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:
- Find the differential dx.
- Use it to estimate dx if r changes by dr and theta by zero.
- Find the differential dy.
- Find dy if r -> r + dr and theta is fixed.
- Find a vector from (x,y) to (x+dx,y+dy).
- Key Point
- We can visualize an area element dA in a non-rectangular coordinate system (r,s) as (nearly) a parallelogram with sides given by vectors.
- Simlarly for a volume element.
- Then dA = .
- 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.
ma215-080-f09 lecture outline 2009-11-06
Created: Fri Nov 6 12:21:26 EST 2009
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©2009 Gavin LaRose