Calculate the work done by a force Vector F = y^2z^3i^+2xyz^3j^+3xy^2z^2k^ in moving an object along the path defined by the parametric equation x=t, y=t, z=t. Between the point A(t=0) and B(t=1).
we need the component of force in the direction of motion.
F dot vector tangent to motion
tangent to motion is velocity vector
dX = Vdt if X is position vector and V is velocity vector
X = t i + t j + t k
V = dX/dt = i + j + k
F dot V = y^2z^3 + 2 xyz^3 + 3x y^2 z^2
= t^5 + 2 t^5 + 3 t^5
= 6 t^5
integrate from t = 0 to t = 1
(6/6)t^6 from 0 to 1 = 1