Mistakes

• 极坐标换元后$\sqrt{ x^{2}+y^{2} }=r$ 而不是 $r^2$
• 看清 $z^2=x^2+y^2$，所以$z^{2}$的值是平方和再开根，不要忘了
• 势能方程需要加常数
• 不要漏掉系数和复合函数求导时乘上内层导数
• 注意$\oint_{C}Pdx+Qdy$里分母为$0$的情况，比如$\frac{\dots}{x^2+4y^2}$在原点不合法，此时无法用格林公式，有可能可以通过化简产生一个合法的公式，见Redo Tasks第二题
• 计算器里calc输变量赋值时，注意变量出现的顺序与其在表达式中出现的顺序一致，而非字母顺序，所以要看清计算器上的标注
• 对于幂级数的收敛半径，如果极限$L$发散，那么收敛半径就是$\infty$（21S1Q10a）。计算出 $|ax+b|<k$ 后，收敛半径即为$\frac{k}{a}$，对于$(ax+b)^{2}<k$，收敛半径为$\frac{\sqrt{ k }}{a}$
• 看清积分区域，是上半部分还是下半部分
• 两曲面交线的曲线积分：参数化然后注意$ds=||\mathbf{r}'||dt=\sqrt{ x'^{2}+y'^{2}+z'^{2} }$，是参数方程求导后求模，见Redo Tasks
• 注意$\lim_{ n \to \infty }(1+\frac{a}{n})^{n}=e^{a}$，括号内是加号，所以如果是减的话结果是$e^{-1}$ (21S1 Q9a)
• $dx\to d(kx)$，积分外除以 $k$
• $x^{k-1}dx\to dx^{k}$，积分外也除以$k$，记忆方法，和上面那一条都是除以$d$后边的新常数
• 方向导数点乘的方向向量必须为单位向量

Redo Tasks

• 两曲面交线的曲线积分 23S1 Q5
• 两曲面交线的曲线积分 21S1 Q5
• 格林公式的特殊运用 23S1 Q7
• 面积分的极坐标换元 18S1 Q2b
• 18S1 Q3a

W1 Partial Derivatives

• contour curve: horizontal plane $z=k$ intersects the surface $z=f(x,y)$ along the contour curve of height $k$
• level curve is the projection of contour curve onto the “ground” $xy-$plane
• the partial derivative of $f$ at $(a,b)$ with respect to (w.r.t.) $y$ : $f_y(a,b)\ \mathrm{or}\ \frac{\partial f}{\partial y}$
  • the use of special symbol $\partial$ instead of $d$ indicates that the derivative is partial and the function actually has more than one independent variable.
• high order partial derivatives. example: $\frac{\partial^2f}{\partial x\partial y}$
• The mixed derivative theorem: under certain condition we have $f_{xy}=f_{yx}$
• normal vector: $\mathbf n=\begin{pmatrix}f_x(u,v)\\f_y(u,v)\\-1\end{pmatrix}$
• normal line: $\mathbf r =\mathbf p+\mathbf n\cdot t\quad t\in\mathbb R$
• equation of the tangent plane: $\mathbf x\cdot\mathbf n=\mathbf n\cdot P$
• ※ equivalent Cartesian equation: $z=f_x(a,b)\cdot(x-a)+f_y(a,b)\cdot(y-b)+f(a,b)$
• chain rule with multiple independent variables: use the tree diagram

  example: for $z=f(x,y),x=g(s,t),y=h(s,t)$ > $\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}\cdot\frac{d x}{d t}+\frac{\partial z}{\partial y}\cdot\frac{d y}{d t}$

			z
			|*>x
			|  *-s
			|  *>t
			|
			|*>y
			   *-s
			   *>t

• directional derivatives: $D_{\mathbf u}f(a,b)=\nabla f(a,b)\cdot\mathbf u$ ($\mathbf u$ must be a unit vector)
  • we can first normalize the direction vector if it isn’t a unit vector
  • $\nabla f$ is called the gradient vector of $f$
• local maximum & minimum $\Rightarrow$ $\nabla f(a,b)=(0,0)$ $\Leftrightarrow$ critical point
• ※ Derivative Test for Nature of Critical Points
  • discriminant: $D=f_{xx}f_{yy}-f^2_{xy}$
  • $D>0$:
    • $f_{xx}<0$: maximum
    • $f_{xx}>0$: minimum
  • $D<0$: saddle point
  • $D=0$: inconclusive
• method of Lagrange Multipliers: solve the equations:$\begin{cases} \nabla f&=\lambda\nabla g\quad(\nabla f\parallel\nabla g)\\ g&=0 \end{cases}$while $f$ is the target function to be optimized and $g$ is the constraint function, the constant/factor $\lambda$ is called a Lagrange Multipliers

W2 Multiple Integrals

• Fubini’s theorem: exchange the order of integration: for continuous function $f$ we have $\int_a^b\int_c^d f(x,y)dydx=\int_c^d\int_a^b f(x,y)dxdy$
• $\iint_D1dA=\mathrm{area\ of\ }D$
• $\int_a^b\int_c^d\ p(x)q(y)\ dydx=(\int_a^bp(x)dx)(\int_c^dq(y)dy)$
• type 1 domain (bound by x-coord x型区域): two curve intersect with two vertical | | lines: $\int_a^b\int_{g(x)}^{h(x)}fdydx$
• type 2 domain (bound by y-coord y型区域): two curve intersect with two horizonal === lines: $\int_a^b\int_{g(y)}^{h(y)}fdxdy$
• polar coordinates: $(r,\theta)\to(x=r\cos\theta,y=r\sin\theta)$
• counter-clockwise: positive, clockwise: negative
• domain: bounded by two polar curves: $\iint_Df(x,y)dA=\int_\alpha^\beta\int_{g(\theta)}^{h(\theta)}f(r\cos\theta,r\sin\theta)\underbrace{rdrd\theta}_{dA}$
  • ※ reason for the $r$ factor is the Jacobian determinant of change of integrating variable (out of MA1511’s scope)
• (Tut 4) for circle not centered at origin, to use polar coordinate, we can first write the eq (e.g. $x^2-2x+y^2=0$) then we substitude the variables with corresponded $r,\theta$ expressions using $x=r\cos \theta,y=r\sin \theta$ then we have $r^2-2r\cos \theta=0$, which can be further simplify to $r=\cos \theta$

W3 Vector-Valued Functions

• vector function (parametric curve) of circles $$\mathbf{r}(t)=\begin{pmatrix}a\cos t\\a\sin t\end{pmatrix}\quad a>0\quad t\in[0,2\pi]$$
• vector function (parametric curve) of straight lines $$\mathbf{r}(t)=\begin{pmatrix}a+mt\\ b+nt\\ c+pt\end{pmatrix}=\begin{pmatrix}a\\ b\\ c\end{pmatrix}+t\begin{pmatrix}m\\ n\\ p\end{pmatrix}\quad t\in\mathbb{R}$$
• orientation of curves: the direction of the movement along the curve corresponding to $t^+$ gives “positive orientation”
• differentiation of a vector function is component-wise $$\mathbf{r}'(x)=\frac{d}{dt}\mathbf r (t)=\begin{pmatrix}f'(t)\\ h'(t)\\ g'(t)\end{pmatrix}$$
• magnitude of the v vector $|\mathbf r'(x)|$ measures the speed
• tangent lines to parametric curve: $\mathbf p= \mathbf r(T_0)+t\mathbf r'(T_0)$
• differentiation rules
  • $(\mathbf u\cdot\mathbf v)'=(\mathbf u'\cdot\mathbf v)+(\mathbf u\cdot\mathbf v')$
  • $(\mathbf u\times\mathbf v)'=(\mathbf u'\times\mathbf v)+(\mathbf u\times\mathbf v')$
• vector function (parametric curve) of segments: $\mathbf r(t)=(1-t)A+tB\quad t\in[0,1]$
• ※ line integral (the first type, scalar) : $\int_Cf(x,y)ds=\int_a^bf(\mathbf r(t)\dots)\underbrace{|\mathbf r'(t)|dt}_{ds}$ : geometric meaning: the surface area of the curtain
• ※ Arc Length: $L=\int_C1ds=\int_a^b|\mathbf r'(x)|dt=\int_a^b\sqrt{(f'(x))^2+(g'(x))^2+(h'(x))^2}dt$
• parametric surfaces: $\mathbf r(u,v)$
  • sphere: $x=a\sin\alpha\cos\beta,\ y=a\sin\alpha\sin\beta,\ z=a\cos\alpha\quad\alpha\in[0,\pi]\ \beta\in[0,2\pi]$

W4 Vector Fields

$ bf F tabtab space

• gradient vector of a scalar function $f$ : $\nabla f=\left( \frac{ \partial f }{ \partial x },\frac{ \partial f }{ \partial y },\dots \right)$

• line integral of the vector field $\mathbf{F}$ along $C$ (the second type, vector) :

  $$\int_{C}\mathbf{F}\cdot d\mathbf{r}=\int_{a}^{b}\mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}'(t)dt$$

• if there exists a scalar function $f$ s.t. $\mathbf{F}=\nabla f$ then $\mathbf{F}$ is said to be conservative

  • $f$ is called the potential function for $\mathbf{F}$
  • to find $f$, solve the pair of differential equation: $\frac{ \partial f }{ \partial x }=\mathbf{F}_{x}\quad \frac{ \partial f }{ \partial y }=\mathbf{F}_{y}$ the 3D case is the same
  • Fundamental theorem of line integrals (FTLI): for conservative vector field $\mathbf{F}$ and smooth curve $C$: $$\int_{C}\mathbf{F}\cdot d\mathbf{r}=f(\mathbf{r}(t_{1}))-f(\mathbf{r}(t_{0}))$$
  • aka the integral is path independent
  • for closed curve $C$ we write $\oint_{C}\mathbf{F}\cdot d\mathbf{r}$ (contour intergral) and if $\mathbf{F}$ is conservative the integral will be $0$
  • ※4.4C 2D $\mathbf{F}$ is conservative if and only if $$\frac{ \partial \mathbf{F}_{x} }{ \partial y }=\frac{ \partial \mathbf{F}_{y} }{ \partial x }$$
  • ※4.4C 3D $\mathbf{F}$ is conservative if and only if $$\frac{ \partial \mathbf{F}_{x} }{ \partial y }=\frac{ \partial \mathbf{F}_{y} }{ \partial x }, \frac{ \partial \mathbf{F}_{x} }{ \partial z }=\frac{ \partial \mathbf{F}_{z} }{ \partial x }, \frac{ \partial \mathbf{F}_{y} }{ \partial z }=\frac{ \partial \mathbf{F}_{z} }{ \partial y }$$
  • Path Independence & Conservative Fields: if component functions of $\mathbf{F}$ are continuous and $\mathbf{F}$ is defined on an open connected and simply connected region then: $$\mathbf{F} \mathrm{\ is\ conservative}\iff \int_{C}\mathbf{F}\cdot d\mathbf{r}\mathrm{\ \ is\ path\ independent}$$

• line integrals in component form

  $$\int_{C}\mathbf{F}\cdot d\mathbf{r}= \int_{a}^{b}\mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}'(t)dt= \underbrace{ \int_{a}^{b}\mathbf{F}_{x}(\mathbf{r}(t))\cdot (\mathbf{r}')_{x}(t)dt }_{ \int_{C}Pdx }+ \underbrace{ \int_{a}^{b}\mathbf{F}_{y}(\mathbf{r}(t))\cdot (\mathbf{r}')_{y}(t)dt }_{ \int_{C}Qdy }$$

• simple closed curve in 2D is the one that doesn’t cross itself (not like "$8$"-shaped)

• counter clockwise is the positive orientation

  • inner boundary: clockwise is the positive orientation instead (e.g. for donut region)

• Green’s theorem - $D$ is a region enclosed by a simple, closed and positively oriented curve $C$, then - add a negative sign if $C$ is negatively oriented - $\mathbf{F}$ should have continuous partial derivative on the entirely of the region $D$ - $P,Q$ should be well-defined in the region $D$ (not have undefined points - singularities) - look out for SINGULARITIES - use by-definition if there exists singularities

  $$\oint_{C}\mathbf{F}\cdot d\mathbf{r}=\oint_{C}Pdx+Qdy=\iint_{D}\left( \frac{ \partial Q }{ \partial x } -\frac{ \partial P }{ \partial y } \right)dA$$

• two important quantities associated with 3D vector fields

  • the vector differential operator or del operator $\nabla$ defined as

    $$\left( \frac{ \partial }{ \partial x } ,\ \frac{ \partial }{ \partial y } ,\ \frac{ \partial }{ \partial z } \right)$$
    • so the gradient field of scalar function $f$ can be written as $\nabla f$

  • The curl vector : the curl of $\mathbf{F}$ (denoted by "$\mathrm{curl}\ \mathbf{F}$") is:

    $$\left( \frac{ \partial \mathbf{F}_{z} }{ \partial y }-\frac{ \partial \mathbf{F}_{y} }{ \partial z } ,\ \frac{ \partial \mathbf{F}_{x} }{ \partial z } -\frac{ \partial \mathbf{F}_{z} }{ \partial x } ,\ \frac{ \partial \mathbf{F}_{y} }{ \partial x } -\frac{ \partial \mathbf{F}_{x} }{ \partial y } \right)$$

  • MNEMONIC: $\nabla \times \mathbf{F}$ (cross product or vector product)

  • using 4.4C we get $\mathrm{curl\ }\mathbf{F}=0$ if $\mathbf{F}$ is conservative

  • The divergence of $\mathbf{F}$ is a scalar (denoted by "$\mathrm{div}\ \mathbf{F}$") is:

    $$\frac{ \partial \mathbf{F}_{x} }{ \partial x } +\frac{ \partial \mathbf{F}_{y} }{ \partial y } +\frac{ \partial \mathbf{F}_{z} }{ \partial z }$$

  • MNEMONIC: $\nabla \cdot \mathbf{F}$ (dot product or scalar product)

  • Laplacian of scalar function $f:\mathbb{R}^3\to\mathbb{R}$ is $\nabla^2f=\nabla\cdot \nabla f=f_{x x}+f_{y y}+f_{z z}$

W5 Infinite Series

two common types of Sequence

• arithmetic sequence (等差数列)
  • $d$ : common difference
• geometric sequence (等比数列)
  • $r$ : common radio

- Limit of a Sequence

• $\lim_{ n \to \infty }a_{n}=0$ or $a_{n}\to0$
• divergent to infinity
• $\{a_{n}\}$ is convergent. It converges to $0$

- poly / poly : indet. form (inf/inf)

eval method: div the numer and denom by the highest power of variable

Some Standard Limit Results

application example:

Limit Laws

• $n^{\mathrm{th}}$ partial sum and its limit: $$\sum_{k=1}^na_{k}\quad \sum_{k=1}^{\infty} a_{k}$$
• the covergence or divergence is unaffected by removing/adding finite terms

Limit of geometric series

• geometric series $\sum ar^{k-1}$ converges iff its common ratio $r\in(-1,1)$ and if so the limit is $\frac{a}{1-r}$

Two Convergence / divergence Tests

• $n^{\mathrm{th}}$ Term Test / Divergence Test
  • if $\lim_{ n \to \infty }a_{n}\neq0$ then its sum series is divergent
  • but if the limit is indeed $0$ the latter isn’t necessarily convergence (counterexample: harmonic series)
  • only meant to test divergence
• $p$-series Test
  • The series $\zeta(p)=\sum \frac{1}{k^p}$ is convergent if $p>1$ and is divergent if $p\le 1$

Power Series

• (regard as functions) $$\sum_{k=0}^\infty c_{k}(x-a)^k$$
  • $c_{0},c_{1},\dots$ are the coeffients and $a$ is the center
  • covergence or divergence behaves exactly one of the following ways:
    • diverges for all $x\neq a$ ($R=0$)
    • converges for all $x$ ($R=\infty$)
    • converges if $x\in(a-R,a+R)$ and diverges otherwise, the maximum $R$ is the radius of convergence
  • Ratio / Root Test for Power Series
    • let $a_{k}=c_{k}(x-a)^k$
    • Ratio Test: $L=\lim_{ k \to \infty }|a_{k+1}/a_{k}|$
    • Root Test: $L=\lim_{ k \to \infty }|a_{k}^{1/k}|$
    • $L<1$ : converge / $L>1$ : diverge
    • $L=1$ or $L$ does not exist: inconclusive

Taylor Series & Maclaurin Series

• (special form of power series) (for a function $f$ ) $$f(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k$$
  • coefficients: $\frac{f^{(k)}(a)}{k!}$
  • Maclaurin Series: Taylor Series centered at $a=0$, hence: $$f(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}x^k$$
  • Taylor series converges to the function it represents. These functions are known as analytic functions
  • Taylor polynomials $P_{n}(x)=\sum^n\mathrm{Taylor}$ provides an approx imation to $f$ $$\begin{align*} \frac{1}{1-x}&=\sum_{k=0}^{\infty}x^k=1+x+x^2+x^{3}+\dots&\mathrm{for}x\in(-1,1) \\ e^{x} & =\sum_{k=0}^{\infty} \frac{1}{k!}x^{k}=1+x+ \frac{x^{2}}{2!}+ \frac{x^{3}}{3!} +\dots & \mathrm{for\ all} x \\ \sin x&=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)!}x^{2k+1}=x- \frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\dots & \mathrm{for\ all} x \\ \cos x&=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k)!}x^{2k}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!} +\dots & \mathrm{for\ all} x \\ \\ \ln(1+x) & =\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} x^{k}=x- \frac{x^{2}}{2}+\frac{x^{3}}{3}+\dots&\mathrm{for}x\in(-1,1) \\ \arctan x&=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2k+1}x^{2k+1}& \mathrm{for\ all} x \\ (1+x)^{p} & =\sum_{n=0}^{\infty}C^{p}_{n} x^n \end{align*}$$ $$\begin{align*} \sqrt{ 1+x } & =1+\frac{\frac{1}{2}}{1!}x+\frac{\frac{1}{2}\cdot-\frac{1}{2}}{2!}x^2+\dots \end{align*}$$

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