Exercise
Given a, b ∈ R^∗ and t is a variable real.
1) Solve in R^2 for x,y this system:
{ ((xsin t−ycos t=−a)),((xcos t+ysin t=b.)) :}
2)/Demonstrate that these solutions can
be written like this ( r and θ ∈ R).
{ ((x=rcos(t+θ))),((y=rsin(t+θ))) :}
3) we suppose now that a=b=1 and θ=(π/(12))
solve this in [0;2π[
{ ((rcos(t+θ)≥−1)),((rsin(t+θ)<−1)) :}
1. Show that:
∫_0 ^( π) ((xdx)/((a^2 sin^2 x+b^2 cos^2 x)^2 )) = ((π^2 (a^2 +b^2 ))/(4a^3 b^3 ))
2.The density at the point (x,y) of a lamina bounded by the circle
x^2 +y^2 −2ax=0 is ϱ =x find its mass.
3.∗
4. If z= ((cos y)/x) and x=u^2 −v , y=e^x find (dz/dv).
We suppose in R^2 the base (i^→ ;j^→ ).
we have these vectors:
u^→ =(m^2 −m)i^→ +2mj^→ ;
v^→ =(m−1)i^→ +(m+1)j^→ m ∈ R^∗
1)Determinate m for which the system
(u^→ ;v^→ ) is linear dependant( det(u^→ ;v^→ )=0)