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#### Coordinate transformation in Maple - derivatives

```Hi,

I have a transformation x+iy=exp(a+ib), where a and b are the
coordinates of the new system. What is the best way to derive
different derivatives in the new system in maple? The first derivative
is quite easy, but when we go, to lets say, second derivative of a
certain function by x and y, getting the derivative by a and b becomes
something tricky and long. And I need third derivatives as well, so
I'd rather the computer do it for me.

Victor

```
 0
Victor
8/27/2007 3:18:42 PM
comp.soft-sys.math.maple 4344 articles. 3 followers.

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```In article <1188227922.245830.163710@r29g2000hsg.googlegroups.com>,
Victor  <Mollari@gmail.com> wrote:

>I have a transformation x+iy=exp(a+ib), where a and b are the
>coordinates of the new system.

Let's see... that would be  a = 1/2*ln(x^2+y^2), b = arctan(y,x),
and y = tan(b) and x = +/- sqrt(exp(2*a) - tan(b)^2)

>What is the best way to derive
>different derivatives in the new system in maple? The first derivative
>is quite easy, but when we go, to lets say, second derivative of a
>certain function by x and y, getting the derivative by a and b becomes
>something tricky and long. And I need third derivatives as well, so
>I'd rather the computer do it for me.

Trying a random function...

> F := 3*x^2*y+cos(x*y):
> simplify(diff(F,x,y)) assuming real;
6 x - cos(x y) x y - sin(x y)
> F2 := subs(x=sqrt(exp(2*a)-tan(b)^2),y=tan(b),F);
F2 := 3*(exp(2*a)-tan(b)^2)*tan(b)+cos((exp(2*a)-tan(b)^2)^(1/2)*tan(b))
> diff(F2,a,b);
6*exp(2*a)*(1+tan(b)^2)-cos((exp(2*a)-tan(b)^2)^(1/2)*tan(b))*(-1/(exp(2*a)-
tan(b)^2)^(1/2)*tan(b)^2*(1+tan(b)^2)+(exp(2*a)-tan(b)^2)^(1/2)*(1+tan(b)^2))/
(exp(2*a)-tan(b)^2)^(1/2)*tan(b)*exp(2*a)-sin((exp(2*a)-tan(b)^2)^(1/2)*tan(b)
)/(exp(2*a)-tan(b)^2)^(3/2)*tan(b)^2*exp(2*a)*(1+tan(b)^2)-sin((exp(2*a)-tan(b
)^2)^(1/2)*tan(b))/(exp(2*a)-tan(b)^2)^(1/2)*(1+tan(b)^2)*exp(2*a)
> simplify(%) assuming real;
-(2*cos(b)*cos(1/cos(b)*(exp(2*a)*cos(b)^2-1+cos(b)^2)^(1/2)*sin(b)/abs(cos(b)
))*sin(b)*(exp(2*a)*cos(b)^2-1+cos(b)^2)^(1/2)*abs(cos(b))-6*(exp(2*a)*cos(b)^
2-1+cos(b)^2)^(1/2)*cos(b)^2*exp(2*a)*abs(cos(b))+6*(exp(2*a)*cos(b)^2-1+cos(b
)^2)^(1/2)*abs(cos(b))-6*(exp(2*a)*cos(b)^2-1+cos(b)^2)^(1/2)*cos(b)^2*abs(cos
(b))-2*cos(1/cos(b)*(exp(2*a)*cos(b)^2-1+cos(b)^2)^(1/2)*sin(b)/abs(cos(b)))*
sin(b)*signum(cos(b))*(exp(2*a)*cos(b)^2-1+cos(b)^2)^(1/2)+cos(1/cos(b)*(exp(2
*a)*cos(b)^2-1+cos(b)^2)^(1/2)*sin(b)/abs(cos(b)))*sin(b)*(exp(2*a)*cos(b)^2-1
+cos(b)^2)^(1/2)*exp(2*a)*cos(b)*abs(cos(b))+sin(1/cos(b)*(exp(2*a)*cos(b)^2-1
+cos(b)^2)^(1/2)*sin(b)/abs(cos(b)))*cos(b)^4*exp(2*a))*exp(2*a)/cos(b)^2/(exp
(2*a)*cos(b)^2-1+cos(b)^2)^(3/2)/abs(cos(b))

Bleh. I'd sure rather have the computer do that!
--
Programming is what happens while you're busy making other plans.
```
 0
roberson
8/27/2007 4:45:54 PM

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