Check if the following map is an immersion (injective or not) and an embedding.
1) $\gamma_1:\mathbb{R}\to\mathbb{R}^2,\gamma_1(t)=(t,|t|)$
2) $ \gamma_2:\mathbb{R}\to\mathbb{R}^2,\gamma_2(t)=(t(t^2-4),t^2-4)$
1) The first curve is not differentiable considering all of $\mathbb{R}$. Therefore I would conclude it is not a immersion.
If restrict the domain into an interval such that [0,1], then we would have 1:1 line which is an injective immersion over a compact hence an embedding.So I would conclude that$\gamma_1$ restricted to any closed interval in $\mathbb{R}+$ or $\mathbb{R}-$ is an embedding. If the interval is not closed it is an injective immersion.
2) Computing the Jacobian we get $J\gamma_2(t)=(3t-3,2t)\neq 0,\:\:\forall t\in\mathbb{R}$. Hence the rank of $\gamma_2$ is 1 which implies that $\gamma_2$ is an immersion. According to the jacobian $\gamma_2$ is monotonous increasing function hence injective. So I have an injective immersion. Since both (t(t^2-4),t^2-4) functions are continuous it is only required to prove that the inverse exists and it is continuous. However I am not seeing how to that last step that would give me the homeomorphism into the image.
Questions:
Are my resolutions insofar correct? How should I finish 2)?
Thanks in advance!
1) Actually if $(a,b)\subset (0,\infty),$ then $\gamma_1$ is an embedding. That's because $t\to (t,t)$ is a homeomorphism of $(0,\infty)$ onto $\{(x,y):x>0,y=x\}.$ Same for $(a,b)\subset(-\infty,0).$ Other than that your answer seems good.
2) Usually the term "Jacobian" alone means the determinant of a square Jacobian matrix. I'll just write $\gamma_2'(t)$ here.
You've made an error in calculating $\gamma_2'(t).$ It should be $\gamma_2'(t) = (3t^2-4,2t).$ Nevertheless we still see $\gamma_2'(t)$ is never $(0,0).$ Therefore $\gamma_2$ is an immersion.
I do not know what you mean by "$\gamma_2$ is monotonous increasing function hence injective". How can a vector valued function be monotonically increasing?
But notice $\gamma_2(-2) = (0,0) = \gamma_2(2).$ Thus $\gamma_2$ is not injective, and therefore $\gamma_2$ is not an embedding.