(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f,...
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...
(a) Let θ : R-+ R be a smooth function. Find the (signed) curvature of the curve a:R- R2 given by cos(θ(t)) dt,I α(s) sin(θ(t)) dt Use your result to give another geometric interpretation to the (signed) curva- ture and its sign? to) rindy,R-- parmetrised with unit speed suchhat y -0and kt) - s for all seR. (a) Let θ : R-+ R be a smooth function. Find the (signed) curvature of the curve a:R- R2 given by cos(θ(t)) dt,I...
The letters rand represent polar coordinates. Write the following equation using rectangular coordinates (x,y). 2 = 14 cos e NICO The equation using rectangular coordinates (x,y) is (x² + y2) 14x =0. r2 = 14cos R(+² ) = K (14 cos ) R² = 14R Coso (R2) 3/4 = 14 Rcoso (x² + y2 %=148 -14 -14 (x² + y2 3%2_14=0 mistake? Did I make a Thank you
V X2 + y2 and θ u(r(z, y), θ(x, y))--sech2 r tanh r sin θ 6. [Sec. I 1.5] Letr tan l (y/z) be the usual polar rectangular coordinates relationships. Furthermore, define and u(r(z, y),θ(z, y)) sech2 r tanh r cos θ Show that tanh r
3. If T2 = r3 cos(0) sin(d) and v2 = sin(0) cos(O)f + r sin(0)θ + r2 sin(d)φ compute the following (a) ▽T, (b) ▽.
The polar equation r = θ defines a spiral in the xy-plane. Let C be the portion of this spiral starting at (x, y) = (−3π, 0) and ending at (x, y) = (0, 0). Let F(x, y) = < (y − 1)^e cos(x−xy) sin(x − xy), xe^cos(x−xy) sin(x − xy) > Find Z C F · dr
Write an equivalent equation using polar coordinates. 3x-y=6 The polar equation is r= : sin cos
The letters x and y represent rectangular coordinates. Write the given equation using polar coordinates (1.0). 5x² + 5y² = 6 Select the correct equation in polar coordinates below. OA. TE OB. 7 = 1/35 OCZ 5(cos + sin ) 6 5(cos + sin ) OD. 6 "- 5
#49,53,57 3- lar coordinates to polar coordinates will Polar Coordinates Convert blar coordinates with r> 0 and the ove describe of the the rectangular con 050<27. 37. (-1,1) be app 39. (V8, V8) 41. (3.4) 38. (3V3,-3) 40. (-V6, -V2) 42. (1,-2) 44. (0, -V3) your a (a) Yo (b) YO 43. (-6,0) Rectangular Equations to Polar Equations Convert the equation to polar form. 45. x = y *.47. y = x² 49. x = 4 46. x² + y2...
Let F(x, y, z) = sin yi + (x cos y + cos z)j – ysin zk be a vector field in R3. (a) Verify that F is a conservative vector field. (b) Find a potential function f such that F = Vf. (C) Use the fundamental theorem of line integrals to evaluate ScF. dr along the curve C: r(t) = sin ti + tj + 2tk, 0 < t < A/2.