The total differential for composite function
Task number: 3201
Consider the function \(H\colon \, \mathbb R^2 \to \mathbb R\) with the form \[ H(r, \alpha) = xe^{x+y}, \] where \(x = r \cos \alpha\) and \(y = r \sin \alpha\).
Variant 1
Evaluate \(\frac{\partial H}{\partial r}\) and \(\frac{\partial H}{\partial \alpha}\), ideally with the help of the chain rule.
Variant 2
Determine the total differential of \(H\).
Variant 3
For small \(\varepsilon\) estimate \(H(1 + \varepsilon, \varepsilon)\) with the help of the total differential.
