How to parameterize the paraboloid z = 9 − x 2 − y 2 ? I know that I can parameterize it in cartesian coordinates as r ( x , y ) = ( x , y , 9 − x 2 − y 2 ) but I see in a book this parameterization of it that is r ( t ) = ( 3 c o s ( t ) , s i n ( t ) , 0 ) with 0 ≤ t ≤ 2 π and I don't know how to deduce it.

Question

How to parameterize the paraboloid   z  =  9  −      x    2    −      y    2  ? I know that I can parameterize it in cartesian coordinates as   r  (  x  ,  y  )  =  (  x  ,  y  ,  9  −      x    2    −      y    2    ) but I see in a book this parameterization of it that is   r  (  t  )  =  (  3  c  o  s  (  t  )  ,  s  i  n  (  t  )  ,  0  ) with   0  ≤  t  ≤  2      π   and I don&#039;t know how to deduce it.

Madden Krause · Accepted Answer

Use polar coordinates to describe the xy plane and let z depend on them.Let   x  =  r  cos  ⁡  θ  ,        y  =  r  sin  ⁡  θ  ,        z  =  z then the parametrization for the parabolid in terms of   r and   θ is   Φ  (  r  ,  θ  )  =  (  r  cos  ⁡  θ  ,        r  sin  ⁡  θ  ,        9  −  (  r  cos  ⁡  θ      )    2    −  (  r  sin  ⁡  θ      )    2    )  =  (  r  cos  ⁡  θ  ,        r  sin  ⁡  θ  ,        9  −      r    2    ) with   θ  ∈  [  0  ,  2  π  ],   r  ∈  [  0  ,  +  ∞  )

How to parameterize the paraboloid z=9-x^2-y^2?

Answered question

Answer & Explanation

New Questions in Integral Calculus