Definition 1 Denote by Lra the straight line that is perpendicular to the direction [cos(a), sin(a)] and at distanc...
Definition 1 Denote by Lra the straight line that is perpendicular to the direction [cos(a), sin(a)] and at distance r from the origin 0 (0,0). Thus (r, y) is on the line Lr,a if and only if r cos(a) +y sin(a) - Common choices are r E R and 0<a< T. Another potential choice might be r 2 0 and -<aT =r. Remark 2 The line Lr,a is a distance r from (0,0) in the direction perpendicular to [cos(a), sin(a)]....
Definition 1 Denote by Lra the straight line that is perpendicular to the direction [cos(a), sin(a) and at distance r from the origin 0= (0,0). Thus (z, y) is on the line Lra if and only if r cos(a)+y sin(a) r Common choices are r E R and 0 a<. Another potential choice might be r2 0 and -T<asT. Remark 2 The line Lra is a distance r from (0,0) in the direction perpendicular to [cos(a), sin(a)] Consequently, the point...
Definition 1 Denote by Lra the straight line that is perpendicular to the direction [cos(a), sin(a) and at distance r from the origin 0= (0,0). Thus (z, y) is on the line Lra if and only if r cos(a)+y sin(a) r Common choices are r E R and 0 a<. Another potential choice might be r2 0 and -T<asT. Remark 2 The line Lra is a distance r from (0,0) in the direction perpendicular to [cos(a), sin(a)] Consequently, the point...
= Pcos 4AEOT 3p sin cos e 4. A field line in the dipole field is described in polar coordinates by the simple equation r=ro sin, where ro is the radius at which the field line passes through the equatorial plane of the dipole. Show that this is true by demonstrating that for any point on the field line curve the tangential direction is the same as that of the electric field of the dipole. 418€gr3 F - P(3 cos?...
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,...
Prob. 12.8 Criticize the following 'proof that r = 0. (a) Apply Green's theorem in a plane to the functions P(x, y) = tan-(y/x) and Q(x, y) = tan-'(x/y), taking the region R to be the unit circle centered on the origin. (b) The RHS of the equality so produced is S SR dxdy, which, either from symmetry considerations or by changing to plane polar coordinates, can be shown to have zero value. (c) In the LHS of the equality,...
The velocity vector of an object is given by y(t) = (* sin(at), 1, a cos(at)). Assume that at t = 1, the object is at the point P(1,1,0). (a) Find the position vector F(t) of the object. (b) Find parametric equations of the line which is tangent to r(t) at P. (c) Find the distance that the object traveled from the point t = 0 to t = 1. (d) Find an equation for the normal plane of r(t)...
(4) Evaluate the line integral F dr where C is the epicycloid with parametrization given by r(t) 5 cos t - gradient of the function f(x, y) = 3 sin(ry) + cos(y2) cos 5t and y(t) = 5 sin t - sin 5t for 0 < t < 2« and F is the (5) EvaluateF dr where F(x, y) with counterclockwise orientation (2y, xy2and C is the ellipse 4r2 9y2 36 _ F dr where F(r, y) = (x2 -...
you can skip question 1 Sketch the graph of x(t) sin(2t), y(t) = (t + sin(2t)) and find the coordinates of the points on the graph where the tangent is horizontal or vertical (please specify), then compute the second derivative and discuss the concavity of the graph. 1. Show that the surface area generated by rotating, about the polar axis, the graph of the curve 2. f(0),0 s asesbsnis S = 2nf(0)sin(0) J(50)) + (r°(®)*)de Find an equation, in both...