# 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.

## Answer

Answers can be viewed only if
1. The questioner was satisfied and accepted the answer, or
2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer

1 Attachment

Erdos
4.4K
• Thank you very much. Would you be able to take a look at my other question? Most recent one in the site.

• I am busy at the moment. I will answer it before the deadline if no one else accepts to answer it.

• Great! Much appreciated.

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to 50% commission on every question your affiliated users ask or answer.
• answered
• 501 views
• \$20.00