Describe the geometric meaning of the following mappings in cylindrical coordinates. (a) (r, θ, z) ↦ (r, θ, −z) There is a reflection across the xz plane. There is a reflection across the yz plane. There is a reflection across the xy plane. There is a rotation by π about the z-axis. There is a rotation by π about the y-axis. (b) (r, θ, z) ↦ (r, θ + π, −z) There is a reflection across the yz plane with a rotation by −π about the z-axis. There is a reflection across the xy plane with a rotation by −π about the y-axis. There is a reflection across the yz plane with a rotation by π about the y-axis. There is a reflection across the xz plane with a rotation by π about the z-axis. There is a reflection across the xy plane with a rotation by π about the z-axis. (c) (r, θ, z) ↦ −r, θ − π 4 , z There is a rotation by − 5π 4 about the x-axis. There is a reflection across the yz plane with a rotation by − 5π 4 about the z-axis. There is a rotation by − 5π 4 about the z-axis. There is a reflection across the yz plane with a rotation by − 5π 4 about the x-axis. There is a reflection across the xy plane with a rotation by − 5π 4 about the z-axis.

Fill in the blanks with the numbers is a measure of the resistance to flow. a high liquid has a high resistance to flow and flows slowly. the ancients thought everything in the world was made of 4 we now know that there are 94 naturally occurring and scientists have created another 24 i am certain they will create even more. honey flows slowly because it has a high to flowing. a can be separated by physical means because it contains more than one pure substance and 2 pure substances are not chemically bonded to each other. a cannot be separated by physical means. all matter is made up of all elements are with the same number of protons. if it is just a single or many bonded together, if all of them have the same number of protons, it is an element. in a piece of pure iron metal, all the are joined together, that piece of iron metal is called elemental iron. a single of iron is called elemental iron. a mixture has differences from place to place. we might need a microscope to see them or they might be obvious to the unaided eye. there are surfaces separating it into different phases. a mixture is the same everywhere. it is uniform. there are no surfaces separating it into different phases. if different kinds of atoms (different elements) are bonded together by their electrons, it is called a there are physical means of to isolate the different pure substances in a mixture and there are chemical means of to isolate the different elements in a compound. 1. element 2. compound 3. mixture 4. heterogeneous 5. homogeneous 6. pure substance 7. atoms 8. separation 9. viscosity 10. resistance