Given a real number find the maximum of the function
where is any region in space.
Solution
Let . Then
In the region we have hence which gives . Thus ,
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Given a real number find the maximum of the function
where is any region in space.
Solution
Let . Then
In the region we have hence which gives . Thus ,
Let be differentiable on and continuous at . If
for then prove that is differentiable at .
Solution
Let us assume without loss of generality that . We will show that as and that means that is differentiable at with .
Fix and choose such that , . Therefore . Now suppose . Let be the point that coincides with on the first coordinates and is elsewhere. Then is a path from to and each vector is parallel to one of the axes. Hence
For we define and . For which does the series
converge?
Solution ( Robert Tauraso )
We may assume that is not the zero vector and otherwise the series is trivially convergent. Then, we show that the series is convergent if and only if there is exactly one component of maximal absolute value.
(a) If the above condition is satisfied then, without loss of generality, let be the component of maximal absolute value and let . Hence , as
and the given series is convergent because .
(b) If the above condition is not satisfied, then there are at least components of maximal absolute value and therefore
and the given series is not convergent because .
Let . Evaluate the integral:
Solution
The key observation to nail the integral is that the domain of integration is symmetric with respect to the axis.
Hence , a symmetry might as well work. So,
since
Let be the volume of the sphere centered at and radius in . Prove that for it holds that
Solution
The volume of the sphere in is given by:
Parametrize the sphere by
taking
It then follows from the Change of Variables formula that the rectangular volume element can be written in spherical coordinates as
Thus,
Hence,
(1)
In particular since we get that:
(2)
Using as well as Wallis’ integral we are able to prove the result. Let us assume that is even, then:
If is odd we work similarly.