Evaluate the line integral
∫CF⋅dr∫CF⋅dr,
where
F(x,y,z)=sin(x)i+cos(y)j+4xzkF(x,y,z)=sin(x)i+cos(y)j+4xzk
and C is given by the vector function
r(t)=t3i−t2j+tkr(t)=t3i−t2j+tk...
Evaluate the line integral
∫CF⋅dr∫CF⋅dr,
where
F(x,y,z)=sin(x)i+cos(y)j+4xzkF(x,y,z)=sin(x)i+cos(y)j+4xzk
and C is given by the vector function
r(t)=t3i−t2j+tkr(t)=t3i−t2j+tk
, 0≤t≤10≤t≤1.
Evaluate the line integral
C
F · dr,
where C is given by the vector function
r(t).
F(x, y, z) = sin(x) i + cos(y) j + xz k
r(t) = t5 i − t4 j + t k, 0 ≤ t ≤ 1
Given the line integral ∫c F(r) · dr where
F(x, y, z) = [mxy − z3 ,(m − 2)x2 ,(1 −
m)xz2 ]
(a) Find m such that the line integral is path independent;
(b) Find a scalar function f such that F = grad f;
(c) Find the work done in moving a particle from A : (1, 2, −3)
to B : (1, −4, 2).
Evaluate the line integral, where C is the given curve.
∫CF(x,y,z)⋅dr where F(x,y,z)=xi+yj+ysin(z+1)k and C
consists of the line segment from (2,4,-1) to (1,-1,3).
Q2. Given the line integral C F (r) · dr where
F(x,y,z) = [mxy − z3,(m − 2)x2,(1 − m)xz2]
∫
(a) Find m such that the line integral is path
independent;
(b) Find a scalar function f such that F = grad f ;
(c) Find the work done in moving a particle from A : (1, 2,
−3) to B : (1, −4, 2).
Compute the derivative of the given vector field F. Evaluate the line integral of
F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )over the path C consisting of line segments joining (1,1,1) to (1,1,2), (1, 1, 2) to (1, 3, 2), and (1, 3, 2) to (4, 3, 2) in 3 different ways, along the given path, along the line from (1,1,1) to (4,3,2), and finally by finding an anti-derivative, f, for F.
using / for integral
Evaluate the double integral //R cos( (y-x)/(y+x) )dA where R is
the trapezoidal region with vertices (1,0), (2,0), (0,2), and
(0,1)
Problem 7. Consider the line integral Z C y sin x dx − cos x dy.
(Please show all work)
a. Evaluate the line integral, assuming C is the line segment
from (0, 1) to (π, −1).
b. Show that the vector field F = is conservative, and find a
potential function V (x, y).
c. Evaluate the line integral where C is any path from (π, −1)
to (0, 1).