Having trouble with error propagation

higgs629

I'm measuring the error of a measurement of the magnetic field in r and theta. I'd like to find the error in the x and y directions.
Let the value of the magnetic field in x be X
Let the value of the magnetic field in y be Y
Let the value of the magnetic field in theta be D
Let the value of the magnetic field in r be R

cos(D) = X / R
sin(D) = Y / R

R ^ 2 = X ^ 2 + Y ^ 2

Thus,

X² = R² * cos²(D)
Y² = R² * sin²(D)

X = R * cos(D)
Y = R * sin(D)

I use the following equations for error propagation:

E_x² = [itex]\frac{dX}{dr}[/itex]²E_r² + [itex]\frac{dX}{dtheta}[/itex]²E_D²

E_y² = [itex]\frac{dY}{dr}[/itex]²E_r² + [itex]\frac{dY}{dtheta}[/itex]²E_D²

[itex]\frac{dX}{dr}[/itex] = cos(D)
[itex]\frac{dX}{dtheta}[/itex] = - R sin (D)
[itex]\frac{dY}{dr}[/itex] = sin(D)
[itex]\frac{dY}{dtheta}[/itex] = R cos (D)
thus,
E_x² = cos²(D)*E_r² + R²sin²*E_D² (1)

E_y² = sin²(D)*E_r² + R²cos²*E_D² (2)


Which is fine and dandy and matches my physical expectation.

The problem I'm having is if I do this in reverse I get a contradictory result.

Suppose I know E_x and E_y and need E_r and E_D.

E_r² = [itex]\frac{dR}{dx}[/itex]²E_x² + [itex]\frac{dR}{dy}[/itex]²E_y²

and

E_D² = [itex]\frac{dD}{dx}[/itex]²E_x² + [itex]\frac{dD}{dy}[/itex]²E_D²


[itex]\frac{dR}{dx}[/itex] = X / R
[itex]\frac{dR}{dy}[/itex] = Y / R

[itex]\frac{dD}{dx}[/itex] = [itex]\frac{-y}{R²}[/itex]
[itex]\frac{dD}{dy}[/itex] = [itex]\frac{x}{R²}[/itex]

giving:

E_R² = (X/R)² * E_x² + (Y/R)²* E_y² (3)

E[SUB]D[/SUB² = (Y/R²)² E_x² + (X/R²) * E_y² (4)

Then solve these equations for E_y and E_x I find that the E_x and E_y in these equations end up disagreeing with the ones in the previous set.

For E_X
Multiply equation (3) through by R²
In equation (3) subtract the E_x term from both sides.
Multiply equation (3) by (R/Y)²
Substitute the resulting expression of E_y into equation (4)

Then solving for E_x² yielding:

E_x² = [itex]\frac{R²sin²(D)*E_D² - cos²(D) * E_r²}{sin²(D) - cos²(D)}[/itex]

Which is off from equation (1) for the addition of two negative signs. (one in the numerator and one in the denominator, I don't believe it to be equal to equation 1 but perhaps I am wrong somehow. Additionally this equation allows for negative values of E_x²! I'm not sure where I went wrong. Is there some reason I can't do what I've done?