Let be continuously differentiable near a point with .

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Conditions of Validity, Ch Differentiability, the implicit function theorem and the licence to change variables

Problem

Let

f(x,y,z)

be continuously differentiable near a point

\mathbf a

with

f(\mathbf a)=0

. ((a)) State what the implicit function theorem needs at

\mathbf a

for each of

x(y,z)

y(z,x)

and

z(x,y)

to exist as a differentiable function nearby. Under those conditions derive the reciprocal relation

(\partial x/\partial y)_z(\partial y/\partial x)_z=1

and the cyclic relation

(\partial x/\partial y)_z(\partial y/\partial z)_x(\partial z/\partial x)_y=-1

. ((b)) Test both relations on the surface

x^2y-z=0

(1,2,2)

by solving for each variable explicitly and differentiating, without using the formula from part (a). ((c)) The point

(0,1,0)

also lies on this surface. Which of the three implicit functions lose their licence there, and does

x(y,z)

exist near

(y,z)=(1,0)

at all?

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