Change of Variables project 

Ruhani Alam  

 

Calculating dV for spherical coordinates 

 

Change of variables:  

x=`ρsin`*`φcos`*theta 

y=`ρsin`*`φsin`*theta 

z=rho*cos*phi 

 

 

 

Image 

 

 

 

delta(x, y, z)/delta(rho, theta, phi)= Matrix(%id = 145160416) 

 

 

delta(x, y, z)/delta(rho, theta, phi) =Matrix(%id = 145901396)  

     

 

Determinant above= a*Typesetting:-delayCrossProduct(b, c) 

 

delta(x, y, z)/delta(rho, theta, phi)=  

 

Determinant above=c*Typesetting:-delayCrossProduct(a, b) 

The two determinants are equal to each other by Theorem 8 of Section 10.4 which states that  

If a, b, c are vectors and c is scalar, then a*Typesetting:-delayCrossProduct(b, c) = c*Typesetting:-delayCrossProduct(a, b) 

 

delta(x, y, z)/delta(rho, theta, phi)= -cos^2*phi*rho^2*sin*phi(sin^2*theta+cos^2*theta)-rho^2*sin^3*phi(sin^2*theta+cos^2*theta) 

 

delta(x, y, z)/delta(rho, theta, phi)= -cos^2*phi^2*rho^2*sin-rho^2*sin^3*phi 

 

delta(x, y, z)/delta(rho, theta, phi)=-rho^2*sin*phi(cos^2*phi+sin^2*phi) 

 

delta(x, y, z)/delta(rho, theta, phi)=-rho^2*sin*phi 

 

abs(delta(x, y, z)/delta(`Á`, `¸`, `Õ`))=abs(-rho^2*sin*phi) 

 

abs(delta(x, y, z)/delta(`Á`, `¸`, `Õ`))rho^2*sin*phi 

 

This is the factor for spherical coordinates 

The dV is now this factor times d*`ρd`*`θd`*phi 

dV=rho^2*sin*phi^2*d*`ρd`*`θd` 

 

Example:  

Find the mass of  a star that is a ball of radius 3 centered at the origin if the density of the star is  

g(x,y,z)=10-x2-y2-z2  

 

g(x,y,z)dxdydz 

 

= 

 

This integral is a mess, so using spherical coordinate would make it a lot easier.   

 

V= 

 

= 

=828/5*Pi 

520.2477 

 

Calculating dV for cylindrical coordinates 

change of variables:  

x=rcos*theta 

y=rsin*theta 

z=w 

 

 

Image 

 

 

 

`´`(x, y, z)/`´`(r, `¸`, w) = Matrix(%id = 151521348) 

 

 

 

Determinant above= a*Typesetting:-delayCrossProduct(b, c) 

 

delta(x, y, z)/delta(r, theta, w) = Matrix(%id = 145755272) 

 

Determinant above= c*Typesetting:-delayCrossProduct(a, b) 

The two determinants are equal to each other by Theorem 8 of Section 10.4 which states that  

If a, b, c are vectors and c is scalar, then  

a*Typesetting:-delayCrossProduct(b, c) = c*Typesetting:-delayCrossProduct(a, b) 

 

 

(1)(cos*theta^2*rcos+rsin^2*theta) 

 

delta(x, y, z)/delta(r, theta, w)= rcos^2*theta+rsin^2*theta 

 

r 

 

The change of variables factor is absolute value of this determinant.   

Then dV is this factor times dzd*`θdr` 

 

abs(delta(x, y, z)/delta(r, theta, w)) = r 

 

dV= rdzd*`θdr` 

 

Example:  Find the volume of cone of height 1 and radius 1.  Bounded by the surface z=sqrt(x*`2`+y*`2`) and plane z=1.   

Using cartesian coordinates, the integral is:  

dzdydx 

 

Since this integral is extremely messy, it can be made a lot easier using cylindrical coordinates:  

V=dzdthetadr 

 

= 

=3.14 

 

To find the volume of an ice cream cone:  

 

 

Image 

 

In Cartesian coordinates:  

dzdydx 

 

Since this is very messy, spherical coordinates can be used to make the integral a lot cleaner 

 

Spherical coordinates: 

rho^2*sin*phi^2*d*`ρd`*`θd` 

 

= 

 

.6134 

 

This same method from above can be applied to any non-standard shape