The surfaces S1 : z = x2 + y2 and S2 : x2 + y2 = 2x + 2y
intersect at a curve gamma
. Find a tangent vector to
at the point (0, 2, 4).
i thought about finding gradients of the two functions and plug in the given point in the gradients and cross product the two.
I get (2,2,8) as the vector tangent to the intersection gamma.
My solution was saying " If we produce two normal vectors n1 ? S1 and n2 ? S2, any
vector perpendicular to n1 and n2 (specifically n1 × n2) would be tangent to
both S1 and S2 and thus tangent to their common intersection "
it didnt make sense to me.
Cause if the vector is tangent to s2, x2 + y2 = 2x + 2y, then the vector should have zero for z component. the intersection is gonna be some kinda curve , so i think there will be infinite number of vectors tangential to the intersectino between s1 and s2. and the cross product of the two gradients is just one of them.
I dont think the vector we get by cross producting the two gradients has to be tangential to s1, and s2