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University Math Lab: Systems & Structures FREE
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Gradient & Directional Derivatives
Learning Objectives
- Calculate gradients
- Find directional derivatives
- Apply to optimization
Key Concepts
Gradient
∇f = ⟨∂f/∂x, ∂f/∂y, ∂f/∂z⟩
Directional Derivative
Rate of change in direction u
Theory
**Gradient Properties:**
- Points in direction of steepest increase
- Magnitude = max rate of change
- Perpendicular to level curves
**Directional Derivative:** D_u f = ∇f · u
Where u is a unit vector in the desired direction.
Examples
∇f for f(x,y) = x² + xy
Solution: ∇f = ⟨2x + y, x⟩