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0x030 Foundation

multivariate analysis version of Notes 0x020. This note follows the first half of Munkres's Analysis of Manifolds

1. Continuity

1.1. Limits of Functions

Limit of \(\mathbb{R}^n\) should be defined by using limit over a metric space:

Definition (functional limit in metric space) Let \(X, Y\) be metric spaces with metrics \(d_x, d_y\), if \(x_0\) is a limit point of \(X\), we said

\[\lim_{x \to x_0} f(x) = L\]

if for each \(\epsilon > 0\), there exists a \(\delta > 0\) such that

\[d_X(x, x_0) \leq \delta \implies d_Y(f(x), L) \leq \epsilon\]

Note in metric space, we all sequences approaching \(x_0\) should exist and have the same limits. Only considering limits along all lines is not enough, the limit still might not exist

line limits does not guarantee limit

Define the function \(f\)

\[f = \frac{||x||}{\theta(x)}\]

where \(\theta(x)\) is the angle measured from the positive part. All line limits is 0, but when taking curve limit \(||x||=\theta(x)\), then the curve limit is 1.

Again to make sure functional limit exist, ALL sequence limit should exist and equal not only line limits.

another counterexample of line limits

Consider the function \(f\)

\[f(x,y) = \frac{x^2y}{x^4 + y^2}\]

It approaches 0 along the any line \(y=kx\), but approaches 0.5 along the line \(y=x^2\)

1.2. Continuity

Recall in topogical space, continuity is defined

Definition (continuity in topological space) a function \(f:X \to Y\) is continuous at point \(x_0 \in X\) for every open set \(V\) of Y

\[f^{-1}(V) = { x | f(x) \in V }\]

is open in \(X\)

In metric space, this reduced to the following definition:

Definition (continuity in metric space) Let \(X, Y\) be metric spaces with metrics \(d_x, d_y\), \(f\) is continuous for each \(\epsilon > 0\), there exists a \(\delta > 0\) such taht

\[d_X(x, x_0) \leq \delta \implies d_Y(f(x), f(x_0)) \leq \epsilon\]

2. Differentiation

2.1. Differentiability

\[\text{differentiable function} \subset \text{ directional defferentiable } \subset \text{ partial differentiable }\]

Definition (differentiability) Let \(E\) be a subset of \(R^n\), \(f: E \to \mathbb{R}^m\) be a function. \(x_0 \in E\) be a point, let \(L: \mathbb{R}^n \to \mathbb{R}^m\) be a linear transformation. We say \(f\) is differentiable at \(x_0\) with derivative \(L\) if we have

\[\lim_{x \to x_0; x \in E - x_0} \frac{|| f(x) - (f(x_0)+L(x-x_0))|| }{|| x-x_0||} = 0\]

where \(L\) is called the total derivative or Jacobian matrix.

it is easy to see that differentiability implies continuity

Each element in Jacobian matrix is a partial derivative

Definition (partial derivative) Let \(E\) be a subset of \(\mathbb{R}^n\), \(f: E \to \mathbb{R}^m\) and \(x_0\) is an interior point of \(E\), \(1\leq j \leq n\). Then the partial derivative of \(f\) with respect to \(x_j\) variable at \(x_0\), denoted \(\frac{\partial f}{\partial x_j}(x_0)\) is defined by

\[\frac{\partial f}{\partial x_j}(x_0) = \lim_{t \to 0; t \neq 0; x_0 + te_j \in E} \frac{f(x_0+te_j)-f(x_0)}{t}\]

By definition, differentiability implies the Jacobian matrix and partial derivative. Unfortunately, partial derivative (or Jacobian matrix) does not imply differentiability

\[\text{differentiable function} \subset \text{ partial differentiable }\]

partial derivative does not imply differentiability

Consider the function

\[f(x,y) = \begin{cases} 0 \text{ if }x=0 \text{ or } y =0 \\ 1 \text{ otherwise} \end{cases}\]

We know \(\frac{\partial f}{\partial x}(0) = 0, \frac{\partial f}{\partial y}(0) = 0\), but \(f\) is not differentiable at \((0,0)\) along any other direction.

Definition (directional derivative) Let \(E\) be a subset of \(\mathbb{R}^n\), \(f: E \to \mathbb{R}^m\) be a function, let \(x_0\) be an interior point of \(E\), and let \(v\) be a vector in \(R^n\). If the limit

\[\lim_{t \to 0; t>0; x_0+tv \in E} \frac{f(x_0+tv) - f(x_0)}{t}\]

exists, we say \(f\) is differentiable in the direction \(v\) at \(x_0\), and we denote the above limit by \(D_v f(x_0) \in \mathbb{R}^m\)

\[(D_v) f (x_0) = \lim_{t \to 0; t>0}\frac{f(x_0+tv)-f(x_0)}{t}\]

Unfortunately, the existence of all directional derivatives does not imply the differentiability:

\[\text{differentiable function} \subset \text{ directional differentiable }\]

directional differentiable does not imply differentiability

Consider the function \(f\)

\[f(x,y) = \frac{x^2y}{x^4 + y^2}\]

It approaches 0 along the any line \(y=kx\). But it is not even continuous as it approaches 0.5 along \(y=x^2\).

2.2. Continuously Differentiable Functions

The mere existence of the partial derivatives (or Jacobian matrix) does not imply differentiability.

By imposing continuity on partial derivatives, we can derive the C1 (thus differentiability). This ensures us that functions such as \(sin(xy)\) are differentiable.

Theorem (continuous partial implies C1) If all the partial derivatives \(\frac{\partial f}{\partial x_j}\) exists and continuous on \(x_0\), then \(f\) is differentiable at \(x_0\) and the linear transformation \(f'(x_0)\) is defined by

\[f'(x_0)(v_j)_{1\leq j \leq n} = \sum_j v_j \frac{\partial f}{\partial x_j}(x_0)\]

This can be proved by telescoping \(h\) with (single-value) mean value theorem.

Recall that mere differentiability does not imply C1, see this stackexchange post

Theorem (mixed partial derivative are equal in \(C^2\) ) For a twice continuously differentiable function \(f\), the mixed second partial derivatives are equal:

\[f_{xy} = f_{yx}\]

2.3. Inverse Function Theorem

The Jacobian at a given point gives important information about the behavior of \(f\) near the point. For example, the continuously differentiable function \(f\) is invertible near \(p\) when the Jacobian determinant at \(p\) is non-zero, which is known as the inverse function theorem

2.4. Implicit Function Theorem

3. Riemann Integral

4. Reference

[1] Tao, Terence. Analysis. Vol. 1. Hindustan Book Agency, 2006.

[2] Tao, Terence. Analysis. Vol. 2. Hindustan Book Agency, 2006.

[3] Abbott, Stephen. Understanding analysis. Vol. 2. New York: Springer, 2001.

[4] Lax, Peter D., and Maria Shea Terrell. Multivariable Calculus with Applications. Springer, 2017.

[5] 杉浦光夫. "解析入門 I." 東京大学出版会

[6] 杉浦光夫. "解析入門 II." 東京大学出版会