Let S be the parametric surface $\vec { r } ( s , t ) = s \cos t \vec { i } + s \sin t \vec { j } + \left( s ^ { 2 } - s \sin t \right) \vec { k }$ for $0 \leq s \leq 2$ , $0 \leq t \leq 2 n$ .
(a)What does the projection of S onto the xy-plane look like?
(b)Show that S is part of the surface $z = x ^ { 2 } + y ^ { 2 } - y$ .
(c)Find a unit vector that is normal to the surface at the point (1,2,3).

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The Answer of Let S be the parametric surface $\vec { r } ( s , t ) = s \cos t \vec { i } + s \sin t \vec { j } + \left( s ^ { 2 } - s \sin t \right) \vec { k }$ for $0 \leq s \leq 2$ , $0 \leq t \leq 2 n$ .
(a)What does the projection of S onto the xy-plane look like?
(b)Show that S is part of the surface $z = x ^ { 2 } + y ^ { 2 } - y$ .
(c)Find a unit vector that is normal to the surface at the point (1,2,3).

Let S Be the Parametric Surface r⃗(s,t)=scos⁡ti⃗+ssin⁡tj⃗+(s2−ssin⁡t)k⃗\vec { r } ( s , t ) = s \cos t \vec { i } + s \sin t \vec { j } + \left( s ^ { 2 } - s \sin t \right) \vec { k }r(s,t)=scosti+ssintj​+(s2−ssint)k

Let S Be the Parametric Surface $\vec { r } ( s , t ) = s \cos t \vec { i } + s \sin t \vec { j } + \left( s ^ { 2 } - s \sin t \right) \vec { k }$