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 only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment
Erdos
4.8K
-
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 a 50% commission on every question that your affiliated users ask or answer.
- answered
- 941 views
- $20.00
Related Questions
- Prove Holder-continuity for $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$
- In what direction the function $f(x,y)=e^{x-y}+\sin (x+y^2)$ grows fastest at point $(0,0)$?
- Use Green’s theorem to evaluate the line integral $\int_C (1+xy^2)dx-x^2ydy$ on the arc of a parabola
- [ Banach Fixt Point Theorem ] $\frac{dy} {dx} = xy, \text{with} \ \ y(0) = 3,$
- Find $\lim\limits _{n\rightarrow \infty} n^2 \prod\limits_{k=1}^{n} (\frac{1}{k^2}+\frac{1}{n^2})^{\frac{1}{n}}$
- Optimization problem
- Matrix Calculus (Matrix-vector derivatives)
- Evaluate $\int_C (2x^3-y^3)dx+(x^3+y^3)dy$, where $C$ is the unit circle.
