1. Section 18.3 Gradient 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