Slope in Any Direction
A partial derivative gives the slope in the \(x\) or \(y\) direction. But what if you want the slope in the direction of a unit vector \(\vec{u}\)? This is the Directional Derivative, denoted \(D_u f\).
\[D_u f = \nabla f \cdot \vec{u}\]
It is the dot product of the gradient and the direction vector.
Worked Examples
Example 1: Calculating Slope
Find the directional derivative of \(f(x, y) = x^2 y\) at \((1, 2)\) in the direction of \(\vec{v} = \langle 3, 4 \rangle\).
- Step 1: Normalize \(\vec{v}\). \(|\vec{v}| = \sqrt{3^2 + 4^2} = 5\). So \(\vec{u} = \langle 0.6, 0.8 \rangle\).
- Step 2: Find the gradient. \(\nabla f = \langle 2xy, x^2 \rangle \implies \langle 4, 1 \rangle\).
- Step 3: Dot product. \(D_u f = \langle 4, 1 \rangle \cdot \langle 0.6, 0.8 \rangle = 2.4 + 0.8 = 3.2\).
- Result: \(3.2\).
The Bridge to Quantum Mechanics
Directional derivatives are used to describe Symmetry Operations. If you rotate a quantum system, you are essentially looking at the directional derivative of the wavefunction along an arc. The operators for Angular Momentum (\(\hat{L}_x, \hat{L}_y, \hat{L}_z\)) are built from directional derivatives. They tell us how the wavefunction changes as we "look" at the particle from different angles.