Q 2 - Calculus

Question

Examples: Construct examples of the thing($s$) described in

the following. Try to find examples that are different than

any in the reading.

(a) A region in $R^3$ that is most easily expressed with rectangular

coordinates.

(b) A region in $R^3$ that is most easily expressed with

cylindrical coordinates.

(c) A region in $R^3$ that is most easily expressed with

spherical coordinates.

Step-by-Step Solution

Answer

Part (a) $\left\{(x,y,z)\in R^3\mid x^2+y^2+z^2=1\right\}.$

Part (b) $\left\{(r,\theta ,z)\in R^3\mid r^2+z^2=1\right\}\text{. }$

Part (c) $\left\{(\rho ,\theta ,\varphi )\in R^3\mid {\rho }^2=1\right\}.$

Part (a) Step 1: Find an example that is different than any in the reading.

A region$R^3$ in which rectangular coordinates can be stated is,

$\left\{(x,y,z)\in R^3\mid x^2+y^2+z^2=1\right\}.$

This is a sphere equation with a radius $1$

Part (b) Step 1: Find an example that is different than any in the reading.

A region$R^3$ in which cylindrical coordinates can be stated is,

$\left\{(r,\theta ,z)\in R^3\mid r^2+z^2=1\right\}\text{. }$

This is a sphere equation with a radius $1$

Part (c) Step 1: Find an example that is different than any in the reading.

A region$R^3$ in which cylindrical coordinates can be stated is,

$\left\{(\rho ,\theta ,\varphi )\in R^3\mid {\rho }^2=1\right\}.$

This is a sphere equation with a radius $1$