Differential Geometry Answers

Two pipes with circular cross-sections are subject to the constraint that the sum of their diameters is 1M. By using careful reasoning find the diameters that give the maximum and minimum possible combined cross-sectional areas. In each case give the combined cross-sectional areas.

how can i find this slution
E=〖(1+u^2+v^2)〗^2

from this
x=u- 1/3 u^3+uv^2
y=-v-u^2 v+1/3 v^3
z=u^2-v^2

A chord of a Circle is equal to the radius of the circle find the angle subtended by it at the centre

Find 2 charts on R which are C^(r-1) compatible but not C^r compatible.

if the equality $a^i_ju_i=Ku_j$ holds for any covariant vector $u_i$ such that $u_iv^i=0$ where $v^i$ is a given contravariant vector, show that $a^i_j=K\delta^{i}_{j} +p_jv^i$

if the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^ip_i$ where $p_i$ is a given covariant vector ,then $a_ij+a_ji=p_iv_j+p_jv_i$ where $v_j$ is some covariant vector