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.