Calculate the line integral
along the curve.
Calculate the line integral along the curve. Question 2. Let F = x + ye”, e®...
Evaluate the line integral f F dr for the vector field F(x, y, z) curve C parametrised by Vf (x, y, z) along the with tE [0, 2 r() -(Vt sin(2πt), t cos (2πi), ?) ,
Evaluate the line integral f F dr for the vector field F(x, y, z) curve C parametrised by Vf (x, y, z) along the with tE [0, 2 r() -(Vt sin(2πt), t cos (2πi), ?) ,
The path integral of a function f(x, y) along a path e in the xy-plane with respect to a parameter r is given by 2. fex,y)ds= f(x),ye) /x(mF +y(t" dr , where a sr sb. (a) Show that the path integral of f(x, y) along a path c(0) in polar coordinates where r=r(0), α<θ<β, is Sf(r cos 0,rsin e) oN+( de. (b) Use this formula to compute the arc length of the path r 1+cos0, 0<0 27
The path integral...
No 3
putin uhd e integral lound a r the val- 0 VIIl, 81. EXERCISES Compute the curve integrals of the vector field over the indicated curves. (x,y)=(x2-2xy,y2-2xy) along the, parabola y=x2 from (-2,4) to 2. 0x, y, xz - y) over the line segment from (0,0, 0) to (1, 2, 4), 3, Let r (x2 y2)1/2 Let F(X)-X. Find the integral of F over the circle of radius 2, taken in counterclock wise direction. 4. Let C be a...
. Let F(a,y)-(3e +secztan,e -90 (a) Show that F is a conservative. (b) Find a function f (potential function) show that F (c) Use above result to evaluate Jc F. V. dr, where C is a smooth curve that begin at the point (2, 1 ) and ends at (0,3). (cos t, sin t) fromtto t particle that moves along the curve. (Write the value of work done without evaluating (d) Find the work done by the force field F(r,...
Line Integral 1. Given F(x, y) = **, 47°: -**!). (a) Calculate DX (b) Determine any region of the plane on which is conservative. (c) Find a potential function for F (d) Find the work done by F in moving a particle along the parabolic curve y=1+ x - xfrom (0, 1) to (1, 1). x2 x? [0, {(x, y)e R?[y+0}, ø = 1] 2 2 y? 1 + + 2 y2
there is first question E then there is the question
of the value of the line integral ,then quwstion A, then question
1, and the last two pictures are one question
Question E (5 points) By Green's theorem, the value of the line integral y 4 is: , where C is the curve given by a) 3 c) 12t d) 27T e) If none of the above is correct, write your answer here in a box rover the line segment...
(3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83, 2) with 0 2 t 1l F=(z-z, 0,2) r(t)-(cost, 0, sin t) with 0 t π F = (-y,2, 2) with r(t) = (-2 cost, 2 sin t, 2t) 0 < t < 2π
(3) For the following velocity fields F on R3, find the flow along the given curve. r(t) = (t, t2, 1) F=(-4xy, 83,...
Calculate the work done by the force F= (x-2y)i+(x+y)j in a) 2. moving from point A at (0,2) to point B at (2,18) along the path y 4x2+2. [5 marks] - Evaluate the line integral(xdy+ydx) along a path C that is b) [5 marks] to t described by x= cos(f), y=2sin(t)+5, from t =: 2
Calculate the work done by the force F= (x-2y)i+(x+y)j in a) 2. moving from point A at (0,2) to point B at (2,18) along the...
(b) Let F(r) be a force vector so that the line integral F(r)r is the work W done by F(r) along the curve C with parametric representation r(r) which is the displacement vector. Let m be the mass of the object. Prove that the work done on the object within te[to4 satisfies the equation W- cF(r).dr-îm v(r 2Г which states the work energy theorem: work done on an object equals to the change of kinetic energy (Hint I: Newton's second...
Question 22 1 pts Compute the path integral of F = (y,x) along the line segment starting at (1,0) and ending at (3, 1). Question 23 1 pts Consider the vector field F= (1, y). Compute the path integral of this field along the path: start at (0,0) and go up 2 units, then go right 3 units, then go down 4 units and stop. Question 24 1 pts Compute Ss(-y+ye*y)dx + (x + xey)dy, where S is the path:...