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practice1 - Math 3210 practice problems for rst prelim Fall...

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Math 3210 practice problems for first prelim Fall 2009 There are more problems here than many of you can do in fifty minutes, but they are representative of the type of problems you might expect on the exam. On the exam you may use a one-sided letter-size crib sheet, but no books, notes, or calculators. From notes: 2.7, 2.8, 2.13(i). (These problems will also be on the next homework assignment.) 1. Let f : R n R be a smooth function and define g ( x ) = sin ( f ( x ) 2 ) . Find the 1-form d f . 2. Let f : R 2 R be a smooth function and assume f ( x , y ) = - f ( y , x ) . Show that f x ( a , b ) = - f y ( b , a ) . 3. Let f : R 2 R and g : R R be smooth functions. Show that d dx Z g ( x ) 0 f ( x , y ) dy = f ( x , g ( x )) g ( x ) + Z g ( x ) 0 f ( x , y ) x dy . (Use the fundamental theorem of calculus, the chain rule and differentiate under the integral sign.) 4. Thermodynamicists like to use rules such as y x z y x z = - 1. Explain the rule and show that it is correct. (Assume that the variables are
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