Calc-3 Discussion 05

Discussion Topics covered: Multivariable functions, domain and range in 2D and 3D, open/closed and bounded/unbounded boundary classification, level curves and contour maps, level surfaces (isosurfaces), vertical cross-sectional traces, first-order partial derivatives, formal limit definitions of partials, cardinal trace tangent slopes, 3D parametric tangent lines, visual estimation from contour maps and sensor grids, equations of planes in 3D, tangent line approximations and differentials, single-variable chain rule, implicit differentiation, 2x2 matrix determinants, Test 1 scope review

Part A: Review This Week's Lecture Topics (~30 mins)

Multivariable Functions, Algebraic Domains, & Boundary Topologies

• Evaluating two-variable scalar functions $z = f(x, y)$ and three-variable functions $w = f(x, y, z)$ at specified coordinate inputs, understanding higher-dimensional domain-to-range mappings ($f: \mathbb{R}^2 \to \mathbb{R}$ and $f: \mathbb{R}^3 \to \mathbb{R}$)
• Formulating algebraic domains in $\mathbb{R}^2$ by enforcing mathematical constraints:
  • Even Radicals ($\sqrt{u}$): Enforcing non-negativity $u(x, y) \ge 0$
  • Fractions ($\frac{P}{Q}$): Excluding denominator roots $Q(x, y) \ne 0$
  • Radical Denominators ($\frac{1}{\sqrt{u}}$): Enforcing strict positivity $u(x, y) > 0$
  • Logarithms ($\ln(u)$): Enforcing strictly positive arguments $u(x, y) > 0$
• Classifying 2D domain regions in the $xy$-plane using topological criteria:
  • Open vs. Closed: An open region excludes all boundary points (strict inequalities $<, >$, drawn with dashed boundary curves), whereas a closed region contains all boundary points (non-strict inequalities $\le, \ge$, drawn with solid boundary curves)
  • Bounded vs. Unbounded: A bounded domain can be entirely enclosed within a circle of finite radius centered at the origin, whereas an unbounded domain extends infinitely across the plane
• Extending domain analysis to 3D space $\mathbb{R}^3$: describing domains of $w = f(x, y, z)$ as solid spatial regions (e.g., the interior of a sphere $x^2 + y^2 + z^2 \le R^2$, solid cylinders, or open half-spaces) and identifying excluded boundary surfaces
• Determining mathematical range in interval notation by evaluating extremal behavior of the radicand, quadratic sum, or denominator across the domain boundary and interior

Topographical Slicing: Level Curves, Level Surfaces, & Vertical Traces

• Constructing 2D level curves (contour lines) by setting the dependent output equal to constant elevation slices: $f(x, y) = c$
• Reading contour maps like topographical terrain surveys: recognizing that closely spaced contour curves signify steep vertical gradients, widely spaced curves indicate gentle slopes, and concentric closed loops identify local peaks, pits, or saddles
• Analyzing 3D level surfaces (isosurfaces) $f(x, y, z) = c$ for four-dimensional scalar fields, mapping physical engineering phenomena such as isothermal shells in heat transfer, isobaric pressure envelopes in aerodynamics, and equipotential surfaces in electrostatics
• Generating vertical cross-sectional traces by slicing a surface $z = f(x, y)$ with coordinate planes ($x = k$ for $yz$-traces, $y = k$ for $xz$-traces, and $z = k$ for horizontal traces) to reconstruct 3D quadric surfaces (elliptic paraboloids, hyperbolic paraboloid saddles, ellipsoids, and cones)

Mechanics of Partial Differentiation & Constant-Variable Isolation

• Differentiating multivariable functions using the fundamental constant-variable rule:
  • When computing $\frac{\partial f}{\partial x}$ (or $f_x$), treat $y$ and $z$ as inert constant scalars
  • When computing $\frac{\partial f}{\partial y}$ (or $f_y$), treat $x$ and $z$ as inert constant scalars
  • When computing $\frac{\partial f}{\partial z}$ (or $f_z$), treat $x$ and $y$ as inert constant scalars
• Deploying single-variable Power, Product, Quotient, and Chain Rules to multivariable combinations, strictly distinguishing between isolated variable terms, additive constant offsets (which differentiate to zero), and multiplicative constant coefficients (which carry through unchanged)
• Formulating and evaluating first-order partial derivatives from the formal limit definition:
  $$\frac{\partial f}{\partial x}(x, y) = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}, \qquad \frac{\partial f}{\partial y}(x, y) = \lim_{h \to 0} \frac{f(x, y+h) - f(x, y)}{h}$$
• Executing partial differentiation across multi-factor rational, exponential, and trigonometric functions without dropping internal chain rule factors

Geometric Meaning: Cardinal Trace Tangent Slopes & 3D Tangent Lines

• Interpreting partial derivatives geometrically:
  • $f_x(x_0, y_0)$ represents the instantaneous slope of the tangent line $T_1$ to the vertical trace curve formed by intersecting the surface $z = f(x, y)$ with the plane $y = y_0$ (traversal purely in the $+x$ cardinal direction)
  • $f_y(x_0, y_0)$ represents the instantaneous slope of the tangent line $T_2$ to the vertical trace curve formed by intersecting the surface with the plane $x = x_0$ (traversal purely in the $+y$ cardinal direction)
• Formulating 3D vector and parametric equations of the trace tangent lines at base point $P_0(x_0, y_0, z_0)$:
  $$T_1: \vec{r}_1(t) = \langle x_0 + t, \, y_0, \, z_0 + f_x(x_0, y_0)t \rangle \implies \vec{v}_1 = \langle 1, 0, f_x(x_0, y_0) \rangle$$
  $$T_2: \vec{r}_2(t) = \langle x_0, \, y_0 + t, \, z_0 + f_y(x_0, y_0)t \rangle \implies \vec{v}_2 = \langle 0, 1, f_y(x_0, y_0) \rangle$$
• Connecting trace tangent direction vectors to surface geometry: taking the cross product $\vec{v}_1 \times \vec{v}_2 = \langle -f_x(x_0, y_0), -f_y(x_0, y_0), 1 \rangle$ (or downward $\langle f_x, f_y, -1 \rangle$) to define the normal vector perpendicular to the local tangent plane

Visual Estimation from Contour Maps & Tabular Sensor Grids

• Reading directional slope signs directly from contour maps:
  • Stepping in the $+x$ direction (due East) across increasing contour values signifies $f_x > 0$; crossing into decreasing contour values signifies $f_x < 0$
  • Stepping in the $+y$ direction (due North) across increasing contour values signifies $f_y > 0$; crossing into decreasing contour values signifies $f_y < 0$
• Identifying contour tangency: when a path moving strictly parallel to the $x$-axis or $y$-axis is momentarily tangent to a level curve, the corresponding partial derivative is zero ($f_x = 0$ or $f_y = 0$)
• Approximating numerical partial derivatives from contour maps and discrete tabular sensor data using two-point difference quotients:
  $$f_x(x_0, y_0) \approx \frac{\Delta z}{\Delta x} = \frac{f(x_0 + \Delta x, y_0) - f(x_0, y_0)}{\Delta x}$$
  $$f_y(x_0, y_0) \approx \frac{\Delta z}{\Delta y} = \frac{f(x_0, y_0 + \Delta y) - f(x_0, y_0)}{\Delta y}$$

Part B: Next Week's Prerequisite Refresher (~30 mins)

Review the following prerequisite skills required for Tangent Planes & Linear Approximations (Session 10) and Chain Rule & Higher Derivatives (Session 11):

• Equations of Planes in 3D (Point-Normal & General Linear Forms)
• Tangent Lines & Single-Variable Linear Approximations ($L(x)$ & Differentials)
• The Single-Variable Chain Rule & Nested Composite Rates
• Implicit Differentiation & Constraint Sensitivity
• Matrix Determinants ($2 \times 2$ Evaluation for Hessian Formulations)

Part C: General Q/A

Open the floor to questions from students regarding any topic