Consider a point P on the x − y plane. Show that ur = cos(θ) ˆi
+ sin(θ) ˆj, uθ = − sin(θ) ˆi + cos(θ) ˆj.
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
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...
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
A curve in polar coordinates is given by: r = 9 + 2 cos θ Point P is at θ = 20π/18 (1) Find polar coordinate r for P, with r > 0 and π < θ < 3π/2. (2) Find cartesian coordinates for point P (3) How may times does the curve pass through the origin when 0 < θ < 2π?
The Cartesian coordinates of a point in the xy-plane are (x, y) = (-3.27, -2.33) m. Find the polar coordinates of this point. r = _____m θ = ______° (b) Convert (r, θ) = (4.62 m, 38.6°) to rectangular coordinates. x = ____m y = ____m EXERCISE HINTS: GETTING STARTED | I'M STUCK! (a) Find the polar coordinates corresponding to (x, y) = (3.12, 1.47) m. r = _____m θ = _____° (b) Find the Cartesian coordinates corresponding to (r, θ) = (4.22...
(a) The Cartesian coordinates of a point in the xy-plane are (x, y) = (-3.44, -2.64) m. Find the polar coordinates of this point. r = m θ = ° (b) Convert (r, θ) = (4.73 m, 36.1°) to rectangular coordinates. x = m y = m
= Consider the vector field F(x, y) (cos y + y cos x)i + (sin x – xsin y)j. Show whether the function f(x,y) = x COS Y – y sin x is a potential function for the vector field, F.
Prob. 4 Assume that on the xy-plane, vectors P and Q make angles θ and φ with respect to the r-axis. Use the basic properties of the dot-product of vectors, show that cos(θ + φ)-cos θ cos φ-sin θ sin φ. Also, use the basic properties of the cross-product, show that sin (e+ φ)-cos θ sin o + cos θ sin o.
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
1) If P and θ that correspond to x andy)? If P = (3.4, 29, 13) in cartesian coordianates, what are the values of r, θ, and φ of this point in spherical polar coordinates? (r, y), x = 3.3.) 4.4, what is P in polar coordinates (i.e., what are the values of r