how to create a symbolic derivative of f(r(x,y),theta(x,y)) and evaluate it

Hello Community,

i have some more or less complicated funtions, e.g. a simple one:

f(r,theta) = r*(pi*cos(theta)+2*sin(theta));
x = x0 +r*cos(theta);
y = y0 +r*cos(theta);

and I need the symbolic derivative of f with respect to x any y: d f(r(x,y),theta(x,y)) / dx = ....

my code look like this:

syms f f1 f2 r theta x x0 y0 y pi

f  = r*(pi*cos(theta)+2*sin(theta));
f1 = x == x0 + r*cos(theta);
f2 = y == y0 + r*sin(theta);

dr_dx = diff(solve(f1,r),x)
dtheta_x = diff(solve(f1,theta),x)

dr_dy = diff(solve(f2,r),y)
dtheta_y = diff(solve(f2,theta),y)

df_r = diff(f,r)
df_theta = diff(f,theta)

but how i could create the desired derivative..

Answer 1

[David Durschlag]

Let's break down the operations required:

First, solve for theta and r in terms of x and y:

theta_r = solve([x == x0 + r*cos(theta), y == y0 + r*sin(theta)], [theta, r]);

theta_r will be a struct with possible substitutions for theta and r as its properties.

Second, choose the substitutions (in this case, the first ones):

theta_r_subs = [theta_r.theta(1), theta_r.r(1)];

Third, perform the substitution:

fxy = subs(f, [theta, r], theta_r_subs);

Fourth, differentiate:

df_x = diff(fxy, x);
df_y = diff(fxy, y);

Further solutions can be obtained by choosing different substitutions.

--David

[Marko]

Hello David, this is elegant solved!