Show that the parametrization of the folium of Descartes given by (x, y) = ( 3Rt / 1 + t^3 ) (1, t) is regular.
Show that the parametrization of the folium of Descartes given by (x, y) = ( 3Rt...
I don't know how to do both a and b 16. The folium of Descartes is the curve C c R2 described by the equation a) Give a parametrization of C. Hint: Use t y/r as parameter b) Compute the area of the "leaf." 16. The folium of Descartes is the curve C c R2 described by the equation a) Give a parametrization of C. Hint: Use t y/r as parameter b) Compute the area of the "leaf."
Thank you! I will rate the answer. a ryWith 5. The folium of Descartes is an algebraic curve given by the equation + v a = 2, the equation reads x + y = 6xy. (a) Find the equation of the tangent line to this equation at the point (2,3). h Find the equation of the normal line to this equation at the point (2.3).
A very important application of Green's theorem is that the bounded domain cut out by a simple closed curve C has area given by: x dy - ydx where C is traversed in the counterclockwise way. Find the area of the of folium of Descartes in the first quadrant given by the equation Hint: You will need to directly compute the above line integral and thus you will need a parametrization for the folium in the first quadrant. The parameter...
3. Let F(x, y) = (-y, x) and let C be the semicircle with parametrization f(t) = (4 cos t, 4 sin t) for 0 <ts. Compute ScF.T ds.
A curve called the folium of Descartes can be represented by the parametric equations shown below. 912 1 + 13 9t and y 1 + t3 (a) Convert the parametric equations to polar form. 9 cos( 0)sin (6) c())3 (sin(0)5 cos(θ) ),+ (sin( (b) Sketch the graph of the polar equation from part (a). 71/2 0.10 0.05 -4 0.05 0.10 0.15 -0.05 0.05 π/2 10 1.0 0.5 0.5 -0.5 10 -0.5 (c) Use a graphing utility to approximate the area...
Name that Parametrization 4. (a) Consider these two sets of parametric cquations: x(1)-1 y(t) = sin t 0<t<oo x (t) sin t y(t) = t 0<t<00 What is the difference between their associated curves? (b) Given any set of equations of the form What does the graph of the set of equations y(r) = 1 x(!) = f (1) 0<t<oo look like? Use a calculator to check that f(x) = x5-3x3 + 5x + 2 is one-to-one. 5, (a) (b)...
Calculus . Let h(x, y) be a smooth parametrization on a region H for a surface S in R3. Suppose there is a continuous transform F :R + H, (u, v) + x(u, v), y(u, v)) such that F is one-to-one on the interior of the region R and r;=ho F is a smooth parametrization on R for S. Show that 9 S/ \ru xroldA= S/ \he x hy|dA= A(s where A(S) is the area of S. (15 pts] 9
13. (5 points) parametrization Find an equation of the tangent line to the curve given by the =t- - y = 1+12 at t=1.
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the unit sphere S2. Let and show that a new parametrization of the coordinate neighborhood x(U) = V can be given by y(u, (sech u cos e, sech u sin e, tanh u Prove that in the parametrization y the coefficients of the first fundamental form are Thus, y-1: V : S2 → R2 is a conformal...
3. 8p] Show that the force field F(x,y, z) sin y, x cos y + cos z, -y sin z) is conservative and use this fact to evaluate the work done by F in moving a particle with unit mass along the curve C with parametrization r(t (sin t, t, 2t), 0 <t<T/2. 4. 8p] A thin wire has the shape of a helix x = sin t, 0 < t < 27r. If the t, y = cos t,...