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.

Difficulty level: Moderate task
Routine calculation training
Cs translation
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