Chapter 10 Partial Derivatives

# Functions of Several Variables

$f: R^2 \to R$ or $f: R^3 \to R$
(1) Domain (2) Range (3) Graph
(4) Level curves (等高線) or Surfaces.

# Limits and Continuity

Given $\epsilon > 0,\exists \delta > 0$ such that $|f(x,y) - L| < \epsilon$ whenever $0 < \sqrt{(x-a)^2 + (y-b)^2}< \delta$ and $(a,b) \in D$

# Partial Derivatives

# Define

1. $f_x(a,b) = \lim_{h \to 0} \frac{f(a+h,b)-f(a,b)}{h}$. (將$y=b$ 固定，把$f$ 看成對$a$ 微分)
2. $f_y(a,b) = \lim_{h \to 0} \frac{f(a,b+h)-f(a,b)}{h}$. (將$x=a$ 固定，把$f$ 看成對$b$ 微分)

簡單來說，偏微就是 固定其他變量 ，對唯一變量作微分。

# Clairaut's theorem

1. If $f_{xy}$ and $f_{yx}$ are continuous on a region $D$, then $f_{xy}(a,b) = f_{yx}(a,b)$ for and $(a,b) \in D$.
2. $f_{xyy} = ((f_x)_y)_y, f_{yxy}, f_{yyx}$ are continuous on a region $D$,then $f_{xyy}(a,b) = f_{yxy}(a,b) = f_{yyx}(a,b)$ for and $(a,b) \in D$.

# Tangent Planes and Linear Approximation

$A(x-x_0) + B(y-y_0) + z-z_0 = 0$
Let $x = x_0$,then $B(y-y_0) + z-z_0 = 0$ 為過 P 點和曲線$z=f(x_0,y)$ 相切的直線方程式$T_1$
$\Rightarrow f_y(x_0,y_0) = T_1$ 的斜率為$-B$.
Similarly, $f_x(x_0,y_0) = T_2$ 的斜率為$-A$.

• 切面法向量
  $\left \langle A,B,1 \right \rangle = \left \langle -f_x(x_0,y_0),-f_y(x_0,y_0),1 \right \rangle$
• 切面方程式
  $z - z_0 = fx(x_0,y_0)(x-x_0)+fy(x_0,y_0)(y-y_0)$

# The Chain Rule

# Define

$z(t) = f(x(t),y(t)) \Rightarrow \frac{dz}{dt} = f_x\frac{dx}{dt} + f_y \frac{dy}{dt}$.
$u(t) = f(x(t),y(t),z(t)) \Rightarrow \frac{du}{dt} = f_x\frac{dx}{dt} + f_y \frac{dy}{dt} + f_z \frac{dz}{dt}$.

# Implicit Differentiation

If $f(x,y,z) = 0$ and $z = g(x,y)$
$\Rightarrow f(x,y,g(x,y)) = 0 \Rightarrow f_x + f_y\frac{dy}{dx} + f_z\frac{\partial z}{\partial x}=0$
Since $\frac{dy}{dx} = 0 \Rightarrow \frac{\partial z}{\partial x} = -\frac{f_x}{f_z}$.
Similarly,$\frac{\partial z}{\partial y} = -\frac{f_y}{f_z}$.

# Directional Derivatives and Gradient Vector

# Define

過點$(x_0,y_0)$，沿著$u$ 方向的微分，$u = \left \langle a ,b \right \rangle ,a^2+b^2 = 1$ 方向的微分

$D_uf(x_0,y_0) = \lim_{h \to 0} \frac{f(\overrightarrow{OP}+hu)-f(\overrightarrow{OP})}{h} \\= \lim_{h \to 0}\frac{f(x_0+ha,y_0+hb) - f(x_0,y_0) - f(x_0,y_0)}{h} = g'(0)$,where $g(h) = f(x_0+ha,y_0+hb)$

# Theorem

$D_uf(x_0,y_0) = \langle f_x(x_0,y_0),f_y(x_0,y_0) \rangle \cdot \langle a,b \rangle = af_x(x_0,y_0) + bf_y(x_0,y_0)$

# Define

$\langle f_x(x_0,y_0),f_y(x_0,y_0) \rangle =: \nabla f$
$D_u f(x_0,y_0) = \nabla f \cdot u$

# Maximum and Minimum Values

$f$ is continuous on a closed and bounded set $D$.

1. There exist absolute minimum and absolute maximum.
2. Those extreme points occur at the critical points of $f$ or the boundary of $D$.

# Lagrange Multipliers

• 想法:

1. 極值產生的地方:
   f 的 level curve or level surface 和 $g$ 的 level curve or level surface 相切。
2. Level curve of f \prep \nabla f

• 結論: Extrma occur when
  1. $\nabla f = \lambda \nabla g$ for some $\lambda \in \mathbb{R}$.
  2. $\nabla f = \lambda \nabla g + \mu \nabla h$ for some $\lambda,\mu \in \mathbb{R}$.

# Reference

• 莊重 - 微積分 (二) Calculus II - 103 學年度