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100note5 - Math100, Vector Calculus 5.1 Vector Fields...

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Unformatted text preview: Math100, Vector Calculus 5.1 Vector Fields Definition 5.1 A vector field in the n dimensional space is a function in which a vector in the n dimensional space is plugged in and a vector in the n dimen- sional space is returned. Example 5.2 Let φ be a function in n variables. ∇ φ is a vector field in the n dimensional space. Recall that ∇ φ ( x 1 , ..., x n ) = ( ∂φ ∂x 1 ( x 1 , ..., x n ) , ..., ∂φ ∂x n ( x 1 , ..., x n )). We plug in a vector in the n dimensional space into ∇ φ and get a vector in the n dimensional space. Thus, ∇ φ is a vector field in the n dimensional space. Definition 5.3 A vector field F in the n dimensional space is conservative if there exists a function φ in n variables such that F = ∇ φ . In this case, φ is often called a potential function of F also. Example 5.4 Let F ( x, y, z ) =- 1 ( x 2 + y 2 + z 2 ) 3 / 2 ( x, y, z ) be a vector field in the three dimensional space. Show that F is a conservative vector field. proof: Let φ ( x, y, z ) = 1 √ x 2 + y 2 + z 2 be a function in three variables. ∇ φ ( x, y, z ) = (- x ( x 2 + y 2 + z 2 ) 3 / 2 ,- y ( x 2 + y 2 + z 2 ) 3 / 2 ,- z ( x 2 + y 2 + z 2 ) 3 / 2 ) =- 1 ( x 2 + y 2 + z 2 ) 3 / 2 ( x, y, z ) = F ( x, y, z ) . Thus F is a conservative vector field. Example 5.5 Let F ( x, y, z ) = (0 , x, 0) be a vector field. Show that F is not conservative. proof: Assume that F is conservative. i.e. there exists a function φ in 3 variables such that F = ∇ φ . So, (0 , x, 0) = F ( x, y, z ) = ∇ φ ( x, y, z ) = ( ∂φ ∂x , ∂φ ∂y , ∂φ ∂z )( x, y, z ) . 1 But ∂ 2 φ ∂x∂y = ∂ ∂x x = 1 ∂ 2 φ ∂y∂x = ∂ ∂y 0 = 0 which is a contradiction. Therefore, we conclude that F is not conservative. Theorem 5.6 If a vector field in the three dimensional space F ( x, y, z ) = ( f ( x, y, z ) , g ( x, y, z ) , h ( x, y, z )) is conservative, then, ( ∂h ∂y- ∂g ∂z , ∂f ∂z- ∂h ∂x , ∂g ∂x- ∂f ∂y )( x, y, z ) = (0 , , 0) . proof: If F = ( f, g, h ) is conservative, there exists a function φ in three variables such that F = ( f, g, h ) = ∇ φ = ( ∂φ ∂x , ∂φ ∂y , ∂φ ∂z ) . So, ∂h ∂y- ∂g ∂z = ∂ 2 φ ∂y∂z- ∂ 2 φ ∂z∂y = 0 ∂f ∂z- ∂h ∂x = ∂ 2 φ ∂z∂x- ∂ 2 φ ∂x∂z = 0 ∂g ∂x- ∂f ∂y = ∂ 2 φ ∂x∂y- ∂ 2 φ ∂y∂x = 0 . Definition 5.7 If F = ( f, g, h ) is a vector field in the three dimensional space, the curl of F is the vector field (in the three-dimensional space) ∇ × F = curl ( F ) = ( ∂h ∂y- ∂g ∂z , ∂f ∂z- ∂h ∂x , ∂g ∂x- ∂f ∂y ) . Corollary 5.8 If a vector field in the three dimensional space is conservative, its curl is zero....
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100note5 - Math100, Vector Calculus 5.1 Vector Fields...

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