5. Vector Calculus

5. Vector Calculus - Math2011 Vector Calculus 5.1 Vector...

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Math2011, 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
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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 the vector field F ( x, y ) = ( f ( x, y ) , g ( x, y )) in the two dimensional space is conservative, then ∂g ∂x - ∂f ∂y = 0 . proof: see below. Theorem 5.7 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 , 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 5.8 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 5.9 If a vector field in the three dimensional space is conservative, its curl is zero. Example 5.10 Determine if the vector field F ( x, y ) = ( x 2 + y 2 , 2 xy ) is conservative. 2
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solution: Define φ ( x, y ) = 1 3 x 3 + xy 2 . Then φ = F so that F is conservative. Example 5.11 Let f be a function in one variable. Evaluate the curl of the vector field F ( v ) = f ( || v || 2 ) v for all three dimensional vector v.
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