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ME681A - Practice Set-1

This document provides problems related to tensors and vector calculus. It includes problems involving vector and tensor operations like dot products, cross products, and tensor transformations under coordinate changes. It also defines common tensor notation and properties.

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tanujg21
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14 views2 pages

ME681A - Practice Set-1

This document provides problems related to tensors and vector calculus. It includes problems involving vector and tensor operations like dot products, cross products, and tensor transformations under coordinate changes. It also defines common tensor notation and properties.

Uploaded by

tanujg21
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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ME681A

Practice Set-1

Note:
• Small case letters denote scalar.

• Small case letters with bold denotes vector.


• Capital letters denote tensors.

Problems:
1. Evaluate u · v where u = ê1 + 3ê2 − 2ê3 , v = 4ê1 − 2ê2 + 4ê3 also find the cross product.
2. Find the projection of vector u = ê1 − 2ê2 + ê3 on the vector v = 4ê1 − 4ê2 + 7ê3 .
3. Find the angle between vector u = 3ê1 + 2ê2 − 6ê3 and v = 4ê1 − 3ê2 + ê3 .

4. Show that u × v = ϵijk ui vj êk .


5. Show that δij ai bj is equivalent to a · b and also verify ϵijk ϵijk = 6 and ϵijk ϵkpq = δip δjq − δiq δjp .
6. Evaluate and simplify the following expressions:
• δkk
• δij δij
• δij δjk
7. Show that using indicial notations:
• v = S · u is equivalent to vi =
P
ij Sij uj .
• (S T )ij = Sji
• (S · T )ij = k Sik Tkj
P

• (a ⊗ b)ij = ai bj
• u ⊗ S · v = u ⊗ v · ST
8. The components
  of a first-and second-order tensor in a particular coordinate frame are given by-
1 1 0 3
ai = 4, aij = 0 2 2,
2 3 2 4
Determine the components of each tensor in a new coordinate system found through a rotation of 60 degree
about the x3 -axis. Choose a counterclockwise rotation when viewing down the negative x3 -axis, as shown in
the figure-1.
9. For the given matrix/vector pairs, compute the following quantities: aii , aij aij , aij ajk , aij bj , aij bi bj , bi bj , bi bi .
For each case, point out whether the result is a scalar, vector, or matrix. Note that aij bj is actually the matrix
product [a][b], while aij ajk is the product [a][a].
   
1 1 1 1
• aij = 0 4 2, bi = 0,
0 1 1 2
   
1 2 0 2
• aij = 0 2 1, bi = 1,
0 4 2 1

1
   
1 1 1 1
• aij = 1 0 2, bi = 1,
0 1 4 0

10. Consider the two-dimensional coordinate transformation (similar shown in the figure-1, but this is 2 − D
case). Through the counterclockwise rotation θ, a new polar coordinate system is created. Show that the
transformation matrix for this case is given by
     
cos θ sin θ b a a21
Qij = , if bi = 1 , aij = 11 , are the components of a first- and second-order tensor
−sin θ cos θ b2 a12 a22
in the x1 , x2 system, calculate their components in the rotated polar coordinate system.

Figure 1: Reference frame

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