Suppose that is given by
and
is given by
. Compute the derivative of
.
We are asked to compute the derivative of (g \circ f) manually. Let’s go step by step. We are given: [ f: \mathbb{R} \to \mathbb{R}^2, \quad f(t) = (t, t^2), ] [ g: \mathbb{R}^2 \to \mathbb{R}^3, \quad g(x,y) = (x^2, xy, y^2 - x^2). ] First, compute (g \circ __________ __________ __________ ______ ___ ___.
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Find the interior, closure, the set of limit points and the boundary of the set
in with the standard metric.
Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set where
Find the interior, closure, the set of limit points and the boundary of the set
in with the standard metric.
) Show that the function f defined by
is not differentiable at (0,0). Do the partial derivatives of f exist at (0,0)? or at any other point in ? Justify your answer.
Which of the following subsets of are compact w.r.t. the metric given against them. Justify your answer.
i) in
of
with standard metric.
ii) with discrete metric.
iii) with standard metric.
1. State whether the following statements are true or false. Justify your answers.
a) The outer measure m^ of the set is 0.
b) A finite subset of a metric space is totally bounded.
c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.
d) The surface given by the equation can also be described by an equation of the form
in a neighbourhood of the point (0, 0).
e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].
Let be given by
Show that f is locally invertible at all points in .