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

question 72

Essay

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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Definitions:

Standard Deviation

A measure of the amount of variation or dispersion of a set of values.

Mean-Variance Criterion

An investment principle where portfolios are selected based on their expected returns (mean) and the variability of those returns (variance), aiming to optimize the trade-off between risk and return.

Expected Return

The average return on an investment anticipated over a specified period based on historical or estimated future performance.

Variance

A statistical measure of the dispersion of a set of data points or portfolio returns, indicating how much the values differ from the average.