) Consider point p = (0, 0, 0) of paraboloid z = x 2 + ky2 , k > 0. (a) Show that the unit vectors of x-axis and y-axis are eigenvectors of −dNp, with eigenvalues 2 and 2k, respectively. Here we choose the orientation of the surface, N, to be the one pointing inwards from the region bounded by the paraboloid. (b) For p = (0, 0, 0), find Gaussian curvature K and mean curvature H at this point.

K = 4k and H=(2+2k) = k+1

                                          2

Step-by-step explanation:

Part(a) :Let's parametrize the given surface-

X (u, v) = ( u²,v²,u²+kv² )

Now : Xu = (1,0,2u) and Xv = (0,1,2kv)

using below formula:

N (u, v) = Xu x X v

               [ Xu x Xv]

N(u,v) = (-2u,-2kv , 1)

          √ {4u²+4k²v²+1}

Now value of normal vector at p=(0,0,0) is N(p)

N(p) = N(0,0 )= (0,0,1)

so finally at p=(0,0,0) : Xu = (1,0,0) , Xv = (0,1,0), Np = (0,0,1)

Clearly Tp(S) is spanned by (1,0,0),(0,1,0)

 dN.Xu = d/dt N(t,0)/ t=0 = (-2,0,0)

 dN.Xv = d/dr N(0,r)/r=0=(0,-2k,0)

since Xu ,Xv ∈Tp(S)

aXu + bXv ∈Tp(S)

-dNp (aXu + bXv) =(2a,2kb,0)

and represent above in matrix form:

   2    0              a

   0    2k           b

-dNp  is symmetric matrix with eigen values (LAMDA) ∧1=2 and ∧2 =2k

Part (b):

Product of Eigen Values is called Gaussian Curvature(K).

Mean of Eigen Values is called Mean Curvature(H).

Therefore K = 4k and H=(2+2k) = k+1

                                             2

answer:   28.56 feet (approx)

step-by-step explanation:

since, triangle abc has measure of angle c equal to 55 degrees, measure of angle abc equal to 90 degrees, and length of bc equal to 20 feet.

we have to find out the measurement of segment ab.

therefore, tan 55° =

but bc = 20 feet.

⇒ tan 55 ° =  

⇒ ab = 20 × tan 55 °

⇒ ab = 20 × 1.42814800674

⇒ ab = 28.5629601348   ≈ 28.56 feet