Multivariable Calculas with Mapple Plot
0 likes81 views
This document contains solutions to problems involving spherical and Cartesian coordinate systems. It begins by presenting the solution to problem 2(a), which defines a function z=f(x,y) for both coordinate systems. Next, it solves problem 2(b) by deriving and plotting the equation r=f(θ) for both systems. Finally, it solves problem 2(c) and 2(d) through similar methods, presenting the equations and Maple plotting commands with figures for each part. Maple code is included to generate 3D plots visualizing the surfaces defined by the equations.
![Problem 2
(a).
Here,
z=r2
For Cartesian Co-ordinate
z=(x2+y2)2 [r2=x2+y2]z=(x +y ) [r =x +y ]
Or, x4+x2y2+y4=z
For spherical Co-ordinate
z=r2
Or, = ( )2 [z= and r= ]
=
φρsinρ
ρ
φ
φ
2
sin
cos
φcos φρ cos φρsin
∴](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-2-320.jpg)
![Maple Plot Command:
(a)with(plots):
> rho:=(phi)-> cos(phi)/sin^2 (phi);
> sphereplot([rho(phi),theta,
:=ρ →φ
( )cos φ
( )sin φ 2
> sphereplot([rho(phi),theta,
phi],theta=0..2*Pi, phi=0..Pi,axes=normal);](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-3-320.jpg)
![Continue…
>Cylinderplot ([r,theta,r^2],r=-2..2, theta=0..2*Pi);
Figure](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-6-320.jpg)
![b.
(b).
Here,
For Cartesian Co-ordinate
[Multiplying both side by r]
Or, x2+y2= 4x [r2=x2+y2 and ]
θcos4=r
θcos42
rr =
θcosrx =Or, x +y = 4x [r =x +y and ]
Or, x2+y2-4x=0
For spherical Co-ordinate
[Putting ]
Or,
θφρ cos4sin = φρsin=r
φ
θ
ρ
sin
cos4
=](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-7-320.jpg)
![Maple Plot Command:
(b)
> restart:
> with(plots):
rho:=(theta,phi)-> 4*cos (theta)/sin(phi);rho:=(theta,phi)-> 4*cos (theta)/sin(phi);
> sphereplot([rho(theta, phi),theta,
phi],theta=0..2*Pi, phi=0..Pi,axes=normal);
:=ρ →( ),θ φ 4
( )cos θ
( )sin φ](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-8-320.jpg)
![Continue..
>cylinderplot([4*cos(theta),theta, z], z=-2..2, theta=0..2*Pi);](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-11-320.jpg)
![c.
(c).
Here,
r2+z2=4, z < -2
For Cartesian Co-ordinate
x2 + y2+z2=4 [r2=x2+y2] as z < -2
For spherical Co-ordinateFor spherical Co-ordinate
Putting
Or,
Or,
Or,
as z < -2 so < 2
φρφρ cos,sin == zr
2
4
4)cos(sin
4)cos()sin(
2
222
22
=
=
=+
=+
ρ
ρ
φφρ
φρφρ
ρΘ](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-12-320.jpg)
![Maple Plot Command with figure
(c)> restarts:
> with(plots):
> Sphereplot ([2,theta, phi],theta=0..2*Pi,
phi=0..Pi,axes=normal);](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-13-320.jpg)
![Continue…
>cylinderplot([sqrt(4-z^2),theta,z],z=-2..2, theta=0..2*Pi);](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-14-320.jpg)
![Maple Plot Command :
(d)> restarts:
> with(plots):
> rho:=(theta,phi)-> -cos(theta)/(sin(phi)^2*cos(2*theta));
:=ρ →( ),θ φ −
( )cos θ
> Sphereplot ([rho(theta, phi),theta, phi],theta=0..2*Pi,
phi=0..Pi,axes=normal);
:=ρ →( ),θ φ −
( )sin φ 2
( )cos 2 θ](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-18-320.jpg)
![Continue…
>cylinderplot([r,theta,-r^2*cos(2*theta)],r=-2..2,
theta=0..2*Pi);](https://image.slidesharecdn.com/multivariablecalculaswithmappleplot-190910154406/85/Multivariable-Calculas-with-Mapple-Plot-20-320.jpg)
More Related Content
Similar to Multivariable Calculas with Mapple Plot (20)
Multivariable Calculas with Mapple Plot
- 1. Welcome On the PresentationOn the Presentation Niroj Dahal School of Education 2011 Feb. Batch
- 2. Problem 2 (a). Here, z=r2 For Cartesian Co-ordinate z=(x2+y2)2 [r2=x2+y2]z=(x +y ) [r =x +y ] Or, x4+x2y2+y4=z For spherical Co-ordinate z=r2 Or, = ( )2 [z= and r= ] = φρsinρ ρ φ φ 2 sin cos φcos φρ cos φρsin ∴
- 3. Maple Plot Command: (a)with(plots): > rho:=(phi)-> cos(phi)/sin^2 (phi); > sphereplot([rho(phi),theta, :=ρ →φ ( )cos φ ( )sin φ 2 > sphereplot([rho(phi),theta, phi],theta=0..2*Pi, phi=0..Pi,axes=normal);
- 4. Figure
- 7. b. (b). Here, For Cartesian Co-ordinate [Multiplying both side by r] Or, x2+y2= 4x [r2=x2+y2 and ] θcos4=r θcos42 rr = θcosrx =Or, x +y = 4x [r =x +y and ] Or, x2+y2-4x=0 For spherical Co-ordinate [Putting ] Or, θφρ cos4sin = φρsin=r φ θ ρ sin cos4 =
- 8. Maple Plot Command: (b) > restart: > with(plots): rho:=(theta,phi)-> 4*cos (theta)/sin(phi);rho:=(theta,phi)-> 4*cos (theta)/sin(phi); > sphereplot([rho(theta, phi),theta, phi],theta=0..2*Pi, phi=0..Pi,axes=normal); :=ρ →( ),θ φ 4 ( )cos θ ( )sin φ
- 9. Figure
- 11. Continue.. >cylinderplot([4*cos(theta),theta, z], z=-2..2, theta=0..2*Pi);
- 12. c. (c). Here, r2+z2=4, z < -2 For Cartesian Co-ordinate x2 + y2+z2=4 [r2=x2+y2] as z < -2 For spherical Co-ordinateFor spherical Co-ordinate Putting Or, Or, Or, as z < -2 so < 2 φρφρ cos,sin == zr 2 4 4)cos(sin 4)cos()sin( 2 222 22 = = =+ =+ ρ ρ φφρ φρφρ ρΘ
- 13. Maple Plot Command with figure (c)> restarts: > with(plots): > Sphereplot ([2,theta, phi],theta=0..2*Pi, phi=0..Pi,axes=normal);
- 15. Continue.. > implicitplot3d(x^2+y^2+z^2=4, x=-4..4, y=-2..3, z=1..3);
- 16. d. (d). Here, Or, Or, Or, For Cartesian Co-ordinate 0sincos 0)sin(cos 02cos 2222 222 2 =−+ =−+ =+ θθ θθ θ rrz rz rz For Cartesian Co-ordinate Putting z+x2-y2=0 x2-y2 +z=0 θθ sin,cos ryrx == Θ
- 17. Continue.. For spherical Co-ordinate Putting Or, θφρφ θφρφρ φρφρ 02cossincos 02cossincos cos,sin 2 22 =+ =+ == zr Or, θφ φ ρ θφρφ 2cossin cos 02cossincos 2 2 − = =+ Θ
- 18. Maple Plot Command : (d)> restarts: > with(plots): > rho:=(theta,phi)-> -cos(theta)/(sin(phi)^2*cos(2*theta)); :=ρ →( ),θ φ − ( )cos θ > Sphereplot ([rho(theta, phi),theta, phi],theta=0..2*Pi, phi=0..Pi,axes=normal); :=ρ →( ),θ φ − ( )sin φ 2 ( )cos 2 θ
- 19. Figure
- 21. Continue.. > implicitplot3d(x^2+y^2+z^2=4, x=-4..4, y=-2..2, z=1..2);
- 22. Thank You