Vectors and Partial Derivatives

In single-variable calculus, we study functions like y = f(x) which represent curves. The derivative tells us the slope of the tangent line. But when we move to 3D space with functions like z = f(x, y), we get a surface. To understand how these surfaces change, we need to combine the geometry of vectors with the logic of derivatives.

1. Review: Vectors in Space

A vector is a quantity with both magnitude (length) and direction. In 3D space, we write vectors as v = <a, b, c>.

Vectors allow us to define "directions" on a surface. While basic calculus only lets us look left/right (x-axis), vectors let us look in any direction (e.g., "North-East" or 45°).

[Image of 3D vector diagram]

2. Partial Derivatives Recap

Before we can move in any direction, we must understand how to move in the two primary directions: X and Y.

• Partial Derivative wrt x (∂f/∂x): Imagine slicing the surface parallel to the x-axis. This is the slope if you walk purely East.
• Partial Derivative wrt y (∂f/∂y): Imagine slicing the surface parallel to the y-axis. This is the slope if you walk purely North.

3. The Gradient Vector (∇f)

The Gradient is the bridge between partial derivatives and vectors. It is a vector composed of the partial derivatives.

This vector has a special property: It always points in the direction of the steepest ascent. The magnitude of the gradient |∇f| tells you how steep that slope is.

[Image of gradient vector field on a contour map]

4. The Directional Derivative

This is where vectors and calculus truly merge. What if you want to find the slope of the mountain if you walk in a random direction given by a unit vector u = <a, b>?

We use the dot product of the Gradient and the direction vector.

Geometric Interpretation

The directional derivative Dᵤf tells you the rate of change of f (height) as you move in the direction of u.

• If the dot product is positive, you are walking uphill.
• If it is negative, you are walking downhill.
• If it is zero, you are walking along a contour line (staying at the same height).

[Image of directional derivative geometric interpretation]

5. Example Problem

Let f(x, y) = x² + y² (A paraboloid/bowl shape). Find the slope at point (1, 2) in the direction of vector v = <3, 4>.

Step 1: Find the Gradient
∂f/∂x = 2x, ∂f/∂y = 2y
∇f = <2x, 2y>
At point (1, 2), ∇f = <2, 4>

Step 2: Find Unit Vector u
Magnitude of v = √(3² + 4²) = 5.
Unit vector u = <3/5, 4/5>

Step 3: Dot Product
Dᵤf = <2, 4> · <3/5, 4/5>
= (2 * 3/5) + (4 * 4/5)
= 6/5 + 16/5 = 22/5 = 4.4

So, the slope in that direction is 4.4.