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M 427L - Practice Test 2

M 427L - Practice Test 2 - M 427L Second Practice Exam 1...

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Unformatted text preview: M 427L Second Practice Exam 1. Let φ : R 3 → R 3 be a smooth function. What is the geometric meaning of | det( D ( φ )( p ) | (the absolute value of the determinant of the derivative at the point p )? (a) This tells you whether or not φ preserves orientation (b) This gives you a vector normal to the image of the boundary of a given region Ω ⊂ R 3 (c) This is roughly the ratio of the volume of the image of a small box around p to the volume of the box itself. (d) This is roughly the rate of change of the function φ if you move in the direction of p (i.e. the directional derivative) (e) This is zero iff the vector field φ is a gradient field. For problems 2-5 let T be the surface you get by rotating a circle of radius 1 in the xy plane centered at (2 , 0) about the y-axis – i.e. a Torus, the surface of a doughnut. 2. Which of the following functions φ : R 2 → R 3 gives a parameterisation of T by the rectangle, E , 0 ≤ u ≤ 2 π, ≤ v ≤ 2 π ? (a) ( cos ( u ) , 2 , sin ( u ) cos ( v )) (b) (2 cos ( u ) cos ( v ) , sin ( u ) , 2 cos ( u ) sin ( v )) (c) (2 − cos ( u )) cos ( v ) , sin ( u ) , (2 − cos ( u )) sin ( v ) (d) (2 + cos ( u )) cos ( v ) , sin ( u...
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M 427L - Practice Test 2 - M 427L Second Practice Exam 1...

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