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Unformatted text preview: ly, we need to compute the volume of a tetrahedron with sides = 3 0 , = 3 0 ,
1
= 3 0 perpendicular to each other. Generally, the volume of a tetrahedron is 3 A ,
where A is the base area and is the height. Since our particular tetrahedron is a right
3
1
0
one, we have A = 2 and = , so its volume us 3 · 9 2 0 · 3 0 = 92 .
:) 2 3 Questions on logical thinking E
Given a function ( 1
) diﬀerentiable at a point a = (
is the maximal growth rate Dv (a) over all unit vectors v?
The direction of the maximal growth rate is u = S direction, that is, the maximal growth rate is Du =
E
+ grad
grad grad
grad ), what 1 , the derivative in this · grad = grad . :) By an appropriate linear substitution, ﬁnd the general solution to the PDE
= 0, where and are some parameters such that 2 + 2 > 0. Let’s use the substitution =
−
and =
+ . Then =
+
and
=−
+
. Substituting it in the equation, we get (
+
) + (−
+
) = 0,
that is
= 0. Hence depends only on
=
−
and the general solution is
(
) = ( − ), where is a function of one variable.
:)
S E
where = Show that if has continuous second order partial derivatives and
cos , = sin , then
∂2
∂ 2 + ∂2
=
∂2 −2 =( ∂2
∂2
+2
∂2
∂ S We have
∂
∂∂
∂∂
∂
∂
=
+
=
cos +
∂
∂∂
∂∂
∂
∂
∂
∂∂
∂∂
∂
∂
=
+
=
(− sin ) +
∂
∂∂
∂∂
∂
∂ sin
cos so
∂2
∂∂
∂
=
=
2
∂
∂∂
∂ ∂
∂ cos +
∂∂
∂∂ ∂
∂ sin cos + = ∂
∂ 3 cos + ∂∂
∂∂ sin + ∂
∂ sin = ), ∂...
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This document was uploaded on 02/10/2014.
 Spring '13

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