The applet on this page uses Sylvester's resultants to reduce systems of polynomial equations. For example, it can be used to reduce a system of 4 polynomials in 4 variables to a single polynomial in one of the variables. The reduction can be done all at once, or step by step.
Equation of a plane
Want to know the equation of a plane through three points? Suppose I want the equation of the plane through (2,3,4), (-1,2,8) and (4,0,-6). Into the 'enter polynomials' part of the applet, I could type in the equation of the plane
So, my 'polynomials' section reads
a*x+b*y+c*z-d. Then, I substitute the three points and type these equations in too (before the equation in x,y and z.
4*a+0*b-6*c-d. Into the 'variables' field, I enter the variables :
x,y,z,a,b,c,d. For this example, it's important that
x,y,z come before a,b,c,d. This is because we want the equation of the plane in terms of x, y and z.
2*a+3*b+4*c-d,-a+2*b+8*c-d,4*a+0*b-6*c-d,a*x+b*y+c*z-d, and the 'variables' section reads
x,y,z,a,b,c,d. Clicking 'Fully Reduce' gives
(-88 + 88*x + -88*y + 44*z)*a, which I can divide by 44a to give the equation
Finding a Differential Equation
Trying to work out the relationship between the mass and velocity of a rocket with exhaust velocity u is tough. Using conservation of momentum in special relativity gives an awful equation :mv/sqrt(1-(v/c)2) = (m-dm)(v+dv)/sqrt(1-((v+dv)/c)2) + dm(v-(1-(v/c)2)u)/sqrt(1-(v/c-(1-(v/c)2)u/c)2)!.
Then, we have to rearrange this to get dv/dm in terms of the other quantities. Multiplying through by c helps a little, but not much.
Then I though, why not write this whole thing as a bunch of polynomials to solve, and let the applet do the donkey work? I typed
(-m*v/a+(m*v-v*dm+m*dv)/b + dm*x/e)*a*b*e, a^2-c^2+v^2, b^2+(v+dv)^2-c^2,e^2-c^2+x^2, c^2*x-c^2*v-(c^2-v^2)*u, dvdm*dm-dv
into the 'polynomials' section. I hope you'll agree this is equivalent to the equation above. For the variables, I tried
c,m,v,u,dvdm,dm,a,b,e,x,dv. Eliminate x first, then e, then b, then a, then dv. This failed completely.