Check the continuity and differentiability of the function at (0,0) where
To check the continuity of ( f(x, y) ) at ((0,0)), consider the limit [ \lim_{(x,y) \to (0,0)} \frac{2x^3 y}{x^2 + y^2}. ] Using polar coordinates, set ( x = r \cos \theta ), ( y = r \sin \theta ). Then [ f(x, y) = \frac{2 (r \cos \theta)^3 (r \sin \theta)}{(r \cos \theta)^2 + (r \sin ______ _______ ____ _________ _____ ___.
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Let and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that
Find the mass of the solid bounded by and
, the density function being
.
State whether the following statements are true or false. Give reasons for your answers.
(i)
(ii) A real-valued function of three variables which is continuous everywhere is differentiable.
(iii) The function , defined by
, is locally invertible at any
.
(iv) , defined by
is integrable.
(v) The function , defined by
, has an extremum at (0, 0).
Check if the following integrals are independent of path and evaluate those which are independent.
निम्नलिखित सीमा ज्ञात कीजिए :
(i)
(ii)
मान लीजिए$
दिखाइए कि (0,0) पर सभी दिशाओं में f दिक् अवकलज होते हैं।
(ख) मान लीजिए और f, x और y का एक संततः अवकलनीय फलन है जिसके आंशिक अवकलज भी संततः अवकलनीय हैं। दिखाइए कि
$
(ग) मान लीजिए के तीन बिन्दु हैं।
|2b - a + 3c| ज्ञात कीजिए।
(0,0) पर निम्नलिखित फलन f के सांतत्य और अवकलनीयता की जाँच कीजिए, जहाँ
(ख) फलन से परिभाषित फलन f का प्रांत और परिसर ज्ञात कीजिए। इस फलन के दो स्तर वक्र भी ज्ञात कीजिए। (4)