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.

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. Write down the wave equation in three dimensional space and explain in words what you think the equation means.

6.1. If you know any special properties of this equation list them.