PHY5200 F06

Chapter 4: Energy

Reading

Taylor 4.3-4.5 (today) and 4.5-4.7 (Wednesday).

Recall

U(r) - U(1) = U(r) = -W(1-->r) = -∫₁^rF(r')⋅dr'.

∇×F = 0 everywhere for a force to be conservative

The Curl

The curl of F is written ∇×F because it behaves like the cross product of the del operator with F, where we use

∇ ≡ ihat ∂/∂x + jhat ∂/∂y + khat ∂/∂z

When written this way, the partial derivatives aren't to be evaluated. They represent the operation that will be performed once the operator acts on a vector function. Note also that these are partial derivatives not full derivatives.

The curl of F is a vector. This assumes that F is a function of the the spatial coordinate, r, F = F(r). So the curl can also be a function of the spatial coordinate. This is a compact notation for a complex object: first F is a function of r, and it is a vector (so it is like three separate functions of r, F(r) = ( F₁(r), F₂(r), F₃(r) ) = ( F₁(r₁, r₂, r₃), F₂(r₁, r₂, r₃), F₃(r₁, r₂, r₃) ); second, the curl is a vector function of partial derivatives of F, ∇×F. To be more explicit, we must choose a coordinate system.

In cartesian coordinates, the curl of F=(Fx,Fy,Fz) is

∇×F = (∂Fz/∂y - ∂Fy/∂z , ∂Fx/∂z - ∂Fz/∂x , ∂Fy/∂x - ∂Fx/∂y)

If F is expressed in different coordinates (cylindrical or spherical) then the curl has a different form. The expressions for the 3 components are printed on the inside-back-cover of the text.

Force as the Gradient of Potential Energy

Let's go back to the definition of potential energy and see if we can come up with the inverse relation between potential energy and force.

U(r2) - U(r1) = ∫_r1^r2 F(r')⋅dr'

Now what if r2 is infinitisemally displaced from r1, r2 = r1 + dr = r + dr so that I can drop the subscript. Then the left hand side is the differential change in potential energy

dU = U(r+dr) - U(r) = U(x+dx, y+dy, z+dz) - U(x, y, z) = F(r)⋅dr = Fx(r)dx + Fy(r)dy + Fz(r)dz

If we have a function of one dimension, g(x), then we write the differential of g in terms of its derivative dg = g(x+dx) - g(x) = (dg/dx)dx. We'd like a generalization of this for two, three, or more dimensions, and that looks like

dU = ∂U/∂x dx + ∂U/∂y dy + ∂U/∂z dz