SET 1

1. Rotation

1.1. Write down the matrix for a rotation about x

1.2. Write down the matrix for a rotation about y

1.3. Write down the matrix for a rotation about z

1.4. Find the rotation matrices that will rotate an object around the x,y,z axes.

1.5. Rotate an
object 90^{o} about x then 90^{o} about y and then 90^{o}
about z

1.6. Find the total rotation matrix

1.7. Do the rotation in reverse order.

1.8. Assume that the object is an eraser aligned and centered on the x-axis.

1.8.1. Draw the transitions as one applies the three transformations for both cases.

2. Find a matrix that performs a boost (transformation for special relativity) and a matrix formulation for a Galilean transformation. Comment on the difference between these two matrices.

3. Describe the Einstein summation convention and give an example that uses it.

3.1. How would one express matrix multiplication in this notation.

3.2. Use the Einstein summation convention to show a 3-d vector rotation.

3.3. Use the Einstein summation convention to depict a 4-d relativistic vector under a 3-d rotation

3.4. Use the Einstein summation convention to depict a 4-d relativistic vector boosted.

4. Find a three dimensional matrix that stretches space.

4.1. How will the transformed vectors behave if one does a dot product or contraction?

5. Find a matrix that curves space.

6. What is a group?

6.1. Find and list a description of a group.

6.2. Show explicitly that rotations form a group.

6.2.1. Give examples for each requirement for rotations.

7. What is the mathematical definition of a vector?