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Chapter 18

Chapter 18. Section 18.2 The Fundamental Theorem of Line Integrals. Fundamental Theorem of Calculus The antiderivative of the continuous function is the function . The definite integral of ( the derivative) is minus . y. Fundamental Theorem of Line (Path) Integrals

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Chapter 18

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  1. Chapter 18 Section 18.2 The Fundamental Theorem of Line Integrals

  2. Fundamental Theorem of Calculus The antiderivative of the continuous function is the function . The definite integral of ( the derivative) is minus . y Fundamental Theorem of Line (Path) Integrals If is a continuous path and a function with a continuous gradient. The integral over of the (the function is the antigradient of ) is found as minus . B: Terminal Point A: Initial Point x Applying the Fundamental Theorem of Line (Path) Integrals: If we are given a and a vector field in order to find the value of the integral we do the following: Determine if , if so continue, if not parameterize and plug into F. Find the antigradient function so that . Determine the initial point A and the terminal point B of the curve .

  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 . Do this by both parameterizing and applying the fundamental theorem of line integrals. Check: Integrate: Derivative: Cancel: Integrate: Substitute: Evaluate the integral by plugging in points and subtracting. Answers are the same!

  4. Example Find the integral where the vector field F is given to the right along with a parameterization of the path . Check: Find the endpoints of the curve . Initial Point: Terminal Point: Integrate: Derivative: Cancel: Integrate: Substitute: Evaluate the integral by plugging in points and subtracting.

  5. Example Find the integral where the 3 dimensional vector field F is given to the right along with a parameterization of the helix path . Check: Integrate: Find the endpoints of the curve . Initial Point: Terminal Point: Derivative: Cancel: Integrate: Substitute: Derivative: Solve: Substitute: Evaluate the integral by plugging in points and subtracting.

  6. Example Find the integral where the vector field F is given to the right along with a parameterization of the path . path from to on the curve . Check: Notice what happens if we attempt to find an antigradient. Does not have antigradient! Must Parameterize. Ugh!!!!! The derivative should depend only on the y variable. In this it dends on the x variable which is not possible!

  7. y Closed Curves (or Paths) A curve (path) is closed (called a closed curve or closed path) if the initial point of and the terminal point of are the same point. If A is the initial point and B is the terminal point then Closed Curve: Initial Point is equal to Terminal Point. x Integrals on Closed Curves If the vector field F has an antigradient (i.e. or in the 2 dimensional case) and is a closed curve then the value of the integral of F on the curve is zero. If and is a closed curve then: This is not difficult to see when you apply the Fundamental Theorem of Line (Path) Integrals and take into account the initial point (A) and terminal point (B) are the same.

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