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May 18

Differential solid angle

Why is the solid angle element d(omega) equal to dl*dm/sqrt(1-l^2-m^2), where l and m are direction cosines with respect to two orthogonal axes? --99.237.234.104 (talk) 03:57, 18 May 2010 (UTC)[reply]

wheels of constant width and variations

A mathematician might be able to give a quicker answer to this question Wikipedia:Reference_desk/Science#Square_wheels than I can. Thanks.77.86.10.27 (talk) 15:58, 18 May 2010 (UTC)[reply]

A neat and elegant proof can be found here. The method can be extended to any regular polygon by truncating the catenary at a point corresponding to the exterior angle. Dbfirs 20:46, 19 May 2010 (UTC)[reply]

Fit an exponential curve

I have two points, and the desired slope at one of the points. How can I translate this information into an equation of the form y=a*e^(b*x)+c ? It seems like it should be possible, but trying to solve the equations is just tying me in knots. 173.52.5.181 (talk) 19:11, 18 May 2010 (UTC)[reply]

The problem here is that b appears in a nonlinear way in the equations. To make this problem a little less bad, you can try to isolate the nolinearity in one eqation, instead it popping up in all of the simultaneous set of the equations. You can do that by considering the quantity:

[y2 - y1]/y1'

where y2 and y1 are the the y-coordinate of the two points and y1' is the slope. This only depends on b and you can solve for it using e.g. Newton Raphson. Once you have b, you have to deal with only linear equations for the two other variables. Count Iblis (talk) 20:06, 18 May 2010 (UTC)[reply]

You have the 3 equations

ae^{bx_{1}}+c-y_{1}=ae^{bx_{2}}+c-y_{2}=abe^{bx_{1}}-y'_{1}=0

in the 3 unknowns, $a,b,c.$ If this is confusing you may rename

(a,b,c,x_{1},y_{1},x_{2},y_{2},y'_{1})\rightarrow (x,y,z,a_{1},b_{1},a_{2},b_{2},c)

getting the 3 equations

xe^{a_{1}y}+z-b_{1}=xe^{a_{2}y}+z-b_{2}=xye^{a_{1}y}-c=0

in the 3 unknowns $x,y,z.$ A standard approach is to eliminate variables to obtain 3 equations in one variable each, and then solve these equations numerically. The unknown $z$ is eliminated by subtraction:

(xe^{a_{1}y}+z-b_{1})-(xe^{a_{2}y}+z-b_{2})=xye^{a_{1}y}-c=0

and simplification

x(e^{a_{1}y}-e^{a_{2}y})-b=xye^{a_{1}y}-c=0

where $b=b_{1}-b_{2}$ . Multiplication and subtraction

ye^{a_{1}y}(x(e^{a_{1}y}-e^{a_{2}y})-b)-(e^{a_{1}y}-e^{a_{2}y})(xye^{a_{1}y}-c)=0

eliminates $x$ after simplification:

ye^{a_{1}y}(-b)-(e^{a_{1}y}-e^{a_{2}y})(-c)=0

(b/c)ye^{a_{1}y-a_{2}y}-(e^{a_{1}y-a_{2}y}-e^{a_{2}y-a_{2}y})=0

Insert $a=a_{1}-a_{2}$ and $A=b/c.$

Aye^{ay}-(e^{ay}-1)=0.

The false solution $y=0$ is discarded

e^{ay}-{\frac {e^{ay}-1}{Ay}}=0.

Simplify by setting $w=ay$ and $B=A/a$

e^{w}-{\frac {e^{w}-1}{Bw}}=0.

The solution $w$ is found numerically and substituted into the former equations which are then easily solved. Then undo the renaming and you are finished. Bo Jacoby (talk) 12:42, 19 May 2010 (UTC).[reply]