To demonstrate multivariate gradient descent in practice, consider the two-dimensional objective function $f(\mathbf{x}) = x_1^2 + 2x_2^2$ with input $\mathbf{x} = [x_1, x_2]^\top$. Its gradient is $\nabla f(\mathbf{x}) = [2x_1, 4x_2]^\top$. Starting from the initial point $[-5, -2]$ and applying the update rule $\mathbf{x} \leftarrow \mathbf{x} - \eta \nabla f(\mathbf{x})$ with a learning rate of $\eta = 0.1$ for $20$ iterations, the trajectory of $\mathbf{x}$ converges steadily toward the minimum at $[0, 0]$. After $20$ steps the parameters reach approximately $x_1 \approx -0.057646$ and $x_2 \approx -0.000073$, confirming well-behaved but relatively slow convergence. The contour plot of $f$ shows elliptical level curves (elongated along $x_1$) because the curvature in the $x_2$ direction is twice that in $x_1$, which causes $x_2$ to converge faster than $x_1$.