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Section 18.3 Gradient Fields and Path-Independent Fields

Section 18.3 Gradient Fields and Path-Independent Fields. Gradient Fields. A vector field is said to be a gradient field if for some function f f is called a potential function

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Section 18.3 Gradient Fields and Path-Independent Fields

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  1. Section 18.3Gradient Fields and Path-Independent Fields

  2. Gradient Fields • A vector field is said to be a gradient field if for some function f • f is called a potential function • If we wanted to find the total change between two points, P and Q, we could use f(P) – f(Q) • Alternatively, if we had a smooth curve C from P to Q, we could break it up into small pieces and estimate the change on each piece • The change on each piece can be estimated by rate of change of f x Distance moved in direction of C on that piece

  3. The Fundamental Theorem of Calculus for Line Integral • Using the ideas from the previous slide we get the following • If C is a piecewise smooth oriented curve starting at P and ending at Q • And f is a function whose gradient is continuous on the path of C then we have • Note: f is a potential function of • Let’s take a look at why this is

  4. Some notes on the FTC for Line Integrals • If Q = P then C is a closed path and the integral will be 0 • When is a gradient field, the value of the line integral is path independent • The integral only depends on the endpoints of C • Using the FTC for line integrals will require that we find the potential function, f

  5. Why do we care about path independent vector fields? • Gravitational fields are path independent • Imagine you have to carry a heavy box from your front door to your bedroom upstairs • Because of the gravity you have to do work to carry the box up (the scientific definition of work) • You have two stairways in your house: a gently sloping front staircase, and a steep back staircase • Since the gravitational field is a path independent vector field, the work you must do against gravity is the same if you take the front or the back staircase • As long as the box starts in the same position and ends in the same position, the total work is the same

  6. Finding the Potential Function for a Vector Field (if one exists) • Example 1 • Example 2 Determine if the vector field could be a gradient field

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