Definition 1 Denote by Lra the straight line that is perpendicular to the direction [cos(a), sin(a) and at distance...
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 (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 (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 -T < a < T. Remark 2 The line Lra is a distance r from (0,0) in the...
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,...
= 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?...
(1) Evaluate the following line integrals in R3. r +yds for C the line segment from (0, 1,0) to (1, 0,0) for C the line segment from (0,1,1) to (1,0,1). for C the circle (0, a cos t, a sin t) for O (iv) 2π, with a a positive constant. t for C the curve (cost +tsint,sint tcost, 0) for Osts v3 (Hint for (i): use the parametrization (z, y, z) = (t, 1-t, 0) for 0 1) t (1)...
(a) Give a set of parametric equations (with domain) for the line segment from (4, -1) to (5,6). (b) Give a set of parametric equations (with domain) for the ellipse centered at (0,0) passing through the points (4,0), (-4,0), (0,3), and (0, -3), traversed once counter-clockwise. (c) Find the (x, y) coordinates of the points where the curve, defined parametrically by I= 2 cost y = sin 2t 0<t<T, has a horizontal tangent.
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 -...
please solve all thank you so much :) Let C be the curve consisting of line segments from (0, 0) to (3, 3) to (0, 3) and back to (0,0). Use Green's theorem to find the value of [ xy dx + xy dx + y2 + 3 dy. Use Green's theorem to evaluate line integral fc2x e2x sin(2y) dx + 2x cos(2) dy, where is ellipse 16(x - 3)2 + 9(y – 5)2 = 144 oriented counterclockwise. Use Green's...