cmi-entrance 2013 QB7

cmi-entrance · India · pgmath 10 marks Not Maths

(a) Show that there exists no bijective map $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 3 }$ such that $f$ and $f ^ { - 1 }$ are differentiable.
(b) Let $f : \mathbb { R } ^ { m } \rightarrow \mathbb { R } ^ { n }$ be a differentiable map such that the derivative $D f ( x )$ is surjective for all $x$. Is $f$ surjective?

(a) Show that there exists no bijective map $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 3 }$ such that $f$ and $f ^ { - 1 }$ are differentiable.\\
(b) Let $f : \mathbb { R } ^ { m } \rightarrow \mathbb { R } ^ { n }$ be a differentiable map such that the derivative $D f ( x )$ is surjective for all $x$. Is $f$ surjective?

Paper Questions

QA1 QA10 QA11 QA12 QA13 QA14 QA15 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB10 QB2 QB3 QB4 QB5 QB6 QB7 QB8 QB9