The chain rule for multivariable functions extends the single-variable concept to handle complex nested functions with multiple variables. It's a powerful tool for analyzing interdependent variables, like temperature changes in heat exchangers or planetary motion in space.

Directional derivatives and gradients help us understand how functions change in specific directions. This is crucial for optimization problems, like finding the steepest path up a mountain or maximizing profit in economics. The gradient points towards the steepest increase, while directional derivatives measure change in any direction.

Chain Rule for Multivariable Functions

Chain rule for composite functions

• Extends single-variable chain rule to multiple variables allowing differentiation of complex nested functions
• General form: $\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$ computes total derivative by summing partial derivatives
• Applies to functions like $f(g(t), h(t))$ where $g$ and $h$ are functions of $t$
• Crucial for analyzing interdependent variables (temperature changes in a heat exchanger)

Calculation of directional derivatives

• Identify outer and inner functions in composite function $f(g(x,y), h(x,y))$
• Calculate partial derivatives of outer function $\frac{\partial f}{\partial u}$ and $\frac{\partial f}{\partial v}$
• Compute derivatives of inner functions $\frac{\partial g}{\partial x}$, $\frac{\partial g}{\partial y}$, $\frac{\partial h}{\partial x}$, $\frac{\partial h}{\partial y}$
• Multiply and sum results: $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial g}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial h}{\partial x}$
• Repeat for $\frac{\partial f}{\partial y}$ to get complete gradient

Direction of steepest ascent/descent

• Chain rule enables calculation of rates of change in complex systems (planetary motion)
• Analyzes motion in multiple dimensions (projectile trajectories)
• Optimizes multivariable functions in machine learning algorithms (gradient descent)
• Models heat transfer in engineering applications (thermal conductivity in materials)

Gradient vs directional derivatives

• Gradient $\nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \rangle$ represents vector of partial derivatives
• Points in direction of steepest ascent with magnitude equal to maximum rate of change
• Directional derivative $D_\mathbf{u}f = \nabla f \cdot \mathbf{u}$ projects gradient onto unit vector $\mathbf{u}$
• Maximum directional derivative occurs when $\mathbf{u}$ aligns with $\nabla f$
• Zero directional derivative when $\mathbf{u}$ perpendicular to $\nabla f$ (tangent to level curves)

Directional Derivatives and Gradients

Calculation of directional derivatives

• Measures rate of change of function $f(x,y)$ in specific direction $\mathbf{u}$
• Formula: $D_\mathbf{u}f(x,y) = \nabla f \cdot \mathbf{u}$ combines gradient and direction vector
• Unit vector $\mathbf{u} = \langle a, b \rangle$ where $a^2 + b^2 = 1$ represents desired direction
• Process:
  1. Compute gradient $\nabla f$
  2. Determine unit vector $\mathbf{u}$
  3. Calculate dot product $\nabla f \cdot \mathbf{u}$
• Interpretation:
  • Positive: function increases in given direction
  • Negative: function decreases in given direction
  • Zero: function constant in given direction (level curve)

Direction of steepest ascent/descent

• Steepest ascent: direction of maximum increase aligns with gradient vector
• Steepest descent: direction of maximum decrease opposes gradient vector
• Calculation:
  1. Compute gradient $\nabla f$ at given point
  2. Normalize gradient to obtain unit vector
• Applications:
  • Optimization in machine learning (gradient descent algorithms)
  • Path finding in computer graphics (terrain navigation)
  • Meteorology (pressure gradients in weather systems)
  • Economics (optimizing profit functions)

Gradient vs directional derivatives

• Gradient $\nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \rangle$ encapsulates all directional derivatives
• Properties:
  • Points toward steepest ascent
  • Magnitude equals maximum rate of change
• Directional derivative as projection: $D_\mathbf{u}f = \nabla f \cdot \mathbf{u} = |\nabla f| \cos \theta$
• Maximum directional derivative when $\mathbf{u}$ parallel to $\nabla f$
• Zero directional derivative when $\mathbf{u}$ perpendicular to $\nabla f$
• Gradient always perpendicular to level curves or surfaces (contour lines in topographic maps)