###### tags: `ClassNote` `必修`
# 微積分搶救中

# V

## Ch.7(14-7)

### Saddle point(7-a)

#### 7.1

if $f(a,b)$ is a local estreme value of $f$, then$f_x(a, b) = 0$ and $f_y(a,b) = 0$
$(a,b)$ is called critical point if:

1. $f_x(a, b)$ does not exist
2. $f_y(a, b)$ does not exist
3. $f_x(a, b) = 0$ and $f_y(a,b) = 0$

#### 7.2

$f(a,b)$ is a local estreme value when $(a, b)$ is either critical point or boundry point

#### 7.3

saddle point is a **critial point** but **not** a local extreme value

### Second derivative test(7-b)

if $f(c)' = 0$ and $f(c)'' > 0$ then $f(c)$ is a local minimum

Let $D = f_{xx}(x,y)f_{yy}(x,y) - {f_{xy}(x,y)}^2$

1. if $D > 0$ and $f_{xx}(x,y) > 0$ then $f(x,y)$ is a local minimum of f
2. if $D > 0$ and $f_{xx}(x,y) < 0$ then $f(x,y)$ is a local maximum of f
3. if $D < 0$ then $f(x,y)$ is saddle point of $f$
4. if $D = 0$, no conclusion
:::info
$D$ is called discriminant or Hessian of $f$
:::
**實作方法**

先一階偏微解聯立求critical points(加上原本的boundry points)
$\rightarrow$再用 $D$ 求極值

### Continuous functions on closed bounded regions(7-c)

#### 7.4

in a close boundry.Suppose R is countainous then:

1. $f$ has both max and min (absolute)
2. absolute extreme value is either at critical point or boundry point

## Ch.8(14-8)

### The method of Lagrange multipliers

#### 8.1

Let $f(x,y)$ and $g(x,y)$ be differentiable functions.

Assume $\nabla g(x,y) \neq \overrightarrow {0}$ whenever $g(x,y) = 0$ If $f$ has local extreme value at (a,b) with $g(a,b) = 0$ Then $\nabla f(a,b)$ and $\nabla g(a,b)$ are parallel. Which exist a Lagrange multiplier $\lambda$ that
$$\Bigg\{ \begin{split} f_x(a,b) = \lambda g_x(a,b) \\
f_y(a,b) = \lambda g_y(a,b)\end{split}$$

**Lagrange mutipliers實作**

1. 找出 $g(x,y)$
2. 代入求出$\lambda$
3. 代入$\lambda$求出 $x,y$ 的關係式，並帶入$g(x,y)$
4. 用求出的 $(x,y)$ 代入原先的 $f(x,y)$ 求值

==Lagrange mutipliers可以套用到三變數函數內==

# VI

## Ch.1(15.1)

### Double integeral(1-a)

==region是三次，area是二次==
For a close region R:
$R$的體積function=$\iint_R \ f(x,y) dA$

### Iterated integrals over rectangles(1-b)

就四把domain的boundry帶入積分(對應的地方要注意:內對內，外對外)

#### 1.2

就四 $dxdy$ 的order可以互換

## Ch.2(15.2)

### Using vertical cross-sections(2-a)

#### 2.1(Fubini's Theorem)

$\iint_R f(x,y) dA = \int_a^b \int_{g_1(x,y)}^{g_2(x,y)} f(x,y) \ dy\ dx$

### Using horizontal cross-sections(2-b)

#### 2.2(Fubini's Theorem)

$\iint_R f(x,y) dA = \int_c^d \int_{h_1(x,y)}^{h_2(x,y)} f(x,y)\ dx\ dy$

## Ch.3(15.3)

### Areas by double integrals(3-a)

If $f(x,y) = 1$ in a close boundry $R$, then volime under the solid (which is area$(R)$) = $\iint_R dA$

### The average value(3-b)

The average value of $f$ over $R$ is $\frac{1}{area(R)} \iint f(x,y)\ dA$

## Ch.4(15.4)

### Double integrals in polar coordinates(4-a)

$\iint_R f(r,\theta)\ dA = \iint_R f(r,\theta)\ r\ dr\ d\theta$

## Ch.5(15.5)

### Triple integrals(5-a)

volume$(D)$ = $\iiint_D dV$
If $f(x,y,z)$ is a density function of $D$ then mass$(D)$ = $\iiint_D f(x,y,z)\ dV$

### Iterated triple integrals(5-b)

$\iiint_D f(x,y,z) dV = \int_{a}^{b}\int_{g_1(x)}^{g_2(x)}\int_{h_1(x,y)}^{h_2(x,y)}\ dz\ dy\ dx$

### The average value(5-c)

The average value of D is $\frac{1}{volume(d)}\iiint_D f(x,y,z)\ dV$

## Ch.6(15.7)

### Triple integrals in cylindrical coordinates(6-a)

==$dV = r\ dz\ dr\ d\theta$==

### Triple integrals in spherical coordinates(6-b)

$\rho = \sqrt{x^2+y^2+z^2}$
$\phi$ = The angle from positive z-axis to $\overrightarrow {OP}$
$\theta$ = 同polar coordinate的 $\theta$

1. If $P = (r,\theta,z)$ and $P = (\rho,\phi,\theta)$ then:
$r = \rho \sin\phi$,
$z = \rho \cos\phi$
$\rho = \sqrt{r^2 + z^2}$
2. If $P = (x,y,z)$ and $P = (\rho,\phi,\theta)$ then
$x = \rho \sin\phi\cos\theta$
$y = \rho\sin\phi\sin\theta$
$z = \rho\cos\phi$

==$dV = r\ dz\ dr\ d\theta = \rho^2\sin\phi \rho\ d\phi \ d\theta$==

## Ch.7(15.8)

### Substitutions in double integrals(7-a)

$\iint_R f(x,y)dA = \iint_S f(x(u,v),y(u,v))\Bigg|\begin{split} x_u\quad x_v\\
y_u \quad y_v \end{split}\Bigg|\ du\ dv$
(Jacobian)$J(u,v) = \frac{\partial(x,y)}{\partial(u,v)} = \Bigg|\begin{split} x_u\quad x_v\\y_u \quad y_v \end{split}\Bigg|$

### Substitutions in triple integrals(7-c)

$\iiint_D f(x,y,z) dV = \iiint_S f(x(u,v,w),y(u,v,w),z(u,v,w))\ J(u,v,w)\ du\ dv\ dw$
> 其實就是三次的版本而已 nothing special

# VII

## Ch.1(16-1)

### Line integrals

The line integral over curve C is
$\int_C f(x,y,z)\ ds$
if C is parametrized by $\overrightarrow{r}(t) = <g(t),h(t),k(t)>$
$\int_C f(x,y,z)\ ds$
= $\int_{a}^{b} f(g(t),h(t),k(t))s'(t)\ dt$
= $\int_{a}^{b} f(g(t),h(t),k(x))\sqrt{(g(t)')^2+(h(t)')^2+(k(t)')^2)}\ dt$

## Ch.2(16-2)

### Vector fields(2-a)

If $z = f(x,y)$ is a differentiable function in N varibles, then gradient of $f$ is a N-dimensional vector field, called gradient field:$$\nabla f(x,y) = <f_x(x,y),f_y(x,y)>$$
>This is two-dimentional gradient field

### Line integrals of vector fields(2-b)

:::info

專有名詞

$T = \dfrac{dr}{ds}$ is the unit vector of a soomth curve
$\kappa = |\dfrac{dT}{ds}|$ is the curvature
$N = \dfrac1{\kappa}\dfrac{dT}{ds}$ is the principal unit normal vector
:::

The line integral of $\overrightarrow{F}$ over $C$ is
$\int_{C} \overrightarrow{F} \cdot \overrightarrow{T} ds$
=$\int_{a}^{b} \overrightarrow{F} \cdot \overrightarrow{r}(t)' dt$
=$\int_{a}^{b} \overrightarrow{F} \cdot d\overrightarrow{r}(t)$

# 一些方便的註解

如果題目給很複雜的 $y$ 請先reverse order成 $x$ 在做
求完極值點記得算實際值
Jacobian的$u,v$自己設
拿$z^2$和$x^2+y^2$可看出$\phi$
要注意到底是甚麼換$dt$(在line integeral)
$\int \ln{x}\ dx = x\ln{x}-x +C$
$\int e^{ax}\ dx = \dfrac{1}{a} e^{ax} + C$
$\dfrac{de^{ax}}{dx} = a e^{ax}$
$\dfrac{da^y}{dy} = a^y \ln a$