Paper 2, Section II, A

Differential Equations | Part IA, 2011

(a) A surface in R3\mathbb{R}^{3} is defined by the equation f(x,y,z)=cf(x, y, z)=c, where cc is a constant. Show that the partial derivatives on this surface satisfy

xyzyzxzxy=1\left.\left.\left.\frac{\partial x}{\partial y}\right|_{z} \frac{\partial y}{\partial z}\right|_{x} \frac{\partial z}{\partial x}\right|_{y}=-1

(b) Now let f(x,y,z)=x2y4+2ay2+z2f(x, y, z)=x^{2}-y^{4}+2 a y^{2}+z^{2}, where aa is a constant.

(i) Find expressions for the three partial derivatives xyz,yzx\left.\frac{\partial x}{\partial y}\right|_{z},\left.\frac{\partial y}{\partial z}\right|_{x} and zxy\left.\frac{\partial z}{\partial x}\right|_{y} on the surface f(x,y,z)=cf(x, y, z)=c, and verify the identity ()(*).

(ii) Find the rate of change of ff in the radial direction at the point (x,0,z)(x, 0, z).

(iii) Find and classify the stationary points of ff.

(iv) Sketch contour plots of ff in the (x,y)(x, y)-plane for the cases a=1a=1 and a=1a=-1.

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