Multivariable Chain Rule
If a function \(z = f(x, y)\) depends on variables that themselves depend on time \(t\), then the total rate of change is:
\[\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\]
You add up the contributions of change from every path.
Worked Examples
Example 1: Total Derivative
Suppose the temperature \(T\) in a room is \(T(x, y) = xy\). A fly is moving along a path \(x = \cos t\), \(y = \sin t\). How fast is the fly's temperature changing?
- \(\frac{\partial T}{\partial x} = y\), \(\frac{\partial T}{\partial y} = x\).
- \(\frac{dx}{dt} = -\sin t\), \(\frac{dy}{dt} = \cos t\).
- \(\frac{dT}{dt} = (y)(-\sin t) + (x)(\cos t)\).
- Substitute \(x, y\): \(\frac{dT}{dt} = (\sin t)(-\sin t) + (\cos t)(\cos t) = \cos^2 t - \sin^2 t = \cos(2t)\).
- Result: \(\cos(2t)\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often switch between coordinate systems (e.g., from Cartesian \(x, y, z\) to Spherical \(r, \theta, \phi\)). The Multivariable Chain Rule allows us to transform the Schrödinger Equation into the form that best fits the symmetry of the problem. For the Hydrogen atom, we use this rule to express the Laplacian in spherical coordinates, which is the only way to solve for the electron's energy levels.