Chain Rule, Partial Differential Equations, Implicit Differentiation, Directional Derivatives, and Gradient

Review of Chain Rule

• General Formulation: Recalled the most general formulation for the chain rule, which encapsulates all previous cases. It states that if u is a differentiable function of n intermediate variables (x1, \dots, xn), and each of these intermediate variables (xk) depends on m other variables (t1, \dots, tm), then the chain rule generates every possible combination for computing the partial derivatives of u with respect to ti.

Partial Differential Equations (PDEs)

• Definition and Importance: Partial differential equations are a significant area of research in almost every scientific field, used extensively for modeling systems that evolve over time.

• Comparison to Ordinary Differential Equations (ODEs): ODEs are the one-variable scenario where derivatives are with respect to a single independent variable (e.g., n''f + k f' = 0). PDEs involve partial derivatives with respect to multiple independent variables. This difference is substantial, making PDEs much harder to solve and understand than ODEs, as many ODE techniques do not directly apply.

• Applications: Examples include river pollution modeling, quantum mechanics (Schrodinger's equation), earthquake dynamics, and general relativity.

• Example: Verifying a PDE Solution:

  • Given a function g(s,t) = f(s^2-t^2, t^2-s^2), where f is differentiable.

  • The task is to show that this function satisfies the PDE: t \frac{\partial g}{\partial s} + s \frac{\partial g}{\partial t} = 0.

  • Method: Introduce intermediate variables: x = s^2 - t^2 and y = t^2 - s^2. Thus, g = f(x,y).

  • Apply the chain rule to find partial derivatives:

    • \frac{\partial g}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s}

    • \frac{\partial g}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t}

  • Calculate intermediate derivatives:

    • \frac{\partial x}{\partial s} = 2s, \frac{\partial y}{\partial s} = -2s

    • \frac{\partial x}{\partial t} = -2t, \frac{\partial y}{\partial t} = 2t

  • Substitute back:

    • \frac{\partial g}{\partial s} = \frac{\partial f}{\partial x} (2s) + \frac{\partial f}{\partial y} (-2s)

    • \frac{\partial g}{\partial t} = \frac{\partial f}{\partial x} (-2t) + \frac{\partial f}{\partial y} (2t)

  • Substitute these into the PDE: t \left( 2s \frac{\partial f}{\partial x} - 2s \frac{\partial f}{\partial y} \right) + s \left( -2t \frac{\partial f}{\partial x} + 2t \frac{\partial f}{\partial y} \right)

  • Simplify: 2st \frac{\partial f}{\partial x} - 2st \frac{\partial f}{\partial y} - 2st \frac{\partial f}{\partial x} + 2st \frac{\partial f}{\partial y} = 0. This confirms the equation is satisfied.

• Popular PDEs:

  • Laplacian (or Laplace Operator): Denoted as \nabla^2u or \Delta u. For a 2D function u(x,y), it's u{xx} + u{yy}, which means the sum of the second partial derivatives with respect to each coordinate. Differential operators are often used to express PDEs.

  • Heat Equation: \frac{\partial u}{\partial t} = \alpha \nabla^2u.

  • Wave Equation: \frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2u.

  • Schrodinger's Equation (Quantum Mechanics): \mathbf{i}\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi. (The term V \psi is a potential function).

    • Wave Function (\psi): A function that governs the probability distribution of a particle in quantum mechanics.

    • Constants: \mathbf{i} is the imaginary unit, \hbar is Planck's constant, m is mass.

    • Behavior: Solutions to Schrodinger's equation describe how probability distributions evolve over time. It's a