 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/12/a/ii/solution

Differentiate the defining function with the other coordinates independent. Put

C = cosh (x + z)

for brevity. Then

F_{x} = yz - C, F_{y} = x z, F_{z} = x y - C .

(\frac{\partial x}{\partial y})_{z} = - \frac{x z}{yz - C}, (\frac{\partial x}{\partial z})_{y} = - \frac{x y - C}{yz - C}, (\frac{\partial y}{\partial z})_{x} = - \frac{x y - C}{x z} .

Each expression applies on the implicit surface wherever its own denominator is nonzero. The hyperbolic cosine appears because the derivative of

sinh (x + z)

with respect to either

x

z

cosh (x + z)

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