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The Directional Derivative

~12 mincode completion

The gradient tells you the steepest direction. The directional derivative tells you the slope along any direction you choose:

— the dot product of the gradient with a unit vector. The unit part is essential: without normalising, doubling the length of $u$ would double the answer, and you would be measuring your arrow rather than the surface.

Three consequences fall straight out of :

Along the gradient, and the slope is — the steepest possible. This is the proof that the gradient is the steepest direction, not merely an assertion about it.

Perpendicular to the gradient, and the slope is zero. Those are the contour lines: move along them and the function does not change.

Directly against the gradient the slope is , the steepest descent, which is the direction gradient descent takes.

grad = [3, 4]     ||grad|| = 5
u = [1, 0]  ->  normalised [1, 0]     D = 3
u = [3, 4]  ->  normalised [0.6, 0.8] D = 5    <- the maximum
u = [-4, 3] ->  normalised [-0.8, 0.6] D = 0   <- along a contour

Your task:

Implement directional_derivative(grad, u), normalising u yourself, and returning the slope as a float.

Example Tests

Along the x axis you feel only the x component of the gradient

Input: {"u":[1,0],"grad":[3,4]}

Expected: 3

Along the gradient itself the slope is its full magnitude

Input: {"u":[3,4],"grad":[3,4]}

Expected: 5

Perpendicular to the gradient the function does not change

Input: {"u":[-4,3],"grad":[3,4]}

Expected: 0

Python

import numpy as np


def directional_derivative(grad, u):
    """
    Slope of f along the direction u.

    Args:
        grad: the gradient at the point, shape (n,)
        u:    a direction, shape (n,), NOT necessarily unit length

    Returns:
        float: grad . (u / ||u||)
    """
    # YOUR CODE HERE
    pass

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