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Proof


The tube surface is given by:
    Yr(t,v) = X(t) + rcos(v)U(t) + rsin(v)N(t)
To find the total Gaussian curvature of the tube surface, we first need to find an expression for the Gaussian curvature. Start by calculating the partial derivative vectors.
    Yrt(t,v) = X'(t) + rcos(v)U'(t) + rsin(v)N'(t)
                = s'(t)T(t) + rcos(v)[-κgs'(t)T(t) + τg(t)s'(t)N(t)] + rsin(v)[-κN(t)s'(t)T(t) - τg(t)s'(t)U(t)]
                = s'(t)[1-rcos(v)κg(t) - rsin(v)κN(t)]T(t) + τg(t)s'(t)rcos(v)N(t) - τg(t)s'(t)rsin(v)U(t)
    Yrv(t,v) = -rsin(v)U(t) + rcos(v)N(t)
Take the cross product of the partial derivative vectors to find the normal to the tube surface. Since the curve X(t) lies on a surface, N(t) is used to denote the normal to this surface at a point on the curve. In order to distinguish this normal from the normal vector for the tube surface, we write the latter vector as N*(t).
    Yrt(t,v) × Yrv(t,v) = -s'(t)[1-rcos(v)κg(t) - rsin(v)κN(t)]rsin(v)N(t) - s'(t)[1-rcos(v)κg(t) - rsin(v)κN(t)]rcos(v)U(t)
                                = s'(t)[1-rcos(v)κg(t) - rsin(v)κN(t)](-rcos(v)U(t) - rsin(v)N(t))
    N*(t) = Yrt(t,v) × Yrv(t,v) / ||Yrt(t,v) × Yrv(t,v)|| = -cos(v)U(t) - sin(v)N(t)
Since Gaussian curvature is given by N*t × N*v = Κ*(t,v)Yrt × Yrv, we need to calculate the partial derivative vectors for N*(t,v):
    N*t(t,v) = -cos(v)U'(t) - sin(v)N'(t)
                = -cos(V)(-κg(t)s'(t)T(t) + τg(t)s'(t)N(t)) - sin(v)(-κN(t)s'(t)T(t) - τg(t)s'(t)U(t))
                = s'(t)[κg(t)cos(v) + κN(t)sin(v)]T(t) - τg(t)s'(t)cos(v)N(t) +τg(t)s'(t)sin(V)U(t)
    N*v(t,v) = sin(v)U(t) - cos(v)N(t)
    N*t(t,v) × N*v(t,v) = s'(t)[κg(t)cos(v) + κN(t)sin(v)](sin(v)N(t) + cos(v)U(t))
To calculate the total Gaussian curvature of the "moldy potato chip", we divided its area into two parts: (1) the circular half-tube around the curve, and (2) the surfaces parallel to the region enclosed by the curve. Here we calculate the total Gaussian curvature over region (1).
    (1)Κ*dA = abπ/23π/2Κ*||Yrt × Yrv||dvdt = abπ/23π/2(N*t × N*v) · N*
                      = abπ/23π/2s'(t)[cos(v)κg(t) + sin(v)κN(t)](cos(v)U(t) + sin(v)N(t)) · (-cos(v)U(t) - sin(v)N(t))
                      = abπ/23π/2-s'(t)[cos(v)κg(t) + sin(v)κN(t)]dvdt
                      = ab[-s'(t)κg(t) π/23π/2cos(v)dv]dt - ab[-s'(t)κN(t) π/23π/2sin(v)dv]dt
Note that in the second term of the last expression, the integral of sin(v) from v = π/2 to v = 3π/2 is equal to 0. So,
    (1)Κ*dA = ab[-s'(t)κg(t) π/23π/2cos(v)dv]dt
                      = 2abκg(t)s'(t)dt
                      = 2Pκgds