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Chapter 18

Chapter 18. Section 18.1 Line Integrals. Vector Fields (or Force Fields) A vector field is function with several variables as inputs and several variables as outputs.

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Chapter 18

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  1. Chapter 18 Section 18.1 Line Integrals

  2. Vector Fields (or Force Fields) A vector field is function with several variables as inputs and several variables as outputs. The function and the function . Since we cannot represent 4 or 6 dimensions on 3 dimensional coordinate system we need a different way to visualize these types of functions. Graphing Vector Fields To graph a vector field at the point draw the vector with its tail located at the point . This represents how energy (gravitation, electrical, thermal, etc.) is being directed though the plane. A similar type of diagram can also be done for a vector field to show how energy is directed through space. Example Graph the constant vector field given by: At every single point the vector field produces the vector . This type of a vector field is sometimes called a river because all force “flows” in the same direction at the same rate.

  3. Example Graph the vector given by: All vectors will end up pointing away from the y-axis. The farther away from the y-axis the smaller the vector. This type of a vector field is sometimes called the red sea because all force “flows” away from the y-axis and gets smaller. Example Graph the vector given by: All vectors will end up pointing away from the origin. The farther away from the origin the larger the vector.

  4. Example Graph the vector given by: All vectors will end up pointing away from the origin. All of the vectors are of unit length. Example Graph the vector given by: All vectors will end up pointing toward the origin. All of the vectors are of unit length.

  5. Example Graph the vector given by: All vectors will end up pointing away from the origin. All of the vectors increase in length as you move toward the origin. (Explosion) Example Graph the vector given by: All vectors will end up pointing toward the origin. All of the vectors increase in length as you move toward the origin. (Black Hole)

  6. Work and Energy Previously you saw that which is a measurement of energy is found by: This is when objects or fluid was lifted or pumped straight up or down. 10 lbs Work Moving on a Path . If is a path from an initial point A to a terminal point B in a vector field F is found by partitioning a parameterization of , say into small vectors, finding the dot product of each one with F (i.e. ), summing them up and taking the limit as the number of partition goes to infinity. Parametric Curve for : This is the limit of a Riemann sum which is an integral. The way this formula is implemented with a parametric curve :

  7. Example Find the work done moving along the curve , given by from the point to the point ,in the vector field: . Use two different parameterizations of the curve . Notice that different parameterizations of the same curve give equal values for the integral.

  8. Example Find the work done moving through the 3 dimensional vector field along the path parameterized to the right. Before we can set up the integral we have to compute . Notice that this formula can also be interpreted for vector fields for dimensions other than 2!

  9. The Path For a path with initial point A and terminal point B the path is the exact same curve but in the opposite direction with initial point B and terminal point A. The energy returning from B is the negative of the energy required to get to B. y y B: Terminal Point B: Initial Point A: Initial Point A: Terminal Point x x A Sequence of Paths If a path is made up a sequence of paths , ,, with the terminal point of one being the initial point of the next then the work on is the sum of the work on each of the curves . y x

  10. Example For the vector field and the path that goes from to along , and from to along , and from to along the y-axis, compute . The curve is made up of the curves . We already found . Find and take negative.

  11. Find and take negative. Adding up the previous values from the other paths we obtain the answer for the integral along the entire path .

  12. Parameterizing curves In order to do these integrals you will need to parameterize a curve. Lines, functions, circles, and ellipses all have a standard method for parameterization. x is a function of y y is a function of x y y d c x x a b A circle with center and radius r A line segment from to z y r y x x

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