Consider the Top Half of the Ellipsoid $x ^ { 2 } + y ^ { 2 } + \frac { 1 } { 4 } z ^ { 2 } = 1 , z \geq 0$

Short Answer

Consider the top half of the ellipsoid $x ^ { 2 } + y ^ { 2 } + \frac { 1 } { 4 } z ^ { 2 } = 1 , z \geq 0$ parametrized by $x = \sin u \sin v , y = \cos u \sin v , z = 2 \cos v , 0 \leq v \leq \frac { \pi } { 2 }$ . Find a normal vector N at the point determined by $u = \frac { \pi } { 4 } , v = \frac { \pi } { 3 }$ , and determine if it is upward and/or outward.