Notes on force, gradient

Consider an element of work in 1-D

• dW = F_x dx
• F_x = d/dt(mv_x) = m dv_x/dt, so
• dW = dx m d v_x/dt.
• We can regard dx as v_x dt, the distance travelled in time dt, so dW becomes
• dW = m v_x dt d v_x/dt = m v_x dv_x.
• since m zdz is the differential of 1/2 m(z^2), we see that
• dW = F_x dx = d ( 1/2 m v_x^2).

Now go to full 3-D, and the full element of work is dW = F_x dx + F_y dy+ F_z dz

• dW = F . dr = F_x dx + F_y dy+ F_z dz = d ( 1/2 m [v_x^2 + v_y^2 + mv_z^2]) = d ( kinetic energy)
• or dW = F.dr = dK
• In systems where energy is conserved K + U = constant, and dK + dU = 0.
• In energy-conserving systems, dK = -dU = F.dr
• What may we then conclude about the relation between F and U, remembering that for a function f, df = grad f . dr ?