# Coordinate Transformation

1. Oct 16, 2004

### Clausius2

Ok, I have a strong problem. It has to do with programming and concepts of mathematics.

I have a coordinate transformation:

$$x=x(\xi,\eta)$$
$$y=y(\xi,\eta)$$;

and I want to calculate the metrics coefficients:

$$\xi_x=\frac{\partial \xi}{\partial x}$$
$$\eta_x=\frac{\partial \eta}{\partial x}$$
$$\xi_y=\frac{\partial \xi}{\partial y}$$
$$\eta_y=\frac{\partial \eta}{\partial y}$$

But I have only the next (I only have the numeric transformation, I don't have any analytical formulae):

x= a matrix of dimensions $$n_{\xi}, n_{\eta}$$;
y= a matrix of dimensions $$n_{\xi}, n_{\eta}$$;

$$\xi$$= a vector of size $$n_{\xi}$$
$$\eta$$= a vector of size $$n_{\eta}$$

Well, I want to compute the metrics coefficients using central diferences. Please, tell me what would you do in order to compute it. I'm goind to show you what i made in Matlab:

for i=2:nxi-1;
for j=2:neta-1;

XI_x(i)=(XI(i+1)-XI(i-1))/(x(i+1,j)-x(i-1,j));
ETA_x(j)=(ETA(j+1)-ETA(j-1))/(x(i,j+1)-x(i,j-1));
XI_y(i)=(XI(i+1)-XI(i-1))/(y(i+1,j)-y(i-1,j));
ETA_y(j)=(ETA(j+1)-ETA(j-1))/(y(i,j+1)-y(i,j-1));

end
end

I'm not sure it works, because I'm not able to imagine the metrics coefficients just acting upon the functions x and y. I don't know If i'm taking correctly the variations in x and y in the denominators.

Any advice will be greatly appreciated.