This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: M 427L Second Practice Exam 1. Let : R 3 R 3 be a smooth function. What is the geometric meaning of | det( D ( )( p ) | (the absolute value of the determinant of the derivative at the point p )? (a) This tells you whether or not preserves orientation (b) This gives you a vector normal to the image of the boundary of a given region 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 if you move in the direction of p (i.e. the directional derivative) (e) This is zero iff the vector field 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 : R 2 R 3 gives a parameterisation of T by the rectangle, E , 0 u 2 , v 2 ? (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...
View Full Document