Unit 8: Multivariable Analysis: Euclidean Topology, Fréchet Derivatives & Multiple Integrals

1. Vector Space Structure of $\mathbb{R}^n$

Euclidean $n$-space is the Cartesian product $\mathbb{R}^n = \mathbb{R} \times \dots \times \mathbb{R}$ of all ordered $n$-tuples of real numbers:

Endowed with componentwise addition $\mathbf{x} + \mathbf{y}$ and scalar multiplication $c \mathbf{x}$, $\mathbb{R}^n$ forms an $n$-dimensional real vector space.

Definition 8.1 (Euclidean Inner Product and Norm):

1. The inner product (or dot product) of $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ is:

1. The standard Euclidean norm (magnitude) is:

2. The Cauchy-Schwarz Inequality

Theorem 8.1 (Cauchy-Schwarz Inequality): For all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$:

with equality holding if and only if $\mathbf{x}$ and $\mathbf{y}$ are linearly dependent (parallel).

Complete Line-by-Line Proof:

If $\mathbf{y} = \mathbf{0}$, both sides equal zero, and equality trivially holds. Assume $\mathbf{y} \ne \mathbf{0}$, so $\|\mathbf{y}\|^2 > 0$. For any scalar $t \in \mathbb{R}$, consider the vector $\mathbf{x} - t \mathbf{y}$. By positive-definiteness of the inner product:

Expanding via bilinearity and symmetry:

This is a quadratic polynomial $q(t) = A t^2 + B t + C$ in the variable $t$, where $A = \|\mathbf{y}\|^2 > 0$, $B = -2\langle \mathbf{x}, \mathbf{y} \rangle$, and $C = \|\mathbf{x}\|^2$. Since $q(t) \ge 0$ for all real $t$, this parabola cannot have two distinct real roots. Therefore, its discriminant $\Delta = B^2 - 4AC$ must be non-positive:

Taking square roots yields:

Corollary 8.1 (Triangle Inequality in $\mathbb{R}^n$): For all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$:

Proof: $\|\mathbf{x} + \mathbf{y}\|^2 = \|\mathbf{x}\|^2 + 2\langle \mathbf{x}, \mathbf{y} \rangle + \|\mathbf{y}\|^2 \le \|\mathbf{x}\|^2 + 2\|\mathbf{x}\|\|\mathbf{y}\| + \|\mathbf{y}\|^2 = (\|\mathbf{x}\| + \|\mathbf{y}\|)^2$. $\blacksquare$

3. Topology of $\mathbb{R}^n$

The metric on $\mathbb{R}^n$ is $d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\|$.

• An open ball centered at $\mathbf{x}_0$ with radius $r > 0$ is $B(\mathbf{x}_0, r) = \{ \mathbf{x} \in \mathbb{R}^n : \|\mathbf{x} - \mathbf{x}_0\| < r \}$.
• A subset $U \subseteq \mathbb{R}^n$ is open if every point is interior: $\forall \mathbf{x} \in U, \; \exists r > 0 \text{ s.t. } B(\mathbf{x}, r) \subseteq U$.
• The Heine-Borel Theorem holds verbatim in $\mathbb{R}^n$: A subset $K \subset \mathbb{R}^n$ is compact if and only if it is closed and bounded.
• The space $\mathbb{R}^n$ is complete: every Cauchy sequence in $\mathbb{R}^n$ converges.

1. Limits and Continuity in Multivariable Spaces

Definition 8.2 (Multivariable Limit): Let $E \subseteq \mathbb{R}^n$, $\mathbf{f}: E \to \mathbb{R}^m$, and $\mathbf{c}$ be a limit point of $E$. We write $\lim_{\mathbf{x} \to \mathbf{c}} \mathbf{f}(\mathbf{x}) = \mathbf{L} \in \mathbb{R}^m$ if:

2. Directional and Partial Derivatives

Definition 8.3 (Directional Derivative): Let $U \subseteq \mathbb{R}^n$ be open, $f: U \to \mathbb{R}$, $\mathbf{x} \in U$, and $\mathbf{v} \in \mathbb{R}^n$ be a unit vector ($\|\mathbf{v}\| = 1$). The directional derivative of $f$ at $\mathbf{x}$ in the direction of $\mathbf{v}$ is:

If $\mathbf{v} = \mathbf{e}_j$ (the $j$-th standard basis vector), this is the partial derivative:

Cautionary Insight: The existence of ALL partial derivatives (and even all directional derivatives) at a point does NOT imply continuity! For example: $f(x, y) = \frac{x y^2}{x^2 + y^4}$ ($f(0, 0) = 0$) has directional derivatives in every direction at $(0, 0)$, yet is discontinuous along the parabola $x = y^2$ where $f(y^2, y) = 1/2 \ne 0$. Differentiability in several variables requires a stronger, linear-algebraic concept: the Fréchet derivative.

3. The Total Fréchet Derivative

Definition 8.4 (Total Derivative / Fréchet Derivative): Let $U \subseteq \mathbb{R}^n$ be open, $\mathbf{f}: U \to \mathbb{R}^m$, and $\mathbf{x}_0 \in U$. The function $\mathbf{f}$ is differentiable at $\mathbf{x}_0$ if there exists a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ (denoted $D\mathbf{f}(\mathbf{x}_0)$) such that:

Equivalently:

Theorem 8.2 (Differentiability Implies Continuity): If $\mathbf{f}$ is differentiable at $\mathbf{x}_0$, then $\mathbf{f}$ is continuous at $\mathbf{x}_0$.

1. Matrix Representation: The Jacobian Matrix

Theorem 8.3 (Uniqueness of Derivative & Jacobian Representation): If $\mathbf{f} = (f_1, f_2, \dots, f_m)^T: U \subseteq \mathbb{R}^n \to \mathbb{R}^m$ is differentiable at $\mathbf{x}_0$, then all partial derivatives $\frac{\partial f_i}{\partial x_j}(\mathbf{x}_0)$ exist, and the matrix representation of $D\mathbf{f}(\mathbf{x}_0)$ with respect to the standard bases is the $m \times n$ Jacobian matrix:

That is, $D\mathbf{f}(\mathbf{x}_0)(\mathbf{h}) = J_{\mathbf{f}}(\mathbf{x}_0) \mathbf{h}$.

2. Sufficient Condition for Differentiability ($C^1$)

Theorem 8.4 (Continuously Differentiable implies Differentiable): Let $U \subseteq \mathbb{R}^n$ be open and $\mathbf{f}: U \to \mathbb{R}^m$. If all partial derivatives $\frac{\partial f_i}{\partial x_j}$ exist on $U$ and are continuous at $\mathbf{x}_0$ (i.e., $\mathbf{f} \in C^1(U)$), then $\mathbf{f}$ is differentiable at $\mathbf{x}_0$.

Proof:

By considering each component function $f_i$ separately, it suffices to prove this for $f: U \subseteq \mathbb{R}^n \to \mathbb{R}$. Let $\mathbf{h} = (h_1, h_2, \dots, h_n)$. Express the difference $f(\mathbf{x}_0 + \mathbf{h}) - f(\mathbf{x}_0)$ as a telescoping sum across coordinate axes:

In each step, only the $j$-th coordinate varies. By the single-variable Mean Value Theorem (Theorem 6.6), there exists $\theta_j \in (0, 1)$ such that:

where $\mathbf{p}_j$ lies on the line segment between $\mathbf{x}_0$ and $\mathbf{x}_0 + \mathbf{h}$. Now subtract the candidate linear approximation $\sum_{j=1}^n \frac{\partial f}{\partial x_j}(\mathbf{x}_0) h_j$:

Since $|h_j| \le \|\mathbf{h}\|$, the ratio $\frac{|h_j|}{\|\mathbf{h}\|} \le 1$. As $\mathbf{h} \to \mathbf{0}$, $\mathbf{p}_j \to \mathbf{x}_0$. Since each partial derivative is continuous at $\mathbf{x}_0$, $\left| \frac{\partial f}{\partial x_j}(\mathbf{p}_j) - \frac{\partial f}{\partial x_j}(\mathbf{x}_0) \right| \to 0$. Thus the error quotient vanishes as $\mathbf{h} \to \mathbf{0}$. Therefore, $f$ is differentiable at $\mathbf{x}_0$. $\blacksquare$

3. The Multivariable Chain Rule

Theorem 8.5 (Multivariable Chain Rule): Let $U \subseteq \mathbb{R}^n$ and $V \subseteq \mathbb{R}^m$ be open sets. If $\mathbf{f}: U \to V$ is differentiable at $\mathbf{x}_0 \in U$ and $\mathbf{g}: V \to \mathbb{R}^p$ is differentiable at $\mathbf{y}_0 = \mathbf{f}(\mathbf{x}_0) \in V$, then the composite mapping $\mathbf{h} = \mathbf{g} \circ \mathbf{f}: U \to \mathbb{R}^p$ is differentiable at $\mathbf{x}_0$, and:

In matrix notation:

The Jacobian matrix of a composite function is the matrix product of their respective Jacobian matrices!

1. The Inverse Function Theorem

The Inverse Function Theorem provides the definitive criterion under which a non-linear mapping can be locally inverted, generalizing the 1D condition $f'(x_0) \ne 0$.

Theorem 8.6 (Inverse Function Theorem): Let $U \subseteq \mathbb{R}^n$ be an open set and let $\mathbf{f}: U \to \mathbb{R}^n$ be a continuously differentiable ($C^1$) mapping. Suppose that at a point $\mathbf{x}_0 \in U$, the total derivative $D\mathbf{f}(\mathbf{x}_0)$ is invertible:

Then:

1. There exists an open neighborhood $V$ containing $\mathbf{x}_0$ and an open neighborhood $W$ containing $\mathbf{y}_0 = \mathbf{f}(\mathbf{x}_0)$ such that $\mathbf{f}: V \to W$ is a bijection (a local $C^1$-diffeomorphism).
2. The inverse function $\mathbf{g} = \mathbf{f}^{-1}: W \to V$ is also continuously differentiable ($C^1$).
3. For every $\mathbf{y} \in W$, the derivative of the inverse is the matrix inverse of the derivative:

2. The Implicit Function Theorem

In applications, equations often define variables implicitly, such as $F(x, y, z) = 0$. Under what conditions can we solve for some variables in terms of the others: $z = g(x, y)$?

Theorem 8.7 (Implicit Function Theorem): Let $U \subseteq \mathbb{R}^n \times \mathbb{R}^m$ be open, and let $\mathbf{F}: U \to \mathbb{R}^m$ be a $C^1$ function, with coordinates $(\mathbf{x}, \mathbf{y}) = (x_1, \dots, x_n, y_1, \dots, y_m)$. Suppose $(\mathbf{x}_0, \mathbf{y}_0) \in U$ satisfies:

1. $\mathbf{F}(\mathbf{x}_0, \mathbf{y}_0) = \mathbf{0}$.
2. The $m \times m$ partial Jacobian matrix with respect to $\mathbf{y}$ is invertible:

Then:

1. There exists an open neighborhood $V \subseteq \mathbb{R}^n$ of $\mathbf{x}_0$ and an open neighborhood $W \subseteq \mathbb{R}^m$ of $\mathbf{y}_0$ such that for each $\mathbf{x} \in V$, there exists a unique $\mathbf{y} = \mathbf{g}(\mathbf{x}) \in W$ satisfying:

1. The function $\mathbf{g}: V \to W$ is continuously differentiable ($C^1$).
2. The derivative of $\mathbf{g}$ is given by:

1. Multiple Integrals in $\mathbb{R}^n$

Let $R = [a_1, b_1] \times [a_2, b_2] \times \dots \times [a_n, b_n]$ be a compact $n$-dimensional rectangle (box) in $\mathbb{R}^n$. Partitions $P = P_1 \times \dots \times P_n$ divide $R$ into sub-rectangles $R_k$ of volume $\operatorname{vol}(R_k) = \prod_{j=1}^n \Delta x_{j, k}$. Darboux upper and lower sums are defined analogously:

A bounded function $f: R \to \mathbb{R}$ is Riemann integrable if $\inf U(f, P) = \sup L(f, P)$, denoted $\int_R f(\mathbf{x}) \, d\mathbf{x}$.

2. Fubini's Theorem: Reduction to Iterated Integrals

Theorem 8.8 (Fubini's Theorem for Continuous Functions): Let $A \subset \mathbb{R}^n$ and $B \subset \mathbb{R}^m$ be compact rectangles, and let $f: A \times B \to \mathbb{R}$ be continuous. Then:

The multiple integral equals the iterated single integrals, and the order of integration can be freely interchanged.

3. The Multivariable Change of Variables Theorem

Theorem 8.9 (Change of Variables Theorem in $\mathbb{R}^n$): Let $U \subseteq \mathbb{R}^n$ be open, and let $\mathbf{g}: U \to \mathbb{R}^n$ be an injective $C^1$ mapping such that $\det J_{\mathbf{g}}(\mathbf{u}) \ne 0$ for all $\mathbf{u} \in U$. Let $E \subset U$ be a compact Jordan-measurable set, and let $f: \mathbf{g}(E) \to \mathbb{R}$ be continuous. Then:

The absolute value of the Jacobian determinant $|\det J_{\mathbf{g}}(\mathbf{u})|$ represents the local volume distortion factor under the non-linear transformation $\mathbf{g}$.