$Let $f:U\subset\mathbb{R} ^3\rightarrow \mathbb{R} ^2$ given by $f(x,y,z)=(sin(x+z)+log(yz^2) ; e^{x+z} +yz)$ where $U = { (x, y, z) ∈ R^3| y, z > 0 }.$ Questions Inside.$

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$Let $f:U\subset\mathbb{R} ^3\rightarrow \mathbb{R} ^2$ given by $f(x,y,z)=(sin(x+z)+log(yz^2) ; e^{x+z} +yz)$ where $U = { (x, y, z) ∈ R^3| y, z > 0 }.$ Questions Inside.$

1. Prove that $f$ is class $C^{∞} $ and calculate the Jacobian Matrix in $(0,1,1)$.

2. Considering $R^3 = R^2×R$, calculate the partial derivatives $D_{i}f(0, 1, 1)$ with $(i = 1, 2)$.

3. Using the Theorem of the Implicit Function, decide if we can see the level set $f(0, 1, 1)$ as the graph of some convenient function.

The more detailed the better, having a tough time understanding the professor's answer.

Calculus Manifolds Functional Analysis

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Erdos

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Thank you very much. Would you be able to take a look at my other question? Most recent one in the site.
Erdos

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I am busy at the moment. I will answer it before the deadline if no one else accepts to answer it.
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Great! Much appreciated.

$Let $f:U\subset\mathbb{R} ^3\rightarrow \mathbb{R} ^2$ given by $f(x,y,z)=(sin(x+z)+log(yz^2) ; e^{x+z} +yz)$ where $U = { (x, y, z) ∈ R^3| y, z > 0 }.$ Questions Inside.$

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