Tutorial05-solutions

Generally the volume of a tetrahedron is 3 a where a

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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.

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