Mathematics 317 Midterm#2 _2009_1

# Mathematics 317 Midterm#2 _2009_1 - Mathematics 317...

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Mathematics 317 Midterm #2: Monday, November 2, 2009 1. Do only two questions. Each question is of equal value. If you attempt more than two questions, only the first two will be marked. 2. No notes, books or calculators are permitted. 3. Time Limit: 50 minutes 1. Consider the vector field given by . ) , ( j i F x y y x = (a) Sketch the flow lines of this vector field. (b) Find the equation of the flow line passing through the point (1,1). 2. Consider the two force fields given by . ) , ( , 1 1 ) , ( 3 2 j i G j i F x y y x y x y x + - = + = For each of these force fields, find the work done in moving a particle from the point Unformatted text preview: (2,1) to the point (1,2) along the smaller arc lying on the circle . 1 ) 1 ( ) 1 ( 2 2 =-+-y x 3. Let k j i a 3 2 1 a a a + + = be a constant vector. Let . | | , 2 2 2 z y x r z y x + + = = + + = r k j i r Find all functions f ( r ) with the property that . ) ) ( ( ≡ × ⋅ ∇ r a r f [Hint: One can show that ) ( ) ( ) ( G F F G G F × ∇ ⋅-× ∇ ⋅ ≡ × ⋅ ∇ for any two differentiable vector fields ). , , ( ), , , ( z y x z y x G F ]...
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