Calc-3 Discussion 07

Discussion Topics covered: The gradient vector in 2D and 3D, differentiate-first rule, geometric orthogonality to level curves and level surfaces, tangent planes to implicit surfaces, directional derivatives along arbitrary vectors, unit vector normalization, bounds on rates of change (steepest ascent, steepest descent, zero change), unconstrained critical points, Hessian matrix and discriminant, local extrema and saddle point classification, Extreme Value Theorem on closed and bounded domains, 3-tier boundary search, Lagrange multipliers, geometric tangency, constraint sensitivity and shadow pricing, single-variable definite integrals, u-substitution, area between curves and planar slicing, integration by parts

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

The Gradient Vector ($\nabla f$) & Surface Normal Geometry

• Formulating general gradient vectors for two-variable functions $\nabla f(x, y) = \langle f_x, f_y \rangle = f_x\hat{i} + f_y\hat{j}$ and three-variable functions $\nabla f(x, y, z) = \langle f_x, f_y, f_z \rangle = f_x\hat{i} + f_y\hat{j} + f_z\hat{k}$
• Enforcing the differentiate-first rule: always compute symbolic partial derivatives across the entire domain before substituting specific numerical coordinates $(x_0, y_0, z_0)$ to avoid collapsing variable dependencies prematurely
• Interpreting the geometric orthogonality of the gradient:
  • In 2D, $\nabla f(x_0, y_0)$ is strictly perpendicular to the tangent vector of the level curve $f(x, y) = c$ passing through $(x_0, y_0)$
  • In 3D, $\nabla F(x_0, y_0, z_0)$ acts directly as the surface normal vector $\vec{n}$ orthogonal to the tangent plane of the implicit level surface $F(x, y, z) = c$
• Constructing tangent planes to implicit surfaces $F(x, y, z) = c$ using the gradient normal $\vec{n} = \langle F_x, F_y, F_z \rangle$:
  $$F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0$$

Directional Derivatives ($D_{\vec{u}} f$) & Extremal Rates of Change

• Computing the instantaneous rate of change of a scalar function along an arbitrary trajectory using the gradient dot product:
  $$D_{\vec{u}} f(x_0, y_0) = \nabla f(x_0, y_0) \cdot \vec{u} = \|\nabla f\|\cos\phi$$
• Strictly enforcing unit vector normalization: if given an arbitrary direction vector $\vec{v}$, always construct the unit direction vector $\vec{u} = \frac{\vec{v}}{\|\vec{v}\|}$; when given an angle $\theta$ measured counterclockwise from the positive $x$-axis, express $\vec{u} = \langle \cos\theta, \sin\theta \rangle$
• Determining the extremal bounds of directional change from the geometry of the dot product:
  • Maximum Rate of Increase (Steepest Ascent): Occurs when $\vec{u}$ is aligned with $\nabla f$ ($\phi = 0$). The direction is $\vec{u} = \frac{\nabla f}{\|\nabla f\|}$ and the maximum rate equals $\|\nabla f\|$
  • Maximum Rate of Decrease (Steepest Descent): Occurs when $\vec{u}$ points opposite to $\nabla f$ ($\phi = \pi$). The direction is $\vec{u} = -\frac{\nabla f}{\|\nabla f\|}$ and the minimum rate equals $-\|\nabla f\|$
  • Zero Instantaneous Change: Occurs when $\vec{u}$ is perpendicular to $\nabla f$ ($\phi = \pi/2 \implies \nabla f \cdot \vec{u} = 0$), corresponding to trajectories tangential to the local contour level curve

Unconstrained Local Extrema & The Second Derivative Test (Hessian Matrix)

• Locating candidate critical points $(a, b)$ across open domains where the gradient vanishes simultaneously:
  $$\nabla f(a, b) = \mathbf{0} \iff \frac{\partial f}{\partial x}(a, b) = 0 \quad \text{and} \quad \frac{\partial f}{\partial y}(a, b) = 0$$
  or where either first-order partial derivative fails to exist (points with horizontal tangent planes $z = f(a, b)$)
• Computing second-order partial derivatives ($f_{xx}, f_{yy}, f_{xy}$) and applying Clairaut's Theorem ($f_{xy} = f_{yx}$) for smooth surfaces to construct the symmetric Hessian matrix:
  $$H = \begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix}, \qquad D(a, b) = \det(H) = f_{xx}(a, b) f_{yy}(a, b) - \Big(f_{xy}(a, b)\Big)^2$$
• Rigorously classifying stationary points via the Second Derivative Test:
  • Local Minimum: $D(a, b) > 0$ and $f_{xx}(a, b) > 0$ (surface curves upward in all directions like a bowl)
  • Local Maximum: $D(a, b) > 0$ and $f_{xx}(a, b) < 0$ (surface curves downward in all directions like a dome)
  • Saddle Point: $D(a, b) < 0$ (principal curvatures have opposite signs; surface bends upward in one cross-section and downward in another)
  • Inconclusive Test: $D(a, b) = 0$ (higher-order analysis or algebraic inspection required)
• Verifying the same-sign condition: when $D > 0$, the inequality $f_{xx}f_{yy} > (f_{xy})^2 \ge 0$ guarantees that $f_{xx}$ and $f_{yy}$ always share the exact same algebraic sign

Global Extrema on Compact Domains & Constrained Optimization (Lagrange Multipliers)

• Applying the Extreme Value Theorem (EVT) on compact (closed and bounded) planar regions $D$: every continuous function $f(x, y)$ is guaranteed to achieve an absolute maximum and absolute minimum on $D$
• Executing the systematic 3-tier candidate search strategy:
  1. Tier 1 (Interior Critical Points): Solve $\nabla f = \mathbf{0}$ within the strict interior of $D$ and discard any roots lying outside the boundary
  2. Tier 2 (Boundary Edge Parameterization): Parameterize each perimeter segment as a 1D function (e.g., $x = g(t), y = h(t)$), find 1D critical points along the edges, or employ Lagrange multipliers on curved perimeter boundaries
  3. Tier 3 (Corner Vertices): Identify all structural intersections and corner points where boundary curves meet
  4. Synthesis: Evaluate $f(x, y)$ at all assembled candidate points to identify the largest (global maximum) and smallest (global minimum) values
• Formulating constrained optimization models using the Method of Lagrange Multipliers:
  • For an objective function $f(x, y)$ subject to a constraint $g(x, y) = k$, extreme values along the constraint occur where level sets of $f$ are tangent to the constraint curve:
    $$\nabla f = \lambda \nabla g \quad \text{and} \quad g(x, y) = k$$
  • Extending to 3D surfaces: $\nabla f(x, y, z) = \lambda \nabla g(x, y, z)$ subject to $g(x, y, z) = k$
  • Resolving non-linear coordinate systems by eliminating $\lambda$ or factoring proportional relationships between variables (avoiding division by zero)
  • Interpreting the Lagrange multiplier $\lambda$ as a quantitative metric of physical system sensitivity and marginal rate of return (shadow pricing): $\lambda = \frac{df_{\text{opt}}}{dk}$

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

Review the following prerequisite single-variable integration and planar geometry skills required for Double Integrals (Session 14):

• Definite Integrals & The Fundamental Theorem of Calculus
  • Evaluating single-variable definite integrals $\int_a^b f(x) \, dx = F(b) - F(a)$ using power, exponential, and trigonometric antiderivatives
  • Holding inactive variables strictly constant during partial integration (e.g., evaluating $\int_c^d f(x, y) \, dy$ while treating $x$ as an inert scalar)
• The Substitution Rule ($u$-Substitution for Definite Integrals)
  • Recognizing composite integrands where an outer factor is proportional to the inner derivative: $\int f(g(x))g'(x) \, dx = \int f(u) \, du$
  • Transforming upper and lower integration limits to the $u$-domain ($u(a)$ and $u(b)$) for clean evaluation in iterated integrals
• Area Between Curves & Planar Region Slicing (Type I & Type II Integration)
  • Partitioning 2D regions into vertical strips ($dx$: $y_{\text{top}} - y_{\text{bottom}}$) vs. horizontal strips ($dy$: $x_{\text{right}} - x_{\text{left}}$)
  • Determining outer constant integration limits by solving simultaneous boundary curve intersections
  • Establishing the geometric foundation for Type I ($g_1(x) \le y \le g_2(x)$) and Type II ($h_1(y) \le x \le h_2(y)$) double integral setups and order-switching protocols
• Integration by Parts (Single-Variable Mechanics for Iterated Integrals)
  • Applying $\int u \, dv = uv - \int v \, du$ using the LIATE selection hierarchy to evaluate algebraic-transcendental products (such as $\int x e^x \, dx$ and $\int x \cos x \, dx$) encountered in outer iterated integrals

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Part C: General Q/A

Open the floor to questions from students regarding any topic