Comprehensive University Study Notes: Advanced Multivariable Calculus, Vector Analysis, and Ordinary Differential Equations

Dimensional Analysis, Similitude, and Physical Modelling

• Foundations of Mathematical Modelling:

  • Mathematical modelling is the process of creating, analysing, and validating mathematical representations of physical, biological, chemical, economic, and engineering phenomena.

  • The modelling cycle proceeds from identifying key variables and interactions in the natural world to formulating equations, solving and analysing those equations, and validating and interpreting predictions against experimental data.

• Units and Base Physical Dimensions:

  • Postulate 1: There exist base quantities providing units in terms of which the units of all other physical quantities can be expressed.

  • In classical mechanics, the three primary base dimensions are:

  • Mass: $[M]$

  • Length: $[L]$

  • Time: $[T]$

  • Additional fundamental base quantities in the International System of Units (SI) include temperature ($\text{Kelvin}$, $[K]$), electric current ($\text{Ampere}$, $[A]$), amount of substance ($\text{mole}$, $[mol]$), and luminous intensity ($\text{candela}$, $[cd]$).

  • The dimensions of any derived physical variable $y$ are denoted $[y]$, expressed as a product of integer powers of the base dimensions: $[y] = M^a L^b T^c$ where $a, b, c \in \mathbb{Z}$.

  • For composite quantities, the dimension operator satisfies the product property: $[yz] = [y][z]$.

  • Fundamental mechanical examples:

  • Velocity: $[v] = \left[\frac{dx}{dt}\right] = LT^{-1}$

  • Acceleration: $[a] = \left[\frac{d^2x}{dt^2}\right] = LT^{-2}$

  • Force: $[F] = [ma] = [m][a] = MLT^{-2}$

  • A quantity is defined as dimensionless if its dimensional representation is $M^0 L^0 T^0$.

• Dimensional Homogeneity and Complete Sets:

  • Postulate 2: Equations representing physical laws are dimensionally homogeneous.

  • An equation is dimensionally homogeneous if and only if the dimensions of every additive term on both the left-hand side and right-hand side are identical: $[\text{LHS}] = [\text{RHS}]$.

  • In transcendental functions (such as $\exp(z)$, $\ln(z)$, $\sin(z)$, $\cos(z)$), the argument $z$ must be strictly dimensionless ($[z] = M^0 L^0 T^0$) for dimensional compatibility, as established by their Maclaurin series expansions (e.g., $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$).

  • A complete set of dimensionless products is a maximal set of independent dimensionless combinations formed from the variables of a model.

• The Buckingham $\Pi$-Theorem:

  • An equation is dimensionally homogeneous if and only if it can be expressed in the form:     $f(\Pi_1, \Pi_2, \dots, \Pi_n) = 0$     where $f$ is a function and $\{\Pi_1, \Pi_2, \dots, \Pi_n\}$ constitutes a complete set of independent dimensionless products corresponding to the mathematical model.

  • The set of exponents yielding dimensionless products is obtained by determining a basis for the null space of the dimensional matrix representing the system of linear equations for the exponents.

• Worked Case Studies in Dimensional Analysis:

  • A Falling Mass:

  • An object of mass $m$ falls from rest in a vacuum near Earth's surface over distance $x$ in time $t$.

  • In metric units (Australia), experiments yield $x = 4.9 t^2$, whereas in imperial units (USA), experiments yield $x = 16.1 t^2$.

  • Dimensional consistency requires $x = c t^2$, where $[x] = [c][t]^2 \implies L = [c] T^2 \implies [c] = LT^{-2}$, revealing that $c = \frac{1}{2}g$.

  • The Simple Pendulum:

  • Period $\tau$ depends on pendulum length $r$, mass $m$, initial release angle $\theta$, and gravitational acceleration $g$: $\tau = f(r, m, \theta, g)$.

  • Dimensions: $[\tau] = T$, $[r] = L$, $[m] = M$, $[g] = LT^{-2}$, $[\theta] = 1$ ($M^0 L^0 T^0$).

  • Product: $\Pi = \tau^a r^b m^c g^e \theta^d$.

  • Dimensional balance: $T^a L^b M^c (LT^{-2})^e 1^d = M^c L^{b+e} T^{a-2e} = M^0 L^0 T^0$.

  • Linear system for exponents:

    • $M: c = 0$

    • $L: b + e = 0 \implies b = -e$

    • $T: a - 2e = 0 \implies e = \frac{a}{2}, b = -\frac{a}{2}$

    • Exponent $d$ is free.

  • Independent basis choices $(a, d) = (1, 0)$ and $(0, 1)$ generate the complete set:       $\Pi_1 = \tau \sqrt{\frac{g}{r}}, \quad \Pi_2 = \theta$

  • Applying the Buckingham $\Pi$-theorem yields $f\left(\tau \sqrt{\frac{g}{r}}, \theta\right) = 0 \implies \tau \sqrt{\frac{g}{r}} = h(\theta) \implies \tau = \sqrt{\frac{r}{g}} h(\theta)$.

  • Archimedes' Buoyancy Law:

  • A box of mass $m$ displacing fluid volume $V$ with fluid density $\rho$ ($[\rho] = ML^{-3}$) under gravity $g$.

  • Testing a proposed equation $m\ddot{x} = mg - mVg$:

    • $[m\ddot{x}] = [mg] = MLT^{-2}$

    • $[mVg] = M \cdot L^3 \cdot LT^{-2} = ML^4 T^{-2}$

    • Since $MLT^{-2} \ne ML^4 T^{-2}$, the proposed law is dimensionally invalid.

  • Testing an alternative equation $m\ddot{x} = mg - \rho V g$:

    • $[\rho V g] = (ML^{-3})(L^3)(LT^{-2}) = MLT^{-2}$

    • The equation is dimensionally homogeneous and physically viable.

  • By the Buckingham $\Pi$-theorem, any valid relationship between total force $F$, $\rho$, $g$, $V$, and $m$ must satisfy:       $F = \rho g V h\left(\frac{m}{\rho V}\right)$       Choosing $h(u) = u - 1$ directly recovers $F = mg - \rho V g$.

  • Stokes' Drag Law:

  • Drag force $F$ ($[F] = MLT^{-2}$) on a sphere of radius $r$ ($[r] = L$) moving at velocity $v$ ($[v] = LT^{-1}$) in fluid of dynamic viscosity $\eta$ ($[\eta] = ML^{-1}T^{-1}$).

  • Dimensionless combination $F^a \eta^b v^c r^d = M^{a+b} L^{a-b+c+d} T^{-2a-b-c} = M^0 L^0 T^0$.

  • Setting $a = 1$ isolates the dependent variable: $b = -1$, $c = -1$, $d = -1$.

  • The solitary dimensionless group is $\Pi = \frac{F}{\eta v r}$, implying $\frac{F}{\eta v r} = k \implies F = k \eta v r$.

  • Stokes' exact theoretical solution yields $k = 6\pi$, giving $F = 6\pi \eta v r$.

  • Fluid and Solute Mixing Dynamics:

  • A tank initially contains $v_0$ litres of water with $m_0$ grams of dissolved salt. Brine containing $n\,g/L$ enters at $x\,L/min$, and well-stirred solution drains at $y\,L/min$.

  • Let $s(t)$ be the mass of salt at time $t$. The volume is $V(t) = v_0 + (x - y)t$, and solute density is $\rho(t) = \frac{s(t)}{v_0 + (x - y)t}$.

  • Rate of mass inflow: $nx\,g/min$. Rate of mass outflow: $y\rho(t) = \frac{ys(t)}{v_0 + (x - y)t}\,g/min$.

  • Governing Initial Value Problem:       $\frac{ds}{dt} = nx - \frac{ys}{v_0 + (x - y)t}, \quad s(0) = m_0$

  • Dimensional consistency check:

    • $\left[\frac{ds}{dt}\right] = MT^{-1}$

    • $[nx] = (ML^{-3})(L^3 T^{-1}) = MT^{-1}$

    • $[v_0] = [(x - y)t] = L^3$

    • $\left[\frac{ys}{v_0 + (x - y)t}\right] = \frac{(L^3 T^{-1})(M)}{L^3} = MT^{-1}$

• Scaling and Similitude:

  • Laboratory scale models preserve dimensionless numbers to match the physical behaviour of full-scale prototypes.

  • Fundamental dimensionless numbers in fluid mechanics:

  • Reynolds number: $Re = \frac{\rho l v}{\mu}$

  • Froude number: $Fr = \frac{v}{\sqrt{gl}}$

  • Mach number: $M = \frac{v}{c}$     where $\rho$ is fluid density, $l$ is characteristic length, $v$ is velocity, $\mu$ is dynamic viscosity, $g$ is gravitational acceleration, and $c$ is speed of sound.

  • Ship Scaling Example:

  • Full-scale vessel: length $l = 20\,m$, cruising velocity $v = 10\,m/s$.

  • Model scale: length $l^* = 0.2\,m$ (scale ratio $l^*/l = 1/100$).

  • Maintaining dynamic similitude via the Froude number:       $Fr = \frac{v}{\sqrt{gl}} = \frac{v^*}{\sqrt{gl^*}} \implies v^* = v \sqrt{\frac{l^*}{l}} = 10 \sqrt{\frac{0.2}{20}} = 10 \sqrt{\frac{1}{100}} = 1\,m/s$

Multivariable Functions, Limits, and Continuity

• Functions of Multiple Variables:

  • A scalar-valued function of several variables is a mapping $f: D \subset \mathbb{R}^n \to \mathbb{R}$, where the domain $D$ is a subset of $\mathbb{R}^n$.

  • The graph of a two-variable function $z = f(x, y)$ represents a surface in $\mathbb{R}^3$.

• Multivariate Limits and Definitions:

  • Formal $\epsilon$-$\delta$ Definition: Let $f: D \subset \mathbb{R}^2 \to \mathbb{R}$ be defined on an open domain $D$ containing points arbitrarily close to $(a, b)$. The limit of $f(x, y)$ as $(x, y) \to (a, b)$ is $L$, written:     $\lim_{(x, y) \to (a, b)} f(x, y) = L$     if for every $\epsilon > 0$, there exists a corresponding $\delta > 0$ such that:     $\text{if } (x, y) \in D \text{ and } 0 < \sqrt{(x - a)^2 + (y - b)^2} < \delta, \quad \text{then } |f(x, y) - L| < \epsilon$

  • Here, $|f(x, y) - L|$ denotes the absolute distance in $\mathbb{R}$, while $\sqrt{(x - a)^2 + (y - b)^2} = \|(x, y) - (a, b)\|$ denotes the Euclidean distance in $\mathbb{R}^2$.

  • Generalisation to $\mathbb{R}^n$: for $f: D \subset \mathbb{R}^n \to \mathbb{R}$, $\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = L$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that $0 < \|\mathbf{x} - \mathbf{a}\| < \delta \implies |f(\mathbf{x}) - L| < \epsilon$.

• The Path-Dependent Non-Existence Criterion:

  • For $\lim_{(x, y) \to (a, b)} f(x, y)$ to exist, $f(x, y)$ must approach the exact same value $L$ along every path in $D$ terminating at $(a, b)$.

  • If $f(x, y) \to L_1$ along path $C_1$ and $f(x, y) \to L_2$ along path $C_2$ with $L_1 \ne L_2$, the limit does not exist.

• Case Evaluations:

  • Function $f(x, y) = \frac{x^2}{x^2 + y^2}$ at $(0, 0)$:

  • Along $y = 0$ ($x \ne 0$): $f(x, 0) = \frac{x^2}{x^2 + 0} = 1 \implies \lim_{x \to 0} f(x, 0) = 1$.

• Along $x = 0$ ($y \ne 0$): $f(0, y) = \frac{0}{0 + y^2} = 0 \implies \lim_{y \to 0} f(0, y) = 0$.

• Since $1 \ne 0$, the limit does not exist.

• Function $f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}$ at $(0, 0)$:

  • Along $y = 0$: $f(x, 0) = 1$.

  • Along $x = 0$: $f(0, y) = -1$.

  • Since $1 \ne -1$, the limit does not exist.

• Function $f(x, y) = \frac{xy}{x^2 + y^2}$ at $(0, 0)$:

  • Along coordinate axes ($x = 0$ or $y = 0$): $f(x, y) = 0$.

  • Along $y = x$ ($x \ne 0$): $f(x, x) = \frac{x^2}{2x^2} = \frac{1}{2}$.

• Since $0 \ne \frac{1}{2}$, the limit does not exist.

• Techniques for Establishing Multivariate Limits:

  • Polar Coordinate Substitution:

  • Evaluate $\lim_{(x, y) \to (0, 0)} \frac{3x^2 y}{x^2 + y^2}$.

  • Set $x = r\cos(\theta)$, $y = r\sin(\theta)$, with $r^2 = x^2 + y^2$:       $f(r\cos(\theta), r\sin(\theta)) = \frac{3(r^2 \cos^2(\theta))(r\sin(\theta))}{r^2} = 3r\cos^2(\theta)\sin(\theta)$

  • Since $|\cos^2(\theta)\sin(\theta)| \le 1$, we have:       $|f(x, y) - 0| \le 3r = 3\sqrt{x^2 + y^2}$

  • For any $\epsilon > 0$, select $\delta = \frac{\epsilon}{3}$. Then $0 < \sqrt{x^2 + y^2} < \delta \implies |f(x, y) - 0| < 3\left(\frac{\epsilon}{3}\right) = \epsilon$.

  • Hence, the limit exists and equals $0$.

  • Algebraic Factorisation:

  • Evaluate $\lim_{(x, y) \to (0, 0)} \frac{x^3 - y^3}{x^2 + xy + y^2}$.

  • Factorise numerator: $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$.

  • For $(x, y) \ne (0, 0)$, cancel common factors:       $\lim_{(x, y) \to (0, 0)} \frac{(x - y)(x^2 + xy + y^2)}{x^2 + xy + y^2} = \lim_{(x, y) \to (0, 0)} (x - y) = 0 - 0 = 0$

• Multivariate Continuity and Properties:

  • A function $f: D \subset \mathbb{R}^2 \to \mathbb{R}$ is continuous at $(a, b) \in D$ if:     $\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$

  • All polynomial functions of $n$ variables are continuous across all of $\mathbb{R}^n$.

  • All rational functions are continuous everywhere across their defined domains.

Differential Calculus of Several Variables: Tangents, Gradients, and Chain Rules

• Partial Derivatives:

  • The first partial derivatives of $f(x, y)$ at $(a, b)$ represent the instantaneous rates of change along cross-sections where the other coordinate is held constant:     $f_x(a, b) = \frac{\partial f}{\partial x}(a, b) = \lim_{h \to 0} \frac{f(a + h, b) - f(a, b)}{h}$     $f_y(a, b) = \frac{\partial f}{\partial y}(a, b) = \lim_{h \to 0} \frac{f(a, b + h) - f(a, b)}{h}$

  • Higher-Order Partial Derivatives and Clairaut's Theorem:

  • Second partial derivatives are denoted $f_{xx} = \frac{\partial^2 f}{\partial x^2}$, $f_{yy} = \frac{\partial^2 f}{\partial y^2}$, $f_{xy} = \frac{\partial^2 f}{\partial y \partial x}$, $f_{yx} = \frac{\partial^2 f}{\partial x \partial y}$.

  • Clairaut's Theorem: If the mixed second partial derivatives $f_{xy}$ and $f_{yx}$ are continuous on an open disc containing $(a, b)$, then $f_{xy}(a, b) = f_{yx}(a, b)$.

  • Example: For $f(x, y) = x\sin(y) + y\cos(x)$, we find:       $f_x = \sin(y) - y\sin(x), \quad f_y = x\cos(y) + \cos(x)$       $f_{xy} = \cos(y) - \sin(x), \quad f_{yx} = \cos(y) - \sin(x) \implies f_{xy} = f_{yx}$       $f_{xx} = -y\cos(x), \quad f_{yy} = -x\sin(y)$

  • Pathology of Partial Derivatives:

  • The existence of partial derivatives $f_x(a, b)$ and $f_y(a, b)$ does not imply continuity at $(a, b)$.

  • For $f(x, y) = \frac{xy}{x^2 + y^2}$ ($(x, y) \ne (0, 0)$) and $f(0, 0) = 0$:       $f_x(0, 0) = \lim_{h \to 0} \frac{f(h, 0) - f(0, 0)}{h} = 0, \quad f_y(0, 0) = \lim_{h \to 0} \frac{f(0, h) - f(0, 0)}{h} = 0$       Both partial derivatives exist at the origin, but the function itself is discontinuous at $(0, 0)$.

• The Tangent Plane and Differentiability:

  • Equation of the Tangent Plane: For a smooth surface $z = f(x, y)$ at $(a, b, f(a, b))$, the tangent plane has equation:     $z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$

  • Vector parametric representation:     $\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a \\ b \\ f(a, b) \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ f_x(a, b) \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 1 \\ f_y(a, b) \end{pmatrix}, \quad \lambda, \mu \in \mathbb{R}$

  • Definition of Multivariable Differentiability: Let $V \subset \mathbb{R}^n$. A function $f: V \to \mathbb{R}$ is differentiable at $\mathbf{a}$ if there exists a vector $\nabla f(\mathbf{a}) \in \mathbb{R}^n$ such that:     $\lim_{\mathbf{h} \to \mathbf{0}} \frac{f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) - \nabla f(\mathbf{a}) \cdot \mathbf{h}}{\|\mathbf{h}\|} = 0$

  • Theorems on Differentiability:

  • If $f$ is differentiable at a point, it is continuous at that point.

  • If all first-order partial derivatives of $f$ exist and are continuous in a neighbourhood of $\mathbf{a}$, then $f$ is differentiable at $\mathbf{a}$.

• Linear Approximations, Total Differentials, and Error Estimation:

  • The linear (first-order) approximation near $(a, b)$ is:     $f(x, y) \approx f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$

  • The total differential of $z = f(x, y)$ is:     $dz = df = f_x(a, b)\,dx + f_y(a, b)\,dy$

  • If independent variables $x, y$ have maximum absolute errors $|\Delta x| \le E_1$ and $|\Delta y| \le E_2$, the estimated worst-case error in $f$ is:     $|E| \approx |f_x(a, b)E_1| + |f_y(a, b)E_2|$

  • Example (Cylindrical Barrel): Volume $V(r, h) = \pi r^2 h$. Nominal dimensions $r = 1\,m$, $h = 2\,m$. Nominal volume $V(1, 2) = 2\pi\,m^3$. Partial derivatives: $V_r = 2\pi r h \implies V_r(1, 2) = 4\pi$; $V_h = \pi r^2 \implies V_h(1, 2) = \pi$.

  • Given a $5\%$ error in radius ($E_r = 0.05\,m$) and height ($E_h = 0.1\,m$):     $|E| \approx |4\pi(0.05)| + |\pi(0.1)| = 0.2\pi + 0.1\pi = 0.3\pi\,m^3$     Relative percentage error: $\frac{|E|}{V(1, 2)} \times 100\% = \frac{0.3\pi}{2\pi} \times 100\% = 15\%$.

• Directional Derivatives and the Gradient Vector:

  • Directional Derivative: For a unit vector $\mathbf{u}$ ($\|\mathbf{u}\| = 1$), the directional derivative of $f$ at $\mathbf{a}$ in direction $\mathbf{u}$ is:     $D_{\mathbf{u}} f(\mathbf{a}) = f_{\mathbf{u}}(\mathbf{a}) = \lim_{t \to 0} \frac{f(\mathbf{a} + t\mathbf{u}) - f(\mathbf{a})}{t}$

  • When $f$ is differentiable:     $D_{\mathbf{u}} f(\mathbf{a}) = \nabla f(\mathbf{a}) \cdot \mathbf{u}$

• For an arbitrary non-zero direction vector $\mathbf{v}$:   $D_{\mathbf{v}} f(\mathbf{a}) = \nabla f(\mathbf{a}) \cdot \frac{\mathbf{v}}{\|\mathbf{v}\|}$

• Properties of the Gradient Vector $\nabla f$:

• $\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n}\right)$.

• By $D_{\mathbf{u}} f = \|\nabla f\| \|\mathbf{u}\| \cos(\theta) = \|\nabla f\| \cos(\theta)$, the directional derivative is maximized when $\theta = 0$ ($\mathbf{u}$ parallel to $\nabla f$). Hence, $\nabla f(\mathbf{a})$ points in the direction of maximal increase of $f$, with maximum rate of change equal to $\|\nabla f(\mathbf{a})\|$.

• The directional derivative is minimized when $\theta = \pi$, pointing in the direction of steepest descent ($-\nabla f(\mathbf{a})$) with rate $-\|\nabla f(\mathbf{a})\|$.

• When $\theta = \frac{\pi}{2}$, $D_{\mathbf{u}} f = 0$. Thus, $\nabla f(\mathbf{a})$ is orthogonal to the level curve (or level surface) of $f$ passing through $\mathbf{a}$.

• Multivariable Chain Rules and Implicit Differentiation:

  • Chain Rule: Let $z = f(x_1, \dots, x_n)$ where each $x_i = x_i(t)$ is differentiable. Then:     $\frac{df}{dt} = \sum_{i=1}^n \frac{\partial f}{\partial x_i} \frac{dx_i}{dt} = \nabla f \cdot \frac{d\mathbf{x}}{dt}$

• If each $x_i$ depends on multiple variables $(s, t)$, then:   $\frac{\partial z}{\partial t} = \sum_{i=1}^n \frac{\partial z}{\partial x_i} \frac{\partial x_i}{\partial t}$

• Implicit Differentiation: Let a plane curve be defined implicitly by $F(x, y) = 0$. Differentiating with respect to $x$ via the chain rule:   $\frac{\partial F}{\partial x} \frac{dx}{dx} + \frac{\partial F}{\partial y} \frac{dy}{dx} = 0 \implies F_x + F_y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{F_x}{F_y} \quad (F_y \ne 0)$

• Example (Folium of Descartes): Curve $x^3 + y^3 - 6xy = 0$. Define $F(x, y) = x^3 + y^3 - 6xy$. Then $F_x = 3x^2 - 6y$, $F_y = 3y^2 - 6x$. Thus:   $\frac{dy}{dx} = -\frac{3x^2 - 6y}{3y^2 - 6x} = -\frac{x^2 - 2y}{y^2 - 2x}$

• Implicit Function Theorem: Let $U \subset \mathbb{R}^2$ be an open set and $f: U \to \mathbb{R}$ be continuously differentiable. Let $(x_0, y_0) \in U$ satisfy $f(x_0, y_0) = 0$ and $\frac{\partial f}{\partial y}(x_0, y_0) \ne 0$. Then there exists an interval $I = (x_0 - r, x_0 + r)$ and a unique continuously differentiable function $g: I \to \mathbb{R}$ such that $g(x_0) = y_0$ and $f(x, g(x)) = 0$ for all $x \in I$.

Multivariable Optimisation, Critical Points, and Constrained Extrema

• Quadratic Approximations:

  • The second-order Taylor approximation of $f(x, y)$ about $(a, b)$ is:     $Q(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) + \frac{1}{2}f_{xx}(a, b)(x - a)^2 + f_{xy}(a, b)(x - a)(y - b) + \frac{1}{2}f_{yy}(a, b)(y - b)^2$

• Classification of Critical Points and the Second Derivative Test:

  • A point $(a, b)$ is a critical point of $f(x, y)$ if $\nabla f(a, b) = (0, 0)$ or if $\nabla f(a, b)$ is undefined.

• The Second Derivative Test: Let $f(x, y)$ have continuous second partial derivatives in a neighbourhood of a critical point $(a, b)$ where $\nabla f(a, b) = (0, 0)$. Define the Hessian determinant (discriminant):     $D = \det \begin{pmatrix} f_{xx}(a, b) & f_{xy}(a, b) \\ f_{yx}(a, b) & f_{yy}(a, b) \end{pmatrix} = f_{xx}(a, b)f_{yy}(a, b) - [f_{xy}(a, b)]^2$

  • If $D > 0$ and $f_{xx}(a, b) > 0$, then $f$ has a local minimum at $(a, b)$.

  • If $D > 0$ and $f_{xx}(a, b) < 0$, then $f$ has a local maximum at $(a, b)$.

  • If $D < 0$, then $f$ has a saddle point at $(a, b)$.

  • If $D = 0$, the test is inconclusive.

• Global Extrema on Closed and Bounded Sets:

  • By the Extreme Value Theorem, a continuous function $f$ on a closed and bounded (compact) set $D \subset \mathbb{R}^2$ attains an absolute maximum and an absolute minimum.

  • Procedure: (1) Find values of $f$ at all critical points inside $D$; (2) Find extreme values of $f$ on the boundary of $D$ using single-variable calculus or Lagrange multipliers; (3) The largest and smallest values from these steps are the absolute maximum and minimum.

• Constrained Optimisation via Lagrange Multipliers:

  • Single Constraint: To maximize or minimize $f(\mathbf{x})$ subject to $g(\mathbf{x}) = 0$, where $\nabla g \ne \mathbf{0}$, solve:     $\nabla f(\mathbf{x}) = \lambda \nabla g(\mathbf{x}), \quad g(\mathbf{x}) = 0$     where $\lambda$ is the Lagrange multiplier. Geometrically, extrema occur where level sets of $f$ are tangent to the constraint manifold $g = 0$.

• Multiple Constraints: To optimize $f(x_1, \dots, x_L)$ subject to $N$ constraints $g_j(x_1, \dots, x_L) = 0$ ($j = 1, \dots, N$ with $N < L$):   $\nabla f = \sum_{j=1}^N \lambda_j \nabla g_j, \quad g_j = 0 \quad (j = 1, \dots, N)$     This is equivalent to rendering stationary the Lagrangian function:     $\mathcal{L}(x_1, \dots, x_L, \lambda_1, \dots, \lambda_N) = f(x_1, \dots, x_L) - \sum_{j=1}^N \lambda_j g_j(x_1, \dots, x_L)$

  • For inequality constraints $g(\mathbf{x}) \le c$, the formulation generalises to the Karush-Kuhn-Tucker (KKT) conditions.

• Envelopes and Economic Applications:

  • Envelope of a Family of Curves: Given a family of curves $y^{(t)}(x) = f(x, t)$ parameterized by $t$, the envelope is a curve $y = E(x)$ tangent to each member of the family.

  • The envelope satisfies the dual conditions: $f_t(x, t) = 0 \implies t = g(x)$, giving the envelope as $E(x) = f(x, g(x))$.

  • Economic Profit Maximisation (Hotelling's Lemma):

  • Let $f(x)$ be a production function ($f(0) = 0$, $f(x) \ge 0$, $f'(x) > 0$, $f''(x) < 0$) yielding output for input $x$.

  • Selling price is $p$, unit input cost is $c$. Profit is $P(p, x) = pf(x) - cx$.

  • Optimal input $x = g(p)$ satisfies $P_x = pf'(x) - c = 0 \implies f'(x) = \frac{c}{p}$.

  • The maximized profit function $P_{\max}(p) = pf(g(p)) - cg(p)$ represents the envelope of the family of profit functions with respect to $x$, possessing the envelope property:       $\frac{dP_{\max}}{dp} = f(g(p))$

  • Example: For $f(x) = 4\sqrt{x}$, $f'(x) = \frac{2}{\sqrt{x}} = \frac{c}{p} \implies x = g(p) = \frac{4p^2}{c^2}$.       Maximized profit: $P_{\max}(p) = p\left(4 \cdot \frac{2p}{c}\right) - c\left(\frac{4p^2}{c^2}\right) = \frac{4p^2}{c}$.       For $c = 4$, $P_{\max}(p) = p^2$, with $\frac{dP_{\max}}{dp} = 2p = f(g(p))$.

Line Integrals, Vector Fields, and Conservative Systems

• Arc Length and Scalar Line Integrals:

  • The arc length of a smooth curve $C$ parameterized by $\mathbf{r}(t) = (x(t), y(t), z(t))$ for $a \le t \le b$ is:     $S = \int_C ds = \int_a^b \|\mathbf{r}'(t)\|\,dt = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}\,dt$

  • In polar coordinates $r = f(\theta)$ for $\alpha \le \theta \le \beta$:     $S = \int_\alpha^\beta \sqrt{[f(\theta)]^2 + [f'(\theta)]^2}\,d\theta$

  • The scalar line integral of $f(x, y)$ along $C$ represents the area of a vertical ribbon of height $f$ above $C$:     $\int_C f(x, y)\,ds = \int_a^b f(x(t), y(t))\|\mathbf{r}'(t)\|\,dt$

• Vector Line Integrals and Work:

  • The work done by a continuous vector force field $\mathbf{F} = F_1\mathbf{i} + F_2\mathbf{j} + F_3\mathbf{k}$ moving a particle along curve $C$ is:     $W = \int_C \mathbf{F} \cdot d\mathbf{r} = \int_C \mathbf{F} \cdot \mathbf{T}\,ds = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt = \int_C F_1\,dx + F_2\,dy + F_3\,dz$

  • Reversing path orientation inverts the sign: $\int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_C \mathbf{F} \cdot d\mathbf{r}$.

• Conservative Vector Fields and Potential Theory:

  • A vector field $\mathbf{F}$ is conservative if $\int_C \mathbf{F} \cdot d\mathbf{r}$ is independent of path, depending solely on the endpoints $A$ and $B$.

  • Fundamental Theorem for Line Integrals: A vector field $\mathbf{F}$ is conservative if and only if it is a gradient field, meaning there exists a scalar potential function $f$ such that $\mathbf{F} = \nabla f$. In this case:     $\int_A^B \mathbf{F} \cdot d\mathbf{r} = \int_A^B \nabla f \cdot d\mathbf{r} = f(B) - f(A)$

  • Component Criterion: On a simply connected domain in $\mathbb{R}^2$, $\mathbf{F} = (F_1, F_2)$ is conservative if and only if:     $\frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x}$

• Conservation of Energy and Central Forces in 3D Space:

  • Under a conservative force $\mathbf{F} = -\nabla V(\mathbf{r})$, Newton's second law reads $m\ddot{\mathbf{r}} = -\nabla V(\mathbf{r})$.

• Taking the inner product with $\dot{\mathbf{r}}$, the multivariable chain rule yields:   $m\ddot{\mathbf{r}} \cdot \dot{\mathbf{r}} = -\nabla V(\mathbf{r}) \cdot \dot{\mathbf{r}} \implies \frac{d}{dt}\left(\frac{1}{2}m\|\dot{\mathbf{r}}\|^2\right) = -\frac{dV}{dt} \implies \frac{d}{dt}\left(\frac{1}{2}m\|\dot{\mathbf{r}}\|^2 + V(\mathbf{r})\right) = 0$

• Hence, total mechanical energy is conserved: $E = \frac{1}{2}m\|\dot{\mathbf{r}}\|^2 + V(\mathbf{r})$.

• Central Force Dynamics:

  • A force is central if $\mathbf{F}(\mathbf{r}) = F(r)\hat{\mathbf{r}}$, where $r = \|\mathbf{r}\|$ and $\hat{\mathbf{r}} = \frac{\mathbf{r}}{r}$.

  • Every continuous central force field is conservative, with potential $V(r) = -\int F(r)\,dr$.

  • Angular Momentum: $\mathbf{L} = m(\mathbf{r} \times \mathbf{v})$.

  • Differentiating with respect to time yields torque: $\frac{d\mathbf{L}}{dt} = m(\dot{\mathbf{r}} \times \mathbf{v}) + m(\mathbf{r} \times \dot{\mathbf{v}}) = \mathbf{0} + \mathbf{r} \times \mathbf{F}$.

  • For central forces, $\mathbf{r} \times \mathbf{F} = \mathbf{r} \times (F(r)\hat{\mathbf{r}}) = \mathbf{0}$, proving that angular momentum $\mathbf{L}$ is strictly conserved.

  • Planar Motion: Since $\mathbf{r}(t) \cdot \mathbf{L} = \mathbf{r}(t) \cdot [m(\mathbf{r}(t) \times \mathbf{v}(t))] = 0$, the position vector $\mathbf{r}(t)$ must remain orthogonal to the constant vector $\mathbf{L}$, confining the motion strictly to a two-dimensional plane.

• Kepler's First Law: In an inverse-square gravitational field $F(r) = -\frac{GMm}{r^2}$, the planar trajectory is a conic section:   $r = \frac{l}{1 + e\cos(\theta)}$   where $e$ is the eccentricity and $l$ is the semi-latus rectum. Bound orbits (planets) have $0 \le e < 1$ (ellipses/circles with maximum distance apoapsis), whereas unbound trajectories have $e \ge 1$ (parabolas/hyperbolas).