Are partial derivatives of parametric surfaces always orthogonal?For a surface r = r ( s , t ) are the partial derivates r s and r t in general orthogonal?

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Are partial derivatives of parametric surfaces always orthogonal?For a surface       r    =      r    (  s  ,  t  ) are the partial derivates             r        s   and             r        t   in general orthogonal?

Quinn Everett · Accepted Answer

Step 1Your example is apparently      r    (  θ  ,  ϕ  )  =  (  cos  ⁡  θ  sin  ⁡  ϕ  ,  sin  ⁡  θ  sin  ⁡  ϕ  ,  cos  ⁡  ϕ  )where it is indeed true that             r        θ    ⋅            r        ϕ    =  0 . But notice that this a standard coordinate parametrization of the sphere. This coordinate system was chosen exactly for the property that the   θ and   ϕ coordinate directions are perpendicular to each other.In general, this is not true. Another familiar way of parametrizing half the sphere is      r    (  x  ,  y  )  =  (  x  ,  y  ,      1    −          x      2        −          y      2        )for which we find            r        x    =  (  1  ,  0  ,            −      x              1      −              x        2            −              y        2              )              r        y    =  (  0  ,  1  ,            −      y              1      −              x        2            −              y        2              )and thus            r        x    ⋅            r        y    =            x      y              1      −              x        2            −              y        2            which is only 0 when either x or y is 0.

Are partial derivatives of parametric surfaces always orthogonal? For a surface <mrow class="MJ

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