Let $F$ from $\mathbf{R}^3$ to $\mathbf{R}^3$ defined by
$F(x, y, z)=(x^3+y+z, y^3+y, x^3-y^3+z)$.
(a) Calculate the Jacobian of $F$. Determine the rank of $F'(x)$. On what set is the rank constant?
(b) Describe the range of $F$, and describe the level set $F^-1(30, 10, 20)$.
I have calculated the Jacobian to be $0$. I have some troubles on determining the rank and range.
Here is how I calculate the Jacobian: Jacobian Calculate
And I just realized that if the Jacobian is $0$, this matrix is not invertible, right? Can someone check my Jacobian please?
System won't let me write a comment, so I have to write a solution instead.
Turns out the Jacobian is $0$. I think this means that your operator is not invertible, but that's perfectly fine for calculating range and rank.
Notice that the three terms are linearly correlated, that is why you have Jacobian to be $0$. And that is why you are asked to do the rank and range. (If the operator is invertible, than range is $\mathbb{R}^3$ and rank is $3$, it's trivial then.