20e-lecture22

# 20e-lecture22 - Lecture 22 Friday May 22nd [email protected]

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Unformatted text preview: Lecture 22 - Friday May 22nd [email protected] Key words: Line integral, Conservative vector field, parametrization 22.1 Line Integrals The Riemann integral R b a f ( x ) dx denotes the integral of the function f along the interval [ a,b ]. More generally, we could define integrals along curves instead of the straight line consisting of the interval [ a,b ]. These integrals are generally referred to as line integrals , and we will see via the major integral theorems of calculus how they relate to multiple integrals. Moreover they arise very naturally in applied mathematics and physics, for example, the notion of the work done by a force field on a particle moving through was seen to be a line integral in the last lecture. We begin the the definition of line integrals. Let γ be a curve in R n connecting points p and q . We assume γ ∈ C 1 – so γ has tangent lines at every point. Let f ( x ) = ( f 1 ( x ) ,f 2 ( x ) ,...,f n ( x )) be a function from R n to R n defined at all points of γ . Let P be a partition of γ into curves, and for a curve I ∈ P let δ j ( I ) denote the difference between the j th co-ordinate of the first point in I and the last point in I . Partition of γ into curves. γ I δ 1 ( I ) δ 2 ( I ) Then we define the Riemann Sum n X j =1 X I ∈ P f j ( x I ) δ j ( I ) . 1 where x I is an arbitrary point on the curve I . Let k P k denote the largest of all the values of δ j ( I ) for I ∈ P and j = 1 , 2 ,...,n . If the limit lim k P k→ n X j =1 X I ∈ P f j ( x I ) δ j ( I ) exists, then the limit is called the line integral of f along γ and denoted by n X j =1 Z γ f j dx j or n X j =1 Z q p f j dx j . If we let r ( x ) = ( x 1 ,x 2 ,...,x n ) denote the position vector of x , and we define formally dr = ( dx 1 ,dx 2 ,...,dx n ), then we may rewrite the line integral in the more succinct form Z γ f · dr where the dot denotes the scalar product of f and dr as vectors. If γ is the path taken by a particle moving through the force field f , then this line integral is exactly the definition of work given in § 21.3. In fact, the evaluation of line integrals is done in a similar way, using parametrizations of γ . If γ is a curve in R n , then γ can be viewed as a function from some interval [ a,b ] to R n . If r ( t ) = ( x 1 ( t ) ,x 2 ( t ) ,...,x n ( t )) is such a function, then r is said to parametrize γ . For example, the unit circle γ centered at the origin can be parametrized by r ( t ) = (cos t, sin t ), and the straight line from (0 , , 0) to (1 , 1 , 1) in R 3 can be parametrized by r ( t ) = ( t,t,t ). Note that parametrizations are not unique. They will play a major rˆole in evaluating line integrals....
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20e-lecture22 - Lecture 22 Friday May 22nd [email protected]

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