Question

Give a formula for the area element in the plane in rectangular coordinates x and y....

  1. Give a formula for the area element in the plane in rectangular coordinates x and y. (Answer: dx dy, or more properly |dx ∧ dy|; either is acceptable, as are dy dx and |dy ∧ dx|.)
  2. Give a formula for the area element in the plane in polar coordinates r and θ.
  3. Give a formula for the volume element in space in rectangular coordinates x, y, and z. (Answer: dx dy dz, or more properly |dx ∧ dy ∧ dz|; either is acceptable, as are the the other orders.)
  4. Give a formula for the volume element in space in cylindrical coordinates r, θ, and z.
  5. Give a formula for the volume element in space in spherical coordinates ρ, φ, and θ (using the U.S. mathematicians' convention for which of these is which).
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