Overview
This lecture introduces differential calculus, focusing on ordinary derivatives, the concept of the gradient (directional derivative), and practical examples, especially gradients in three-dimensional space.
Ordinary Derivatives
- The derivative measures how a function changes with respect to its variable (domain).
- Mathematically, the average rate of change: Δf(x) / Δx = [f(x₂) - f(x₁)] / [x₂ - x₁].
- The slope is the rise (change in y) over run (change in x) between two points.
- As Δx → 0, the difference quotient becomes the derivative: df/dx = lim (Δx→0) [f(x₂) - f(x₁)] / [x₂ - x₁].
- The slope of a horizontal (x-axis) line is zero; the slope of a vertical (y-axis) line is undefined.
- Space derivatives (with respect to x, y, z) are called spatial derivatives; with respect to time, the derivative is called the temporal or time derivative.
Partial Derivatives and Spatial Variables
- For functions of multiple variables (e.g., t(x, y, z)), use partial derivatives: ∂t/∂x, ∂t/∂y, ∂t/∂z.
- The total differential: dt = (∂t/∂x)dx + (∂t/∂y)dy + (∂t/∂z)dz.
Gradient (Directional Derivative)
- The gradient (∇t or grad t) gives the direction and rate of maximum increase of a scalar function.
- Gradient is a vector operator represented by "del" (∇): ∇ = (∂/∂x)𝑥̂ + (∂/∂y)ŷ + (∂/∂z)ẑ.
- The differential can be rewritten: dt = (∇t) · d𝐫, where d𝐫 is a small displacement vector.
- Gradient points in the direction of steepest ascent for the function.
Example: Gradient of Position Magnitude
- Given r = √(x² + y² + z²), the gradient ∇r = 𝐫̂ (the unit vector in the radial direction).
- The gradient of a scalar function always yields a vector in the direction of greatest change.
Dot and Cross Products with Del Operator
- ∇t (gradient of a scalar) gives a vector.
- ∇·𝐯 (divergence) measures how much a vector field spreads out from a point (translational dynamics).
- ∇×𝐯 (curl) measures the rotation of a vector field (rotational dynamics).
Key Terms & Definitions
- Derivative — The rate of change of a function with respect to its variable.
- Slope — The ratio of vertical change to horizontal change between two points.
- Partial Derivative — Derivative of a multivariable function with respect to one variable, holding others constant.
- Gradient (∇) — Vector indicating direction and rate of fastest increase of a scalar function.
- Divergence — Measure of how much a vector field spreads out from a point.
- Curl — Measure of the rotational tendency of a vector field.
- Del Operator (∇) — Vector differential operator used to define gradient, divergence, and curl.
Action Items / Next Steps
- Solve problems 1.11, 1.12, and 1.13; focus especially on problem 1.12.
- Prepare for upcoming discussions on divergence, curl, and related theorems.