Multivariate Vector-valued functions.qmd

book/chapters/01 Numeric Structures/04 Functions/04 - Multivariate Vector-valued functions.qmd

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+# Functions: Multivariate Vector-Valued Functions
+
+A multivariate vector-valued function is one of the most expressive types of mathematical entities — it takes in a vector and returns another vector. This is the essence of **transformations**, **vector fields**, and **flows**.
+
+---
+
+## 1. Definition
+
+A function $f: \mathbb{R}^n \to \mathbb{R}^m$ is called a **multivariate vector-valued function** if:
+
+$$
+f(x_1, x_2, \dots, x_n) =
+\begin{bmatrix}
+f_1(x_1, \dots, x_n) \\
+f_2(x_1, \dots, x_n) \\
+\vdots \\
+f_m(x_1, \dots, x_n)
+\end{bmatrix}
+$$
+
+Each $f_i$ is a scalar-valued function. Together, they form an $m$-dimensional output vector.
+
+---
+
+## 2. Example
+
+$$
+f(x, y) =
+\begin{bmatrix}
+x + y \\
+x^2 - y^2
+\end{bmatrix}
+\quad \text{is a function from } \mathbb{R}^2 \to \mathbb{R}^2
+$$
+
+If $x = 2, y = 1$:
+
+$$
+f(2, 1) =
+\begin{bmatrix}
+3 \\
+3
+\end{bmatrix}
+$$
+
+> 🟡 **Key Idea**  
+> Each input is a vector, and each output is a vector. These functions describe transformations — they rotate, scale, twist, and morph the input space.
+
+---
+
+## 3. Domain and Range
+
+- **Domain**: $\mathbb{R}^n$
+- **Codomain**: $\mathbb{R}^m$
+- **Range**: Subset of $\mathbb{R}^m$ defined by the actual function
+
+**Example:**
+
+$$
+f(x, y) = \begin{bmatrix}
+\sin(xy) \\
+\cos(x^2 + y)
+\end{bmatrix}
+$$
+
+- Domain: $\mathbb{R}^2$
+- Range: Lies within $[-1, 1]^2$ due to bounded sine and cosine values
+
+---
+
+## 4. Component Function View
+
+You can always express a vector-valued function as a stack of scalar-valued functions:
+
+$$
+f(x) = \begin{bmatrix}
+f_1(x) \\
+f_2(x) \\
+\vdots \\
+f_m(x)
+\end{bmatrix}
+\quad \text{where each } f_i: \mathbb{R}^n \to \mathbb{R}
+$$
+
+> 🔵 **Insight**  
+> Think of a vector-valued function as a collection of $m$ scalar functions, one for each output dimension.
+
+---
+
+## 5. Jacobian Matrix
+
+The **Jacobian matrix** of $f$ at point $x$ is the matrix of all partial derivatives:
+
+$$
+J_f(x) =
+\begin{bmatrix}
+\frac{\partial f_1}{\partial x_1} & \dots & \frac{\partial f_1}{\partial x_n} \\
+\vdots & \ddots & \vdots \\
+\frac{\partial f_m}{\partial x_1} & \dots & \frac{\partial f_m}{\partial x_n}
+\end{bmatrix}
+\in \mathbb{R}^{m \times n}
+$$
+
+It generalizes the concept of a derivative — describing how the output vector changes as the input vector is perturbed.
+
+![Jacobian Matrix Intuition](path/to/jacobian_visual.png)
+
+---
+
+## 6. Visualization
+
+Visualizing vector-valued functions depends on their dimensions:
+
+- For $f: \mathbb{R}^2 \to \mathbb{R}^2$, we can plot how a grid of points is transformed.
+- For $f: \mathbb{R}^2 \to \mathbb{R}^2$, we can also draw vector fields.
+
+![Vector Field Example](path/to/vector_field_plot.png)
+
+> 🔍 Think of $f(x)$ as arrows assigned to every point $x$ — this is the vector field interpretation.
+
+---
+
+## 7. Applications
+
+🧭 **Where these functions show up:**
+
+- **Physics:**  
+  Velocity fields, electromagnetic fields, force maps  
+- **Machine Learning:**  
+  Each layer of a neural network performs a transformation from $\mathbb{R}^n \to \mathbb{R}^m$  
+- **Economics:**  
+  Multivariate utility and production functions; optimization in high dimensions  
+
+> ✅ **Example: Flow Field**  
+> Let $f(x, y) = \begin{bmatrix} -y \\ x \end{bmatrix}$. This defines a rotation field — imagine a whirlpool, where each point spins around the origin in circular motion.
+
+![Circular Flow Field](path/to/rotation_field.png)
+
+---
+
+## 8. Summary Table
+
+> 🧾 **Key Summary**
+
+| Aspect             | Description                                       |
+|--------------------|---------------------------------------------------|
+| Type               | $f: \mathbb{R}^n \to \mathbb{R}^m$                |
+| Output             | $m$-dimensional vector                            |
+| Structure          | Composed of $m$ scalar-valued functions           |
+| Jacobian Matrix    | $m \times n$ matrix of partial derivatives        |
+| Visual Analog      | Vector fields, geometric transformations          |
+
+---
+
+> 🧠 **Mathematician’s Lens**  
+> Understanding how inputs warp into outputs gives rise to differential geometry, machine learning transformations, and even quantum mechanics. This chapter is the gateway to describing the world not as it is, but as it moves.
+