) Does the function
satisfy the requirement of Schwarz's theorem at (1, 1)? Justify your answer. (b) Locate and classify the stationary points of the following:
(i)
(ii)
For part (a), we check if the function satisfies Schwarz's theorem (equality of mixed partial derivatives) at ((1,1)). Let the function be (f(x,y)). Compute the second-order partial derivatives: [ f_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right), \quad f_{yx} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) ] Evaluate both at ((1,1)). If (f_{xy}(1,1) = f_{yx}(1,1)), the theorem is satisfied. If not, it fails. Without the explicit function given _____ __________ _______ ________ ______.
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) Does the function
satisfy the requirement of Schwarz's theorem at (1, 1)? Justify your answer. (b) Locate and classify the stationary points of the following:
(i)
(ii)
Check the continuity and differentiability of the function at (0, 0) where
(b) Find the domain and range of the function f, defined by . Also find two level curves of this function. Give a rough sketch of them.
Evaluate , where C is the curve given by
.
(b) Use double integration of find the volume of the ellipsoid
.
Find the values of a and b, if
(b) Suppose S and C are subsets of . S is the unit open sphere with centre at the origin and C is the open cube
.
Which of the following is true. Justify your answer.
(i)
(ii)
(c) Identify the level curves of the following functions: (i)
(ii)
(iii) x - y
(iv) y / x
swers.
(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).
Find the centre of gravity of a thin sheet with density , bounded by the curves
and
.
(b) Find the mass of the solid bounded by and
, the density function being
.
5. (a) State Green's theorem, and apply it to evaluate
Where C is the ellipse .
(b) Find the extreme values of the function
on the surface
.
State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . Verify this theorem for the functions f and g, defined by
(b) Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that and
in a neighbourhood of (2, 1), where
defines the function F. Also find g'(y).
(c) Check the local invertibility of the function f defined by at (1, -1). Find a domain for the function f in which f is invertible.