Several Variable Calculus UNSW Lecture Notes

Chapter 7: Several Variable Calculus - Introduction

• Conceptual Overview: Many mathematical problems require more than one input variable to be described adequately.
• Finance Application Example: In mathematical finance, the value $\text{V}$ of an option depends on several parameters: current price ($\text{S}$), volatility ($\omega$), time value ($\epsilon$), risk-free interest rate ($\text{r}$), and more, expressed as V = V(S, heta, ho, r, K, q, ext{…}).
• The "Greeks": These are sensitivities of $\text{V}$ to underlying parameters, represented by Greek letters. Mathematically, they are derivatives of $\text{V}$ with respect to specific variables.
• Complexity of Several Variables:     * The space of allowable parameters may be more complex than a simple interval in $\mathbb{R}$.     * Visualization is more difficult.     * In $\mathbb{R}$, movement is limited to left or right (larger or smaller); in $\mathbb{R}^n$, a point can be approached from infinitely many directions, meaning limits are no longer restricted to left and right.

Graphs of Multi-Variable Functions

• Graph Definitions:     * For a function $f : \mathbb{R} \rightarrow \mathbb{R}$, the graph is the subset $\text{graph}(f) = {(x, y) \in \mathbb{R}^2 : y = f(x)}$.     * For a function $g : \mathbb{R}^n \rightarrow \mathbb{R}$, the graph is $\text{graph}(g) = {(x, z) \in \mathbb{R}^{n+1} : z = g(x)}$.
• Dimensional Constraints: Graphs where n > 2 cannot be drawn using standard 3D visualization techniques.     * Two variables ($x, y$) typically represent a curve in a plane, such as the circle $x^2 + y^2 = 4$.     * Three variables ($x, y, z$) typically represent a surface in 3D, such as the sphere $x^2 + y^2 + z^2 = 4$.
• Level Curves and Contour Maps: Surfaces can be represented via contour maps where $h(x, y) = \text{height at position } (x, y)$. The level curves are equations $h(x, y) = c$ for various constants $c$.
• Example Function: For $f(x, y) = \sin(x^2 + y^2)$, the graph is the surface $\text{graph}(f) = {(x, y, z) \in \mathbb{R}^3 : z = f(x, y)}$.

Partial Differentiation

• Informal Definition: For a function of several independent variables $x_1, x_2, ext{…}, x_n$, the partial derivative with respect to $x_i$ ($f_{x_i}$ or $\frac{\partial f}{\partial x_i}$) is calculated by treating all variables except $x_i$ as constants.
• Notation Distinction: The partial symbol $\partial$ is used for functions of more than one independent variable. The symbol $d$ is used for single variable or implicit functions (e.g., $x^2 + 3xy + y^4 = 4$ defines $y$ implicitly as a function of $x$).
• Geometric Interpretation:     * If we fix $y = y_0$, we obtain a cross-section function $c_{y_0}(x) = f(x, y_0)$. The partial derivative $\frac{\partial f}{\partial x}(x_0, y_0)$ is the slope of this curve in the $x$-direction.     * If we fix $x = x_0$, we obtain a cross-section $d_{x_0}(y) = f(x_0, y)$. The partial derivative $\frac{\partial f}{\partial y}(x_0, y_0)$ is the rate of change in the $y$-direction.
• Formal Definition:     * $\frac{\partial f}{\partial x_i}(\mathbf{x}) = \lim_{h \to 0} \frac{f(x_1, x_2, \dots, x_i + h, \dots, x_n) - f(x_1, x_2, \dots, x_n)}{h}$, provided the limit exists.
• Example 1 Calculation:     1. $f(x, y) = x \cos(xy)$.     2. $g(x, y, z) = xy + \sin(x^2 z) + y^2 e^z$.
• Higher Order Derivatives:     * Fractional notation: $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x} (\frac{\partial f}{\partial y})$.     * Subscript notation (sequential from inside out): $f_{xy} = (f_x)<em>y = \frac{\partial^2 f}{\partial y \partial x}$.     * Schwarz’s Theorem: If the partial derivatives of $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ up to and including second order are continuous, then $f</em>{xy}(x, y) = f_{yx}(x, y)$.

Continuity in Several Variables

• Definition: A function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous at $\mathbf{x}<em>0 = (x_0, y_0)$ if $\lim</em>{\mathbf{x} \to \mathbf{x}_0} f(\mathbf{x})$ exists and equals $f(\mathbf{x}_0)$.
• Path Dependency: Limits in $\mathbb{R}^n$ are more complex. For a limit to exist, it must be the same regardless of the path taken to reach the point.
• Example 4 (Discontinuity): Let $f(x, y) = \frac{2x^2 y}{x^4 + y^2}$ at $\mathbf{x}_0 = (0, 0)$.     * Approaching along $y = x$: $f(x, x) = \frac{2x^3}{x^4 + x^2} = \frac{2x}{x^2 + 1} \rightarrow 0$ as $x \rightarrow 0$.     * Approaching along $y = x^2$: $f(x, x^2) = \frac{2x^4}{x^4 + x^4} = \frac{2x^4}{2x^4} = 1$.     * Since different paths yield different limits, the function is not continuous at $(0, 0)$.
• General Rule: Any function composed of continuous single-variable functions is continuous on its domain, excluding points like division by zero or negative square roots.

The Tangent Plane

• Concept: Just as a single variable function is approximated by a tangent line, a function of two variables $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ is approximated by a two-dimensional tangent plane.
• Cartesian Equation: If $f_x$ and $f_y$ exist and are continuous at $(x_0, y_0)$, the tangent plane is:     * $z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$.
• Parametric Vector Form: The plane passes through $\mathbf{a} = (x_0, y_0, f(x_0, y_0))^T$ and is spanned by vectors $\mathbf{v}_1 = (1, 0, f_x(x_0, y_0))^T$ and $\mathbf{v}_2 = (0, 1, f_y(x_0, y_0))^T$:     * $\mathbf{x} = \begin{pmatrix} x_0 \ y_0 \ f(x_0, y_0) \end{pmatrix} + \omega_1 \begin{pmatrix} 1 \ 0 \ f_x(x_0, y_0) \end{pmatrix} + \omega_2 \begin{pmatrix} 0 \ 1 \ f_y(x_0, y_0) \end{pmatrix}$.
• Point-Normal Form: The normal vector $\mathbf{n} = \mathbf{v}_1 \times \mathbf{v}_2 = (-f_x(x_0, y_0), -f_y(x_0, y_0), 1)^T$.     * $\mathbf{n} \cdot (\mathbf{x} - \mathbf{a}) = 0$.

The Gradient Vector and Optimization

• Definition: The gradient of $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is the vector $\nabla f(\mathbf{x}) = (\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n})^T$.
• Total Differential Approximation: For $\mathbf{x}$ near $\mathbf{x}_0$, $f(\mathbf{x}) \approx f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0) \cdot (\mathbf{x} - \mathbf{x}_0)$.     * $f(\mathbf{x})$ changes fastest when $(\mathbf{x} - \mathbf{x}_0)$ is parallel to $\nabla f(\mathbf{x}_0)$.     * $f(\mathbf{x})$ is constant when $(\mathbf{x} - \mathbf{x}_0)$ is orthogonal to $\nabla f(\mathbf{x}_0)$.
• Local Maxima and Minima: Similar to looking for $f'(x) = 0$ in single variables, we seek points where the tangent plane is flat: $\nabla f(x, y) = (0, 0)$, which implies $f_x = 0$ and $f_y = 0$.

Chain Rules

• Case 1 (Function of one variable $t$): If $z = f(x, y)$ where $x = g(t)$ and $y = h(t)$, then:     * $\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$.
• Case 2 (Function of two variables $s, t$): If $z = f(x, y)$ where $x = g(s, t)$ and $y = h(s, t)$, then:     * $\frac{\partial z}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s}$.     * $\frac{\partial z}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t}$.

Directional Derivatives

• Definition: The directional derivative of $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ in the direction of a unit vector $\mathbf{v} = (a, b)^T$ is:     * $D_{\mathbf{v}} f(x, y) = \lim_{h \to 0} \frac{f(x + ah, y + bh) - f(x, y)}{h}$.
• Formula using Gradient: For a unit vector $\mathbf{v}$, $D_{\mathbf{v}} f = \nabla f \cdot \mathbf{v}$.
• Basis Vectors: The directional derivatives in the direction of standard basis vectors $e_1$ and $e_2$ correspond exactly to the partial derivatives $f_x$ and $f_y$.

Level Curves and Hypersurfaces

• Normal to Level Curve: For $F(x, y) = 0$, the gradient $\nabla F(x_0, y_0)$ is normal to the curve at the point $(x_0, y_0)$.
• Level Surface: The graph $z = f(x, y)$ can be written as $F(x, y, z) = f(x, y) - z = 0$. The normal vector is $\nabla F = (f_x, f_y, -1)^T$.

The Jacobian Matrix

• Definition: For a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, the Jacobian matrix $Df(\mathbf{x})$ is the matrix of partial derivatives where $[Df]_{ij} = \frac{\partial f_i}{\partial x_j}$.
• General Chain Rule: $D(f \circ g)(\mathbf{x}) = Df(g(\mathbf{x})) Dg(\mathbf{x})$.
• Applications:     * Inverse Function Theorem: $D(g^{-1})(\mathbf{x}) = [Dg(g^{-1}(\mathbf{x}))]^{-1}$.     * Change of Variables: When transforming a region $A \subset \mathbb{R}^n$ via $g$, the integral is adjusted by the determinant of the Jacobian: $\int_A f(g(\mathbf{x})) |\det(Dg)(\mathbf{x})| d\mathbf{x} = \int_{g(A)} f(\mathbf{u}) d\mathbf{u}$.

Leibniz’s Rule

• Theorem: If $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous and $\frac{\partial f}{\partial y}$ is continuous, then the derivative of the integral with respect to the parameter $y$ is:     * $\frac{d}{dy} \int_a^b f(x, y) dx = \int_a^b \frac{\partial f}{\partial y} (x, y) dx$.
• Example Application: This rule helps calculate complex integrals by differentiating under the integral sign relative to a constant parameter $y$.

Discussion of Example 15

• Problem: Find $I = \int_0^{\pi/4} \frac{\sin^2(x)}{(\cos^2(x) + 4\sin^2(x))^2} dx$.
• Method:     1. Consider $A(y) = \int_0^{\pi/4} \frac{1}{\cos^2(x) + y^2 \sin^2(x)} dx$.     2. Note that $\frac{\partial}{\partial y} \frac{1}{\cos^2(x) + y^2 \sin^2(x)} = \frac{-2y \sin^2(x)}{(\cos^2(x) + y^2 \sin^2(x))^2}$.     3. By Leibniz's Rule: $A'(y) = \int_0^{\pi/4} \frac{-2y \sin^2(x)}{(\cos^2(x) + y^2 \sin^2(x))^2} dx$.     4. Evaluating $A(y)$: Divide num/den by $\cos^2(x)$ to use $\int \frac{\sec^2(x)}{1 + y^2 \tan^2(x)} dx$.     5. Let $u = y \tan(x)$, then $A(y) = \frac{1}{y} \int_0^y \frac{1}{1+u^2} du = \frac{\tan^{-1}(y)}{y}$.     6. Finding $A'(y) = \frac{\frac{y}{1+y^2} - \tan^{-1}(y)}{y^2}$.     7. Setting $y = 2$, the integral $I = -\frac{A'(2)}{4}$.     8. Final calculation: $I = \frac{1}{8} [\tan^{-1}(2) - \frac{2}{5}]$.