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- find an equation of the form ρ = f (θ , φ) in spherical coordinates for the following surfaces x2 − y2 = 4we have the equation z=c, convert it to cylindrical coordinates, and then convert to spherical cooedinatesDetermine whether the following trajectory lies on either a circle in ℝ2 or a sphere in ℝ3 centered at the origin. If so, find the radius of the circle or sphere, and show that the position vector and the velocity vector are everywhere orthogonal. r(t)=(10 cos 2t,10 sin2t), for 0≤t≤π The trajectory lies on a circle with radius_?
- Find a parametrization, using cos(?) and sin(t), of the following curve:The intersection of the plane y=4 with the sphere ?^2 + ?^2 + ?^2 = 80 ?(t)=⟨ , , ⟩Obtain the following expression for the cylindrical coordinates....The position of an object subject to uniform circular motion around the origin is given by ?(?) = ? (cos(??) sin(??)(a) Calculate the velocity and acceleration vectors for this trajectory.(b) Show that ?(?) is perpendicular to ?(?) for all ?.(c) Show that ?(?) · ?(?) = −?2R2 for all ?. Hint: Recall that sin2(?) + cos2(?) = 1