The level surface of the function f ( x , y , z ) = ( x 2 + y 2 ) − 1 / 2 area) Circles centered at the origin b) spheres centered at the origin c) cylinders around the z-axis d) upper halves of spheres centered at the originI had chosen a) asmy answer, but it turns out the answer is c). Could someone help me understand why?

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The level surface of the function   f  (  x  ,  y  ,  z  )  =  (      x    2    +      y    2        )          −      1              /            2       area) Circles centered at the origin b) spheres centered at the origin c) cylinders around the z-axis d) upper halves of spheres centered at the originI had chosen a) asmy answer, but it turns out the answer is c). Could someone help me understand why?

Cameron Benitez · Accepted Answer

Step 1The answer is c because the function only depends on x and y. You can move freely on the z direction without changing the value of f.

Christina Matthews · Accepted Answer

Step 1  f  (  x  ,  y  ,  z  )  =  K    ⇒        1                  x        2            +              y        2              =  K    ⇒        x    2    +      y    2    =      1          K       Therefore the level surfaces of f have equation       x    2    +      y    2    =      1          K       , which is indeed a circle in             R        2   or a cylinder in             R        3   . Since f is defined on a subset of             R        3   , it is necessarily a cylinder here.If the function was   f  (  x  ,  y  )  =      1                  x        2            +              y        2              , , your answer would be correct.

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