{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

midterm1 2007 - HKUST MATH 102 Midterm One Examination...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
HKUST MATH 102 Midterm One Examination Multivariable and Vector Calculus 30 Oct 2007 Answer ALL 5 questions Time allowed – 120 minutes Problem 1 Identify the following surfaces (a) r · b u = 0. (b) ( r - a ) · ( r - b ) = k . (c) k r - ( r · b u ) b u k = k . [Hint: What are the vectors ( r · b u ) b u and r - ( r · b u ) b u ?] Here k is fixed scalar, a , b are fixed 3D vectors and b u is a fixed 3D unit vector and r = ( x, y, z ). Problem 2 (a) Find the velocity, speed and acceleration at time t of the particle whose position is r ( t ). Describe the path of the particle. r = at cos ωt i + at sin ωt j + b ln t k (b) Find the required parametrization of the first quadrant part of the circular arc x 2 + y 2 = a 2 in terms of arc length measured from (0 , a ), oriented clockwise. (c) Let C be the curve x 2 / 3 + y 2 / 3 = a 2 / 3 on the xy -plane, find the parametric equation of the curve C . Hence find the tangent line to the curve C at ( a, 0). Problem 3 (a) Assume a , b , c , x and y are three dimensional vectors and if ( a × b ) · ( b × c ) × ( c × a ) = [ x · y ] 2 .
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}