20e-lecture23

# 20e-lecture23 - Lecture 23 Wednesday May 27th...

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Unformatted text preview: Lecture 23 - Wednesday May 27th [email protected] Key words: Curl, Conservative vector field, formal determinant 23.1 Conservative vector fields Recall that f : R n → R n is conservative if f = ∇ F for some F : R n → R . A test for conservativity of a given vector field in R 2 or R 3 is done using the curl of the vector field. Definition. The curl of a vector field ( f,g,h ) in R 3 , denoted curl( f ) or ∇ × f , is the vector ( h y- g z ,f z- h x ,g x- f y ). An easy way to remember the formula for the curl is to write it as a formal determinant: ∇ × f = fl fl fl fl fl fl i j k ∂ ∂x ∂ ∂y ∂ ∂z f g h fl fl fl fl fl fl . Here we use the notation i = (1 , , 0), j = (0 , 1 , 0) and k = (0 , , 1), and when we multiply ∂ ∂x by g for example, we formally obtain g x . As an exercise, verify that this gives the above formula from the definition. We now come to a simple test for a vector field f : R 3 → R 3 to be conservative. An oriented curve γ in R 3 is simple if it the image of a one-to-one piecewise C 1 map r : [ a,b ] → R 3 . It is closed if it is one-to-one on ( a,b ] and r ( a ) = r ( b ). Test for Conservativity. 1. A vector field f is conservative if and only if ∇ × f = 0. 2. A vector field f is conservative if and only if for any two simple oriented curves γ and δ with the same endpoints, R γ f · dr = R δ f · dr . 3. A vector field f is conservative if and only if for any simple oriented closed curve γ , R γ f · dr = 0. Example 1. Let f ( x,y,z ) = ( yz,xz,xy ). We saw that this is conservative since f = ∇ F where F = xyz . Now we check it using the test ∇ × f = fl fl fl fl fl fl i j k ∂ ∂x ∂ ∂y ∂ ∂z yz xz xy fl fl fl fl fl fl = i ‡ ∂xy ∂y- ∂xz ∂z ·- j ‡ ∂xy ∂x- ∂yz ∂z · + k ‡ ∂xz ∂x- ∂yz ∂y · = ( x- x,y- y,z- z ) = 0 ....
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20e-lecture23 - Lecture 23 Wednesday May 27th...

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