Consider the function given by . Suppose is an inverse function to f, which is defined and has total differential on an open set . Find the total differential of g at the point .
Find all the partial derivatives of the function up to order 3.
Find the tangent plane to the surface in given in parametric form by
at the point s=2, t=3.
Let and be two functions given by:
Compute Df, Dg, , .
If f(0,0)=0 and
prove that the partial derivatives of f exist at every point of , although f is not continuous at (0,0).
Suppose where U is an open set in
. Prove that if all partial derivatives exist and
are bounded in U (i.e. all
their values are in an interval (-M,M) for some
constant M;SPMgt;0), then f is continuous.
[Hint: We proved in 295 that if is a one-variable function
in an interval satisfying everywhere, then
|f(x)-f(y)|;SPMlt;M|x-y|. Use this fact.]