can anyone at all do this question please. rep will be given. :
find Gx, Gy, Gxx, Gyy, Gxy for G(x,y) = tan^-1(y/x) ... note tan^-1(u) = 1/(1+u^2).
Hi, hope this helps:
G(x,y) = z = arctan(y/x) (arctan = tan^-1, just different notation)
therefore tan(z) = y/x;
differentiate both sides with respect to y:
sec^2(z) dz/dy = 1/x (sec^2(z) = tan^2(z) + 1)
and tan^2(z) = (y/x)^2;
so dz/dy = 1/(x(1 + (y/x)^2));
as for dz/dx, same procedure reveals:
dz/dx = -1/(y(1 + (x/y)^2));
differentiating again with respect to the same variables gives:
d^2z/dx^2 = 2xy/(x^2 + y^2)^2;
d^2z/dy^2 = -2xy/(x^2 + y^2)^2;
and the cross derivative gives:
d^2z/dxdy = 1/(x^2 + y^2) - 2x^2/(x^2 + y^2)^2;
I believe these to be correct, however there may be sign errors in my workings out as I rattled through these quickly. Any that aside, this is the general jist of how to do these derivatives,
Hope this helps.
thanks!! i believe ur answers are correct!
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