps unchanged**.- Let $−C$ denote the
**same curve**as $C$ but the**orientation is reversed**- But in the first kind, the orientation will not be affected.

**Steps**:

- Get the parametrization of $C$;
- Get the $x^{\prime}(t)$, $y^{\prime}(t)$, $z^{\prime}(t)$;
- Rewrite the integral.

**[Example]**

**[Example]**

### 2.6.4 Properties of Line Integrals

For both first kind and second kind:

- Linearity:

- Domain Additivity:

where $C=C_1\cup C_2$, and are **disjoint**.

## 2.7 Line Integral of a Vector Field

For $\bold{F}=P\hat{i}+Q\hat{j}+R\hat{k}$, a vector field in R3,

For $C$, a curve in R3, $\bold{r(t)}=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$, $t\in [a,b]$

Then the **Line Integral of a Vector Field $\bold{F}$ on $C$** is:

- It is independent of the parametrization of C , as long as the orientation keeps unchanged.
- If $C$ is closed, then we write $\oint_C\bold{F}d\bold{r}$

- it is just like the
**second kind**of line integral of a**scalar field**.

**[Example]**

### 2.7.1 Line Integral of a Conservative Vector Field

For $\bold{F}$, a **Conservative** Vector Field in R3 with potential function $f(x,y,z)$

For $C$, a curve in R3, $\bold{r(t)}=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$, $t\in [a,b]$

Two end points: $\bold{r}(a)=P_0$ and $\bold{r}(b)=P_1$

Then the **Line Integral of a Conservative Vector Field $\bold{F}$ on $C$** is:

Path independence

- For a conservative field $F$ ,$\int_C \bold{F}\cdot dr$
**depends only on**its**initial point**$P_0$ and its**terminal point**$P_1$; - integral is independent of the path between $P_0$ and $P_1$;
- If $C$ is closed, then $\int_C \bold{F}\cdot dr=\oint_C\bold{F}\cdot d\bold{r}=0$

### 2.7.2 Path Independence and Conservativity

Let $F$ be a **vector field** in a **simply connected** region $D$.

- No hole inside $D$.

The following are **equivalent**.

- $F$ is conservative: $\bold{F}=\triangledown f$ for some function $f$.
- $F$ is curl-free: $\triangledown \times \bold{F}=\bold{0}$.
- $\int_C \bold{F}\cdot dr$ is independent of the path $C$;
- For closed curve $C$, $\oint_C \bold{F}\cdot dr=0$.

### 2.7.3 Line Integral of a Conservative Vector Field

For a conservative field $\bold{F}$, to calculate $\int_C \bold{F}\cdot dr$:

- As usual: $\int_C \bold{F}\cdot dr= \int_a^b \bold{F(\bold{r(t)})}\cdot \bold{r^{\prime}(t)} dt$;
- Find the potential function $f$, $\int_C \bold{F}\cdot dr=f(r(b))-f(r(a))$
- Replace the $C$ with any other path $C^\prime$, starting form $r(a)$ and ending at $r(b)$, then $\int
*C \bold{F}\cdot dr=\int*{C^\prime} \bold{F}\cdot dr$, which is easy to calculate.

**[Example]**

**[Example]**

### 2.7.4 Green’s Theorem

For curve $C$ as simple **closed** and oriented **counterclockwise**.

- Pay attention to the
**order**of the partial derivatives.

If $\bold{F}$ is a conservative vector field, then:

thus:

Note for Green’s Theorem:

- the curve must be
**closed**; - the curve must be oriented
**counterclockwise**, otherwise the sign of the integral will be**negative**. **counterclockwise**is called**positive orientation**,**clockwise**is called**negative orientation**.

**[Example]**

**[Example]**

**[Example]**

Let $C$ be a **simple closed curve**, and $D$ is the region bounded by $C$.

**[Example]**

## 2.8 Surface Integral

### 2.8.1 Parametrized surfaces

$\bold{r(u,v)}=x(u,v)\hat{i}+y(u,v)\hat{j}+z(u,v)\hat{k}$ represents a **parametrized surface**.

**[Example]**

**[Example]**

**[Example]**

**[Example]**

**[Example]**

**[Example]**

### 2.8.2 Surface Integral of a Scalar Function

For a surface $S$: $\bold{r}(u,v)=x(u,v)\hat{i}+y(u,v)\hat{j}+z(u,v)\hat{k}$, the **surface integral** of a **scalar function** $f(x,y,z)$ is:

where

$\bold{r}_u=\frac{\partial \bold{r}}{\partial u}$ = $\frac{\partial x}{\partial u}\hat{i}+\frac{\partial y}{\partial u}\hat{j}+\frac{\partial z}{\partial u}\hat{k}$, $\bold{r}_v=\frac{\partial \bold{r}}{\partial v}$ = $\frac{\partial x}{\partial v}\hat{i}+\frac{\partial y}{\partial v}\hat{j}+\frac{\partial z}{\partial v}\hat{k}$.

$\bold{r}_u\times \bold{r}_v$ is the ** cross product** of $\bold{r}_u$ and $\bold{r}_v$.

surface integral of the **first kind**.

**Cross product**:

$\bold{a}=(a_1,b_1,c_1),\bold{b}= (a_2,b_2,c_2)$, then:

$\iint_Sf(x,y,z)dS=\iint_Df(r(u,v))\left|\frac{\partial \bold{r}}{\partial u}\times \frac{\partial \bold{r}}{\partial v}\right|dudv$

Steps:

- Get the range of the $uv$;
- Replace $u$ and $v$ with $x,y,z$;
- $dS=\left|\frac{\partial \bold{r}}{\partial u}\times \frac{\partial \bold{r}}{\partial v}\right|dudv$;

3.1. Calculate the $\frac{\partial \bold{r}}{\partial u}\times \frac{\partial \bold{r}}{\partial v}$ and get the length, or

3.2 $\left|\frac{\partial \bold{r}}{\partial u}\times \frac{\partial \bold{r}}{\partial v}\right|dudv=\sqrt{||\bold{r_u}||^2\cdot ||\bold{r_v}||^2-(\bold{r_u}\cdot \bold{r_v})^2}$

**[Example]**

For $f(x,y,z)$ and $z=z(x,y)$

Thus,

General steps:

- Get the $\bold{r(u,v)}$;
- Get the $\bold{r}_u$ and $\bold{r}_v$, $E=||\bold{r}_u||^2, F=||\bold{r}_v||^2, F=\bold{r}_u \cdot \bold{r}_v$, then $||\bold{r}_u\times \bold{r}_v||=\sqrt{E\cdot F-E^2}$;
- $\iint_Sf(x,y,z)dS=\iint_Df(r(u,v))\sqrt{E\cdot F-E^2} \; dudv$

**[Example]**

### 2.8.3 Surface Area

For $S$: $\bold{r}(u,v)$, area of $S$ is:

**[Example]**

### 2.8.4 Surface Integral of a Vector Field

#### 2.8.4.1 Normal Vctors of a Parametrized Surface

- A vector is
**upward**(respectively, downward) if its**third component (i.e., the coefficient of k) is positive**(respectively, negative).

For a S: $\bold{r}(u,v)=x(u,v)\hat{i}+y(u,v)\hat{j}+z(u,v)\hat{k}$, the **normal vector** of $S$ is:

The **unit normal vector** is:

#### 2.8.4.2 Normal Vector of Surface S : F(x,y,z)=constant

- Unit sphere: $x^2+y^2+z^2=1$
- Plane: $Ax+By+Cz=D$;
- Graph of a function $z=z (x,y)$: $z-z(x,y)=0$

for each point $(x, y, z)$ on $S$, the **normal vector** is: $\triangledown F(x,y,z)$

the **unit normal vector** is:

or

**[Example]**

**[Example]**

#### 2.8.4.3 Two Methods to Compute Surface Integral of Vector Field/Flux

**Method1**:

- By using parametrization and calculating the double

integral with respect to two parameters.

Let the surface $S$ be parametrized by $\bold{r}(u,v)$, on $D$ then

Step:

- get the $\bold{r}(u,v)$; and its range;
- $dS=\left(\frac{\partial \bold{r}}{\partial u}\times \frac{\partial \bold{r}}{\partial v}\right)dudv$;

- $\frac{\partial \bold{r}}{\partial u}\times \frac{\partial \bold{r}}{\partial v}$ is in the
**same direction**as $\bold{n}$; - $-$ will be added if in the opposite direction of $\bold{n}$;

**[Example]**

**[Example]**

**Method2**:

Using:

- Then
**reformulate**to the

integral of the scalar function $F·n$ over $S$ - Hope $\iint_S(\bold{F}\cdot \bold{n})dS$ is
**easy to evaluate**. - This is a particular method that works
**only for integrals with special structures**, e.g., integrals over**spheres**or**planes**.

**[Example]**

**[Example]**

**[Example]**

## 2.9 Divergence Theorem of Gauss

For a closed surface $S$, boundary of a region $D$ in $R^3$, $S$ is positively oriented w.r.t $D$ (outward)

**[Example]**

**[Example]**

**[Example]**

## 2.10 Stokes’s Theorem

For curve $C$ is the boundary of a surface $S$ in $R^3$, $C$ is positively oriented w.r.t $S$.

If $\bold{F}$ is conservative, $\triangledown\times \bold{F}=\bold{0}$, then $\oint_C\bold{F}\cdot d\bold{r}=0$, path independence.

**[Example]**

**[Example]**

## 2.11 Three IMPORTANT theorems

# 3 Fourier Series

## 3.1 Fourier Series with Period $2\pi$

Let $f$ be a periodic function with period $2\pi$. The Fourier series of $f$ is defined as:

Where

When the $f$ is ** even**, only get $cos$ term:

When the $f$ is ** odd**, only get $sin$ term:

**[Example]**

**[Example]**

## 3.2 Fourier Series with Period $2T$

for a periodic function $f$ with period $2T$, the Fourier series is defined as:

Where

When the $f$ is ** even**, only get $cos$ term:

When the $f$ is ** odd**, only get $sin$ term:

**[Example]**

## 3.3 Half-Range Fourier Series

$f$ is defined on $[0,\pi]$, the Fourier series is defined as:

- Extend $f$ to $2T$
**even**function with**cosine**series:

$f(-x)=f(x), f(x+2T)=f(x)$

the Fourier series of $f$ is defined as:

where

**[Example]**

- Extend $f$ to $2T$
**odd**function with**sine**series:

$f(-x)=-f(x), f(x+2T)=f(x)$

the Fourier series of $f$ is defined as:

where

**[Example]**

# 4 PDE

## 4.1 An IBVP of the Heat Equation

**Solving the IBVP by Separation of Variables**

**[Example]**

## 4.2 A Second IBVP of Heat Equation

**Solving the IBVP by Separation of Variables**

**[Example]**

## 4.3 A PDE of Wave Equation

**[Example]**

# References

Slides of AMA2111 Mathematics II, The Hong Kong Polytechnic University.

*个人笔记，仅供参考，转载请标明出处**PERSONAL COURSE NOTE, FOR REFERENCE ONLY*

*Made by Mike_Zhang*