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Math 296 problems 10

Igor Kriz

Regular problems:


The equations u=f(x,y), x=X(s,t) and y=Y(s,t) define u as a function of s and t, say u=F(s,t).

(a) Use an appropriate chain rule to express the partial derivatives of tex2html_wrap_inline224 and tex2html_wrap_inline226 in terms of tex2html_wrap_inline228 , tex2html_wrap_inline230 , tex2html_wrap_inline232 , tex2html_wrap_inline234 , tex2html_wrap_inline236 , tex2html_wrap_inline238 .

(b) If f has continuous partial derivatives of second order, prove that


(c) Find similar formulas for the partial derivatives tex2html_wrap_inline244 and tex2html_wrap_inline246 .


Let, in this problem, all functions have continuous partial derivatives of all orders. For a function tex2html_wrap_inline248 , a function tex2html_wrap_inline250 is given by


For a function tex2html_wrap_inline254 where


a function tex2html_wrap_inline258 is given by


Finally, for a function tex2html_wrap_inline262 where


a function tex2html_wrap_inline266 is given by


Prove that






Calculate tex2html_wrap_inline276 , tex2html_wrap_inline278 .


Consider the function tex2html_wrap_inline280 given by tex2html_wrap_inline282 . Determine the tangent plane to the graph of the function f (which is a surface in tex2html_wrap_inline286 at a point tex2html_wrap_inline288 .

Challenge problem:


In the xz plane in tex2html_wrap_inline286 , consider the circle C given by th equation


Let M be the set obtained by rotating the circle C around the z axis (``the surface of a donut'').

(a) By finding local coordinate (=parametrizing) functions at all points, prove that M is a manifold.

(b) Using your parametrizations, calculate the vector tangent space to M at a point tex2html_wrap_inline308 .

(c) Consider the function tex2html_wrap_inline310 given as follows: Let tex2html_wrap_inline312 be the image of C under rotation by the angle tex2html_wrap_inline316 around the z-axis. Then f maps tex2html_wrap_inline322 to itself by rotating it by the same angle tex2html_wrap_inline316 . (In measuring these angles, the angles from the positive part of the x axis to the positive part of the y axis and to the positive part of the z axis is considered to be tex2html_wrap_inline332 .) Calculate Df.

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Igor Kriz
Wed Mar 18 20:39:27 EST 1998