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Chapter 18

Chapter 18. Section 18.4 Another Notation for Line Integrals. The Differential Form of Line Integrals (2 dimensional) Let the functions and be the x component and y component of the 2 dimensional vector field F (i.e. ). . The curve is parameterized by . , .

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Chapter 18

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  1. Chapter 18 Section 18.4 Another Notation for Line Integrals

  2. The Differential Form of Line Integrals (2 dimensional) Let the functions and be the x component and y component of the 2 dimensional vector field F (i.e. ). The curve is parameterized by . , The Differential Form of Line Integrals (3 dimensional) Let the functions , and be the x,y , z components of the 3 dimensional vector field F (i.e. ). The curve is parameterized by . ,

  3. Example Find the value of the integral to the right for the given vector field F where goes from the point to the point along the curve , and also along the curve given by . For let: , For let: , If a vector field F has an antigradient (i.e. ) and and have the same initial and terminal points then: Notice that and the answers are the same!

  4. Path Independence An integral is said to be path independent (or independent of path) if any two paths with the same initial and terminal points have the value of the integral over those paths equal. In the previous example we saw that if the vector field has an antigradient (potential function) the integral is independent of path since the value is the difference of the end points. Conservative Vector Fields A vector field is conservative if energy is conserved along any path in the vector field. That is the energy expended moving to any position along any path is equal to the energy required to return to the same position along any other path possibly different path. The function that has the vector field as its gradient represents the potential energy and is called the potential function.

  5. The Following statements are equivalent: The vector field is conservative. We have the follow equality: . An antigradient (potential function) exists for . (i.e. ) The integral is independent of the path . If is any closed curve . (All loop integrals are zero!) All of these have analogues statements for 3 dimensions. The Following statements are equivalent: The vector field is conservative. We have the follow equality: . An antigradient (potential function) exists for . (i.e. ) The integral is independent of the path . If is any closed curve . (All loop integrals are zero!)

  6. Example Find the value of the integral to the right along the parabolic path . For let: Notice that this integral is not independent of path since: and This means that we needed to find the value using the parameterization.

  7. Example The triangular path goes on straight lines from the point to then to and back to . Find the value of the following integral: y is on for is on for x is on for

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