Gradient Magnitude

The Gradient

The gradient of $f(x, y)$ is the vector of its partial derivatives:

$\nabla f = \left(\frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y}\right)$

The Gradient Points Uphill

The gradient vector always points in the direction of steepest ascent. Its magnitude tells you how steep that ascent is.

$|\nabla f| = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2}$

Key Facts

• $\nabla f = \mathbf{0}$ at critical points (local minima, maxima, saddle points)
• The gradient is perpendicular to level curves (contour lines)
• Moving against the gradient is steepest descent — the basis of gradient descent in machine learning

Examples

For $f(x,y) = x^2 + y^2$:

• $\nabla f = (2x,\; 2y)$
• At $(3, 4)$: $|\nabla f| = \sqrt{36 + 64} = \sqrt{100} = 10$

For $f(x,y) = xy$:

• $\nabla f = (y,\; x)$
• At $(1, 1)$: $|\nabla f| = \sqrt{1 + 1} = \sqrt{2} \approx 1.4142$

Your Task

Implement double gradient_magnitude(double (*f)(double, double), double x, double y, double h) that computes $|\nabla f|$ at $(x, y)$ using central differences for the partial derivatives.