calculus - SECTION 15.2: Conservative Fields 811 Exercises...

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SECTION 15.2: Conservative Fields 811 Exercises 15.1 In Exercises 1–8, sketch the given plane vector field and determine its field lines. 1. F ( x , y ) = x i + x j 2. F ( x , y ) = x i + y j 3. F ( x , y ) = y i + x j 4. F ( x , y ) = i + sin x j 5. F ( x , y ) = e x i + e x j 6. F ( x , y ) =∇ ( x 2 y ) 7. F ( x , y ) ln ( x 2 + y 2 ) 8. F ( x , y ) = cos y i cos x j In Exercises 9–16, describe the streamlines of the given velocity fields. 9. v ( x , y , z ) = y i y j y k 10. v ( x , y , z ) = x i + y j x k 11. v ( x , y , z ) = y i x j + k 12. v ( x , y , z ) = x i + y j ( 1 + z 2 )( x 2 + y 2 ) 13. v ( x , y , z ) = xz i + yz j + x k 14. v ( x , y , z ) = e xyz ( x i + y 2 j + z k ) 15. v ( x , y ) = x 2 i y j 16. 3 v ( x , y ) = x i + ( x + y ) j Hint: Let y = x v( x ) . In Exercises 17–20, determine the field lines of the given polar vector fields. 17. F = ˆ r + r ˆ θ 18. F = ˆ r + θ ˆ θ 19. F = 2 ˆ r + θ ˆ θ 20. F = r ˆ r ˆ θ 15.2 Conservative Fields Since the gradient of a scalar field is a vector field, it is natural to ask whether every vector field is the gradient of a scalar field. Given a vector field F ( x , y , z ) , does there exist a scalar field φ( x , y , z ) such that F ( x , y , z ) x , y , z ) = ∂φ x i + y j + z k ? The answer in general is “no.” Only special vector fields can be written in this way. DEFINITION 1 If F ( x , y , z ) x , y , z ) in a domain D , then we say that F is a conservative vector field in D , and we call the function φ a (scalar) potential for F on D . Similar definitions hold in the plane or in n -space. Like antiderivatives, potentials are not determined uniquely; arbitrary constants can be added to them. Note that F is conservative in a domain D if and only if F φ at every point of D ; the potential φ cannot have any singular points in D . The equation F 1 ( x , y , z ) dx + F 2 ( x , y , z ) dy + F 3 ( x , y , z ) dz = 0iscalledan exact differential equation if the left side is the differential of a scalar function x , y , z ) : d φ = F 1 ( x , y , z ) + F 2 ( x , y , z ) + F 3 ( x , y , z ) . In this case the differential equation has solutions given by x , y , z ) = C (constant). (See Section 17.3 for a discussion of exact equations in the plane.) Observe that the differential equation is exact if and only if the vector field F = F 1 i + F 2 j + F 3 k is conservative and that φ is the potential of F . Being scalar fields rather than vector fields, potentials for conservativevector fields are easier to manipulate algebraically than the vector fields themselves. For instance, a sum of potential functions is the potential function for the sum of the corresponding vector fields. A vector field can always be computed from its potential function by taking the gradient. Example 1 (The gravitational field of a point mass is conservative) Show that the gravitational field F ( r ) =− km ( r r 0 )/ | r r 0 | 3 of Example 1 in Section 15.1 is
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812 CHAPTER 15 Vector Fields conservative wherever it is defned (i.e., everywhere in 3 except at r 0 ), by showing that φ( x , y , z ) = km | r r 0 | = p ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 is a potential ±unction ±or F .
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This note was uploaded on 03/21/2012 for the course STAT 101 taught by Professor Graham during the Spring '08 term at Iowa State.

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calculus - SECTION 15.2: Conservative Fields 811 Exercises...

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