1. Line integrals (10/22/04) :vector function of position in 3 dimensions. :space curve With each point P is associated a differential distance vector Definition of the line integral of along space curve C, from point P 1 to P 2. Example: represents the force on a particle then the line integral represents the work done by the force in moving the particle from point P 1 to P 2.)

2. In general, the line integral will depend on the path that is taken There are special physical cases (conservative fields) for which the line integral is independent of path. The line integral around a closed loop is called the circulation of the vector. then If and

3. Example: Find the work for a particle taken through the force field from (0,0,0) to (1,1,1) along the curve C defined by: Take the force in Newtons and the distances to be in meters. x y z (1,1,0) (1,1,1)

4. For the first part of the path, relate the distances and differential to each another. The final integral is For the second part of the curve, x=y=1, dx=dy=0.

5. What if the path were the straight line from (0,0,0) to (1,1,1)? Then the curve is described by x=y=z, and dx=dy=dz In this example the integral depended on the path chosen. This is an example of what is known as a non-conservative force.

6. The line integral between two points will not depend on the path taken. (This can be a representation of a force due to a scalar potential field.) The line integral depends only on start and finish point. If the integral is a closed loop, then. Special case - Conservative forces The vector field can be expressed as the gradient of a scalar field

7. 1. 2. A c b a d Green’s and Stokes theorems Let be a function with continuous first partial derivatives in a certain region. Consider a small rectangle in the (x,y) plane 1. Green’s theorem Surface integral: Counter-clockwise line integral: Only vertical lines contribute since dy – horizontal = 0 Comparing 1 and 2 gives:

8. Let be a function with continuous first partial derivatives in a certain region. Repeat the above but interchanging the x and y operations. Combining Let P and Q be the x and y components of a vector

9. This is also true for an arbitrary 3-d surface bounded by a curve Why is this important in physics applications? Stay tuned. Stokes Law