 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013/11/a/solution

Let

r (s)

ϕ = c

. Differentiating the constant value along it gives

0 = \frac{d}{d s} ϕ (r (s)) = \nabla ϕ \cdot \frac{d r}{d s} .

\nabla ϕ ⊥ T_{P} Σ, \hat{n} = \pm \frac{\nabla ϕ}{∣\nabla ϕ ∣} when \nabla ϕ \neq = 0.

The nonzero-gradient condition matters for determining a normal direction. For example,

ϕ = z^{3}

has the plane

z = 0

as a level set, but its gradient vanishes there and supplies no direction, even though the geometric plane has a well-defined normal.

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