## The total differential for composite function

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.