1when sketching a vector field what should you do first?
select points and plug them in to find the shape at the points
2how to find the parametric description of the line segment "from (x0,y0,z0) to (x1,y1,z1)"
r(t)=<x0+t(x1-x0),y0+t(y1-y0),z0+t(z1-z0)>
3when doing a line integral xy ds, r(s)=<cos(s),sin(s)>
int(cos(s)sin(s))ds
4when evaluating line integral (x-y)ds and r is in terms of t what do you integrate
int (r(x)-r(y))*|r'(t)|
5what is int f ds in terms of t?
f(x(t),y(t),z(t))*|r'(t)|
6how do you find length L of curve r(t)
integrate |r'(t)| in terms of t
7what is F*Tds?
F*r'(t) dt
8when calculating work, what must you integrate by?
F*r'(t) (for r plug in whenever possible to share like variables)
9how do you find circulation of a vector field
use F*r'(t) dt
10how do you compute outward flux of a vector field?
use F*nds (n is the derivative of x and y of f)
11how do you determine if a vector field inR^2 is conservative?
f'(y) must be equal to g'(x)
12how do you find the potential function f in a conservative Vector field?
integrate the partial of Fx and integrate x then add c(y,z)
13how do you use a line integral to compute the ara of a plane region>
1/2 * int(x(dy)-y(dx))
14what is divergence
f'(x) +g'(y)+h'(z)
15what is curl?
the cross product of <x',y',z'> and <f,g,h>
16when a 2 dimentional vector has 0 divergence what does it look like?
quadrants 1 and3 are getting mor positive and opposite of eachother and more negative respectively and quads 2 and 4 are pointed toward eachother getting larger as the higher numbers stack up
17what do you integrate by using greens theorem?
g'x-f'y
18what do you integrate in flux problems? also shown as F*nds
int <F>*<F'x,F'y,F'z>
19when making parametric descriptions, when should you use u and v?
when dealing with curves such as parabolas, cones, and cylinders
20when making parametric descriptions, when should you use x and y?
planes and other non curved equations, such as planes
21when making parametric descriptions, when should you use lambda and theta?
when dealing with hemispheres
22when using stokes theorem what is the curl of F?
plug in the variable that has an integer limit and turn x and y into sin and cos respectively inside the original function F
23when using stokes theorem what is n?
r'(thet)