Topology (Part 1): Lagrange Multiplier

3-Sphere = 2 dimensional 
(站在locale 看周围 neighbourhood) 
eg. 站在地球一点上, 看脚下似平面 (2-D)

Differentiable 可微性 ( => 连续性 continuity ) = 平滑 (smooth) 线性变化 [Intuitively]

全微分 {f: R^{2} \to R^{1}}
represented by matrix (Linear Transformation):
{ \big[\frac {\partial f } {\partial x} , \frac {\partial f } {\partial y}\big] } .\begin{pmatrix} x\\ y \end{pmatrix} = { \big[\frac {\partial f } {\partial x}.x + \frac {\partial  f } {\partial y}.y\big]}

f: R^{n} \to R^{m} => Matrix (n,m)

微分 = 线性变化
\boxed { \text {Differentiation} = \text{ Linear Transformation} }

Find “mini – max ” <=> Kernel { \big[\frac {\partial f } {\partial x} , \frac {\partial f } {\partial y}\big] } .

梯度 (gradient): 地形图 等高线 f(x,y)

垂直方向 = { \big(\frac {\partial f } {\partial x}.i + \frac {\partial f } {\partial y}.j\big)}

路人行走的路线: { \big[\frac {\partial g} {\partial x}, \frac {\partial g } {\partial y}\big]}

路线 g(x,y) 的 (max.)极大值 if and only if

f 和 g 的梯度平行: \iff
{ \boxed { \bigg(\frac {\partial f } {\partial x}.i + \frac {\partial f } {\partial y}.j\bigg)= \lambda. \bigg(\frac {\partial g } {\partial x}.i + \frac {\partial g} {\partial y}.j\bigg) }}

Lagrange Multiplier : \lambda

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