ma215-080-f09 outline 2009-11-06

Last time we talked about spherical coordinates and coordinate transformations.
Example: How do we fine the area element dA for a coordinate system? In rectangular:
(Example, cont.) In Polar:
Game: For polar coordinates:
1. Find the differential dx.
2. Use it to estimate dx if r changes by dr and theta by zero.
3. Find the differential dy.
4. Find dy if r -> r + dr and theta is fixed.
5. Find a vector from (x,y) to (x+dx,y+dy).
Key Point
1. We can visualize an area element dA in a non-rectangular coordinate system (r,s) as (nearly) a parallelogram with sides given by vectors.
2. Simlarly for a volume element.
3. Then dA = .
4. And dV = .
These ideas will reappear in section 17.4
Now: Vector Fields. A Vector field is just a vector valued function of multiple variables; e.g., F(x,y) = (1/2)<y,x>.
Game: Sketch this F
Other vector fields: Gradient Vector Fields: grad f = <f_x, f_y>.
If we can find an f such that grad f = F, we say that F is conservative.

2009-11-06: Coordinate Transformations and Vector Fields