Session 4 - Analysis II

Adding dimensions: several inputs, several outputs, and several equations at once, where the linear algebra pays off a second time.

Topics covered

• Functions of several variables: contour maps and partial derivatives
• The gradient and gradient descent; critical points
• Vector-valued functions: trajectories, velocity, and the Jacobian
• Systems of differential equations: one mode per eigenvector
• Stability from the eigenvalues (node, saddle, spiral, centre)
• Nonlinear systems: linearising at a fixed point, and nullclines

Materials

• 📊 Slides
• ✏️ Exercise sheet
• ✅ Solutions