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Unformatted text preview: 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 : Rn → R be a smooth function and deﬁne g(x) = sin( f (x)2 ). Find the 1-form d f . 2. Let f : R2 → R be a smooth function and assume f ( x, y) = − f (y, x ). Show that ∂f ∂f ( a, b ) = − ( b, a ). ∂x ∂y 3. Let f : R2 → R and g : R → R be smooth functions. Show that
g( x ) ∂ f ( x, y ) g( x ) d f ( x, y) dy = f ( x, g( x )) g ( x ) + dy. dx 0 ∂x 0 (Use the fundamental theorem of calculus, the chain rule and differentiate under the integral sign.) 4. Thermodynamicists like to use rules such as ∂y ∂z ∂ x = −1. ∂ x ∂y ∂z Explain the rule and show that it is correct. (Assume that the variables are subject to a relation F ( x, y, z) = 0 deﬁning functions x = f (y, z), y = g( x, z), z = h( x, y), and differentiate implicitly.) Naively cancelling numerators against denominators gives the wrong answer! 5. Let α = x1 dx2 + x3 dx4 , β = x1 x2 dx3 dx4 + x3 x4 dx1 dx2 and γ = x2 dx1 dx3 dx4 be forms on R4 . Calculate (a) αβ, αγ; (b) d β, dγ; (c) ∗α, ∗γ. 6. Suppose that f : Rn → R is a smooth function satisfying f ( t a1 x1 , t a2 x 2 , . . . , t a n x n ) = t k f (x) for all x. Here k is a positive constant and a1 , a2 , . . . , an are arbitrary real conn ∂f stants. Deduce from the chain rule that ∑ ai xi (x) = k f (x). ∂ xi i =1 ...
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This note was uploaded on 04/14/2010 for the course MATH 3210 at Cornell University (Engineering School).