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Homework 2 Due Friday, 8 September
Basic Vectors and Indicial Notation 1. (3 points) The following basis vectors form a set, sketch them and determine whether or not they form an orthonormal set.
𝒆𝟏 =22 𝑬𝟏 +
22 𝑬𝟐
𝒆𝟑 = 𝑬𝟑
𝒆𝟐 =22 𝑬𝟏 −
22 𝑬𝟐
2. (10 points) Let two vectors, a, b, c, and d be specified by,
a = 4E1 + 2E2 + 2E3
b = 2E1 − 2E2 + 4E3
c = E1 +E2 +E3
d = E2 −5E3
where Ei{ } represents a fixed, orthonormal basis. Sketch the vectors and evaluate the following
expressions,
a ⋅b =
aiai =
b ⋅b =
aibia jbj
bkbk
=
aibjaibj
bkbk
=
a × b =
a + b( ) ⋅c =d− b( ) ⋅a =
3. (5 points) Simplify the following expressions,
Θik = ηδ ijδ jkδmm
Ψ = 6δ ijδ jkδklδ li
Ξk = 4βδ ijε ijk
Grad/Div/Curl
4. (6 points) Suppose we have a scalar field (which may define a surface, an electric potential, a
temperature field, etc.) defined in three dimensions by f x1,x2 ,x3( ) = Ax1
2x2x32 −17 = 0 (Assume
A = 1m−5 ). Determine the unit normal to the surface at the points:
1,1, 17( )1,17,1( )
17,1,1( )
5. (10 points) For the scalar function on E3
φ = αx1x2x32 + βcos γx2( )
where α,β, γ are arbitrary real numbers. Give formulae for the following fields on E3
v = gradφα = divvw = curlv
6. (3 points) Calculate the divergence of the vector field given by,
b = 3γ x1E1 + 6α x1x3E2 − βx1x2x33E3 .
7. (3 points) Calculate the divergence of the vector field given by, b = 3γ x2x3tE1 − 6α x1x3E2 + 7βx1x2x3
3E3 . 8. (6 points) Find a vector field whose divergence is given by, x12x2 .
N.B. this answer will not be unique. 9. (5 points) Let the vector field, 𝒃 𝑥!, 𝑥!, 𝑥!, 𝑡 = 2𝑥!𝑥!!𝑥! 𝑬𝟏 + 2𝑥!𝑥!!𝑥! + 𝑐𝑜𝑠𝛾 𝑬𝟐 +𝑥!!𝑥!! 𝑬𝟑. Find the function, f(x1,x2,x3), that satisfies the relation,
b = ∇f . 10. (20 points) You may assume that 𝑬𝟏,𝑬𝟐,𝑬𝟑 form a standard orthonormal basis. Let 𝑨 = 5𝑬𝟏⨂𝑬𝟐 + 2𝑬𝟏⨂𝑬𝟏 − 𝟑𝑬𝟐⨂𝑬𝟐 + 𝑬𝟏⨂𝑬𝟑 − 2𝑬𝟑⨂𝑬𝟐 + 5𝑬𝟐⨂𝑬𝟑 − 6𝑬𝟑⨂𝑬𝟏 +2𝑬𝟑⨂𝑬𝟑 − 𝑬𝟐⨂𝑬𝟏 and have units of MPa. Let 𝑩 = −5𝑬𝟏⨂𝑬𝟐 + 𝟑𝑬𝟐⨂𝑬𝟐 + 𝑬𝟏⨂𝑬𝟑 + 4𝑬𝟑⨂𝑬𝟐 + 5𝑬𝟐⨂𝑬𝟑 + 2𝑬𝟑⨂𝑬𝟑 − 𝟑𝑬𝟐⨂𝑬𝟏 and have units of m/s. Let 𝑪 = 2𝑬𝟏⨂𝑬𝟏 − 𝟑𝑬𝟐⨂𝑬𝟐 + 2𝑬𝟑⨂𝑬𝟑 with no dimensions. For each one, do the following:
1. Write the tensor in matrix form. 2. Find its transpose. 3. If 𝒃 = 𝑬𝟏 + 2𝑬𝟐 − 𝑬𝟑, what happens when each tensor acts on it? 4. Find the principal invariants of the tensor. 5. Decompose the tensor into symmetric and skew-symmetric parts.