# 2009-11-06: Coordinate Transformations and Vector Fields

Stewart section 16.9--17.1

• 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 = <fx, fy>.
• If we can find an f such that grad f = F, we say that F is conservative.
ma215-080-f09 lecture outline 2009-11-06
Created: Fri Nov 6 12:21:26 EST 2009