3. Let C be the curve r(t) = < sint, cost, t>,0 sts 1/2. Evaluate the line integral S ry ryds. 1/V2. 1/2, V2, 0,
Given the path C: x(t) = (cost, sint, t), 0<t<2n. Let f(t, y, z) = x2 + y2 + 22. Evaluate (12 pts) f(,y,z)ds.
Let C be the helix parametrized by r(t) = (cost, sint,t), 0 <t<7/2 in R3. Compute the flow of the vector field (x – yz sin xyz, zey? – zx sin xyz, yeyz – xy sin xyz) along C.
:) IS (x+y+z)ds X-1 (b): Find the work done by F over the curve in the direction of increasing t, where F =< x² + y, y2 + 1, ze >, r(t) =< cost, sint,t/27 >, Osts 27. y-2=2-3 =+ C) -1-2 I-3
Given ř(t) =< 2 cost, t, 2 sint > as a trace of a moving object. (a) Find the curvature of K(t). (b) Find the arc length when 0<t <31. (c) Find the unit normal and binormal vectors of F(t).
(7.5 points) Let C be the oriented closed space curve traced out by the parametrization r(t) = (cost, sint, sin 2t), 0<t<27 and let v be the vector field in space defined by v(x, y, z) = (et - yº, ey + r), e) (a) Show that C lies on the cylinder x2 + y2 = 1 and the surface z = 2cy. (b) This implies that C can be seen as the boundary of the surface S which is...
1. Find 12 + y² + 22 ds where is the helix r(t) = (a cost, a sint, bt) and 0 <t<l. 2. Evaluate |(2.84 +248, +16) - dr where C' is a curve that begins at (0,1) and ends at (1,2).
Problem 3 The curve C is given by the parameterization r(t) = (t?,t) for 0 <t< 1. Find the midpoint of this curve.
(a) Sketch the curve r(t) = (e cost, e sint) in R2 and compute its are length for 0 < t < 87. For the sketch, use of software is acceptable, but the graph should be drawn by hand and the right features should be present.] (b) The vector v makes an angle of with the positive -axis. Write the vector v in component form. Furthermore, write the equation of the line lt') passing through the origin with direction vector...
13.4.1 TT ht Find T, N, and k for the plane curve r(t) = 2+ i +2 In (cost)j, - 3<t<z