is a 2nd-rank tensor. It can be
written: δij = .Problem Set 2: Vectors and Tensors
Due September 18, 2006
Show all derivations and do your own work.
1-1. Expand the following expressions into their elemental forms for i,j,k = 1,2,3; where eijk is the permutation symbol and δij is the Kronecker delta symbol.
| a) | δij δij |
| b) | eijk eijk |
| c) | eijk Aj Ak |
| d) | δij δjk |
| e) | δij eijk |
Prove the following identities for the permutation symbol eijk:
| f) | eijk = ejki = ekij |
| g) | eijk eist = δjs δkt - δjt δks |
1-2. Verify that the vector cross product u x v of two vectors u=(u1, u2, u3) and v=(v1, v2, v3) can be written in the indicial form as
| wi = eijk uj vk |
where eijk is the permutation symbol.
1-3. Show that:
| a) | The gradient of a scalar is a first rank tensor, i.e., that the gradient transforms according to the 1st-rank tensor transformation rule. |
| b) | The gradient of a vector is a 2nd-rank tensor. |
| c) | It follows from (b) that is a 2nd-rank tensor. It can be
written: δij = . |
| d) | Write out the components δij and prove that δij transforms into itself under the 2nd-rank tensor transformation rule. |
1-4. Answer the following using Matlab or Maple programs and turn in your program listings and scripts.
| a) | Express in terms of i,j,k a unit vector parallel to a = 4i - 2j + 4k. |
| b) | A vector of magnitude 100 m is directed along the line from point A (10, -5, 0) to point B(9, 0, 24). Express the vector in terms of i,j,k. |
| c) | Find: u · v, u x v, and v x u for u = 6i - 4j - 6k, v = 4i - 2j - 8k |
| d) | Find the projection of the vector c = 18i - 27j + 81k onto d = i + 2j - 2k. |
| e) | Show that the vector Ai + Bj + Ck is normal to the plane whose equation is Ax + By + Cz = D. Hint: Show that it is perpendicular to the vector joining any two points (x1,y1,z1) and (x2,y2,z2) lying in the plane. |
| f) | Find the angle between the two vectors from the origin to the two points A (4, 3, 2) and B (-2, 4, 3). |