Check if the following integrals are independent of path and evaluate those which are independent.
(i) Consider the vector field: [ \mathbf{F}(x,y) = (P, Q) = (6xy - y^3, 3x^2 - 3xy^2) ] Check for path independence by verifying if (\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}). [ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(3x^2 - 3xy^2) = 6x - 3y^2 ] [ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(6xy - y^3) = 6x - 3y^2 ] Since (\frac{\partial Q}{\partial x} = ____ __________ _____ ____ ___ ________.
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ध्रुवीय निर्देशांकों का प्रयोग करते हुए दिखाइए कि है। दो पुनरावृत्ति सीमाएँ ज्ञात कीजिए।
ख) प्रदेश D पर एक समाकल के रूप में लिखिए। प्रदेश D का चित्र बनाइए और दिखाएं कि यह टाईप 1 और टाईप 2 दोनों है। समाकल के क्रम को उलटिए और इसका मूल्यांकन कीजिए।
Evaluate , where S is the solid region between the spheres
and
, by using spherical coordinates.
Find the following limits:(i)
(ii)
Using polar coordinates, show that . Also, find the two repeated limits.
tate Green’s theorem, and apply it to evaluate
Where C is the ellipse .
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).
वक्रों और
से परिबद्ध और
के घनत्व वाले एक पतली शीट का गुरुत्व केन्द्र ज्ञात कीजिए।
(ख) और
से परिबद्ध ठोस घनाकृति का द्रव्यमान ज्ञात कीजिए, जबकि घनत्व फलन,
हो।