How do I show that the direction of second differential vector is normal to the surface?

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- Jul 8th 2013, 03:32 PMshyamalshukladirection of second differential vector
How do I show that the direction of second differential vector is normal to the surface?

- Jul 8th 2013, 03:57 PMHallsofIvyRe: direction of second differential vector
The position vector is a function of

**two**variables. The "second differential" with respect to which variable? If, for example, we take, as parametric equations for the unit sphere, then and then the**mixed**derivative is and the dot product of that with is so the mixed second derivative is perpendicular to the surface.

But the second derivative with respect to both times is and that is NOT perpendicular to the surface.