Let us consider the vector field
Now, we have to find
where C is the line joining A(-1,0,2) and B(3,-4,0)
So, the position vector of any point on C is given by
, where t varies from 0 to 1
So, we have, differentiating with respect to t
and,
So, we have
Thus, the required integral becomes
Hence, we have
Calculate the Line Integral: S**.dř = f(x,y,z)|8c3 -4 0) – f(x,y,z) A(-1 0 2) A Work...
Line Integral & Path Independency Problem 1 Prove that the vector field = (2x-3yz)i +(2-3x-2) 1-6xyzk is the gradient of a scalar function f(x,y,z). Hint: find the curl of F, is it a zero vector? Integrate and find f(x,y,z), called a potential, like from potential energy? Show all your work, Then, use f(x,y,z) to compute the line integral, or work of the force F: Work of F= di from A:(-1,0, 2) to B:(3,-4,0) along any curve that goes from A...
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), ?) ,
(5 points) 2. Find the line integral of f (x, y, z) = 2x – 3y + 5z along the straight line segment from (1,0, 2) to (3, 2, -1).
This is all one question please answer asap Line Integral & Path Independency Problem 1 Prove that the vector field F = (2x – 3yz?) { +(2 – 3xz) j-6xyzk is the gradient of a scalar function f(x,y,z). Hint: find the curl of F, is it a zero vector? Integrate and find f(x,y,z), called a potential, like from potential energy? Show all your work. Then, use f(x,y,z) to compute the line integral, or work of the force F: Work of...
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
1. Evaluate the line integral S3x2yz ds, C: x = t, y = t?, z = t3,0 st 51. 2. Evaluate the line integral Scyz dx - xz dy + xy dz , C: x = e', y = e3t, z = e-4,0 st 51. 3. Evaluate SF. dr if F(x,y) = x?i + xyj and r(t) = 2 costi + 2 sin tj, 0 st St. 4. Determine whether F(x,y) = xi + yj is a conservative vector field....
13. (10 points) (a): Find the line integral of f(x, y, z) = x+y+z over the straight-line segment from (1,2,3) to 0,-1,1). (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 >, 0<t<27.
→ (1 point) Let Vf-6xe-r sin(5y) +1 5e* cos(Sy) j. Find the change inf between (0,0) and (1, n/2) in two ways. (a) First, find the change by computing the line integral c Vf di, where C is a curve connecting (0,0) and (1, π/2) The simplest curve is the line segment joining these points. Parameterize it: with 0 t 1, K) = dt Note that this isn't a very pleasant integral to evaluate by hand (though we could easily...
Use Stokes' Theorem to evaluate the line integral $cF. dr, where F(x, y, z) = (-y+z)i + (x – z)j + (x – y)k. S is the surface z = V1 – 22 – y2, and C is the boundary of S with counterclockwise orientation (from above).
4. (5 points) Express the integral JI f(x, y, z) dV as an iterated integral in 6 different ways, where E is the solid region bounded by y2 + z-9. x--2, and x-2. 4. (5 points) Express the integral JI f(x, y, z) dV as an iterated integral in 6 different ways, where E is the solid region bounded by y2 + z-9. x--2, and x-2.