Classic Analyssi

Limit

Definition: The Limit of a Function over a Base.
A topology space . A function on to . A neighborhood of point is a subset of containing an open set containing . A base is a family of sets s.t.

• .
• ${B{1}, B{2} \in \mathcal{B}}\implies{\exists\,B \in \mathcal{B}\,(B \in B{1} \cap B{2}))}$. $${ (\lim_{\mathcal{B}} f = a) := \forall\,V(a)\,(\exists\,B \in \mathcal{B}\,(f(B) \subset V(a))). }$$

---

Differential

Definition: Differential.
Let be an open region on and be a map on to and a point ${x{0} \in \Omega}{\exists\,A(\text{linear operator}): \mathbb{R}^{n} \to \mathbb{R}\,(f(x) - f(x{0}) = A(x - x{0}) + o(\lvert x - x{0} \rvert))}f{A}$ is its differential.

Definition: Directional derivative.
Let be a vector on . is a map on to . The directional derivative of w.r.t. on point ${x{0} \in \Omega}{\lim{ h \to 0 } \frac{f(x{0}+hv)-f(x{0})}{h}}{\nabla{v} f (x{0})}$.

Claim: If is differentiable, then for all , ${\nabla{v} f(x{0})}{x{0} \in \Omega}v{x{0}}$ fixed. And

$${ \nabla_{v}f(x_{0}) = \mathrm{d}f(x_{0})(v). }$$

This equation implies that

$${ \mathrm{d}f(x_{0})(v) = \sum_{i=1}^{n} v_{i} \nabla_{i}f(x_{0}), }$$

here ${\nabla{i}}f{e{i} = \delta_{i} \in \mathbb{R}^{n}}$.

Claim: If ${\nabla{i}f(x{0})}i \in [1, n] \cap \mathbb{N}xff{L:\,v \mapsto \sum{i=1}^{n} \nabla{i}f(x{0}) v{i}}$. It’s sufficient to verify the definition of differential. And we need Lagrange mid-value theorem.

Claim: is differentiable at iff

$${ f(x_{0} + h) - f(x_{0}) = \operatorname{J} f(x_{0}) h + \sum_{i=1}^{n} \beta_{i}(h)h_{i}. }$$

Here means Jacobi matrix.(${\operatorname{J}f(x) = (D{1}f(x), D{2}f(x), \dots, D{n}f(x))}{\beta{i}(h) = o(\lVert h \rVert)}$.
Proof: : ${\frac{\sum{i=1}^{n}\beta{i}(h)h{i}}{\lVert h \rVert} = \sum{i=1}^{n} \beta{i}(h) \frac{h{i}}{\lVert h \rVert} \leq \left( \sum{i=1}^{n} \beta{i}(h)^{2} \right)^{\frac{1}{2}} 1 = (\sum{i=1}^{n} o(\lVert h \rVert)^{2})^{\frac{1}{2}} = \sqrt{ n }o(\lVert h \rVert) \to 0}{\lVert h \rVert \to 0}\implies{r(h) \triangleq f(x{0} + h) - f(x{0}) - \operatorname{J}f(x{0}) = o(\lVert h \rVert) = \left( \sum{i=1}^{n} \frac{h{i}}{\lVert h \rVert} h{i} \right) \frac{r(h)}{\lVert h \rVert} = \sum{i=1}^{n} \frac{r(h)}{\lVert h \rVert} \frac{h{i}}{\lVert h \rVert} h{i}}{\beta{i}(h) = \frac{r(h)}{\lVert h \rVert} \frac{h{i}}{\lVert h \rVert}}{\frac{h{i}}{\lVert h \rVert} \lt 1}{\beta{i}(h) \to 0}{\lVert h \rVert \to 0}$.

{align

graph LR
	id1("各阶偏导数存在")
	id2("函数可微")
	id3("函数连续")
	id4("各方向导数存在")

id1 --> id2
id2 --> id3
id2 --> id4

方向导数算子和微分算子是对偶的. ${(\nabla{v}f, x{0}) = (\mathrm{d} f(x_{0}), v)}$.

Integration

Claim: Let be a map on to . And . Then ${\int \chi{A} \mathrm{d}\mu = \int \chi{A} \varphi’ \mathrm{d}\mu}$.