Ch2 - Ch.2 PHYS2004 Quantitative Methods for Basic Physics...

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Ch.2 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 1 Chapter 2 Some Applications of Calculus 2.1 Maxima and Minima of Functions 2.1.1 Stationary Points A point at which the derivative of a function   f x vanishes,  0 0 fx . (2.1.1) A stationary point may be minimum, maximum or inflection point, as shown in below. 2.1.2 Extremum Test Consider a function f x in one dimension. If   f x has a local (relative) extremum at 0 x , then either   0 0 or   f x is not differentiable at 0 x . Either the first or second derivative tests may be used to locate relative extrema of the first kind. A necessary condition for f x to have a minimum (maximum) at 0 x is 0 0 and   0 0      0 0 . (2.1.2) A sufficient condition is 0 0 and   0 0     0 0 . (2.1.3) Generally, let   0 0 , 0 0 , ,   0 0 n , but 1 0 0 n . Then
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Ch.2 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 2  f x has a relative maximum at 0 x if n is odd and   1 0 0 n fx , and f x has a relative minimum at 0 x if n is odd and   1 0 0 n . There is a saddle point at 0 x if n is even. 2.2 Differentiation of Vectors ˆ ˆˆ rx iy jz k  , (position) ˆ dr dx dy dz vi j k dt dt dt dt  , (velocity) 22 2 2 2 2 ˆ dv d r d x d y d z ai j k dt dt dt dt dt  . (acceleration) As an example, we consider vectors in R 2 and may use Polar Coordinates instead of Cartesian Coordinates . The unit vectors in both systems are related as: ˆ cos sin ri j  , ( 2 . 2 . 1 a ) and ˆ sin cos ij   . ( 2 . 2 . 1 b ) Differentiating ˆ r and ˆ with respect to t , we get ˆ ˆ sin cos dr d d d dt dt dt dt , (2.2.2a) and ˆ ˆ cos sin dd d d r dt dt dt dt . ( 2 . 2 . 2 b ) Therefore, given a vector ˆ ˆ r AA rA , where A r and A are functions of t , then ˆ ˆ r r dA dA dA d d A dt dt dt dt dt    . (2.2.3) In general, the direction of the vector dt t f d ) ( is along the tangent at ) ( t f to the curve given by the vector function f . If ) ( t g is constant, then 0 ) ( ) ( dt t g d t g .
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Ch.2 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 3 Some rules for the differentiation of sums and products of vectors: (i)  dt t v d dt t u d t v t u dt d ) ( ) ( ) ( ) ( ; (ii) () () () dd m t d u t mtut ut mt dt dt dt   ; (iii) dt t v d t u t v dt t u d t v t u dt d ) ( ) ( ) ( ) ( ) ( ) ( ; (iv) dt t v d t u t v dt t u d t v t u dt d ) ( ) ( ) ( ) ( ) ( ) ( . Exercises: 1. For a position vector r in two-dimensional space, try to express dr dt in polar coordinates. 2. Find a unit vector along the direction of the velocity for a particle whose position vector is given by ˆ ˆˆ () (2cos) (s in ) rt ti t j t k  at the point given by /2 t .
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This note was uploaded on 10/14/2010 for the course PHYS 2051 taught by Professor Drkwong during the Spring '10 term at CUHK.

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Ch2 - Ch.2 PHYS2004 Quantitative Methods for Basic Physics...

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