math 32b midterm 2

math 32b midterm 2 flashcards mostly for personal use

spherical coordinates

(rho, theta, phi)

rho definition and bounds

distance from point to orign, >= 0

theta bounds

0 <= theta < 2pi

phi bounds

0 <= phi < pi

rho equals

sqrt(x² + y² + z²)

tan(theta)

y/x

cos(phi)

z/rho

x (spherical coordinates)

rhosin(phi)cos(theta)

y (spherical coordinates)

rhosin(phi)sin(theta)

z (spherical coordinates)

rhocos(phi)

volume element for spherical coordinates

rho²sin(phi)drhodphidtheta

joint probability density function of x and y

(1/2pi)exp(-0.5(x²+y²))

terms for pdfs

p(x,y) >= 0; double integral over J of p(x,y)dydx = 1

total mass in R²

double integral over D of delta(x,y)dA

x-moment in R²

double integral over D of ydelta(x,y)dA

y-moment in R²

double integral over D of xdelta(x,y)dA

center of mass of D in R²

(xcm,ycm)

xcm in R²

y-moment/mass

ycm in R²

x-moment/mass

mass in R³

triple integral over W of delta(x,y,z)dV

center of mass in R³

(xcm,ycm,zcm)

xcm in R³

(triple integral over W of xdelta(x,y,z)dV)/(triple integral over W of delta(x,y,z)dV)

image/range G of G(X)

set of all images G(x) where x is in X

injective

one-to-one

surjective

G(X) = Y

bijective

injective and surjective

Jacobian of G

dxdu(dydv) - dxdv(dydu)

linear Jacobian of G

AD - BC

Jacobian of G in 3 dimensions

d(x,y,z)/d(u,v,w)

vector field

map that assigns a vector to every point

unit radial vector field

magnitude is always 1; is a function of r

del operator

gradient

divergence of F

del dot F

curl of F

del cross F

conservative vector field F

F is a gradient of f

potential function f

gradient of f is F

divergence of curl of F

0

if F is conservative

curl of F = 0

if curl(F) is not equal to 0

F is not conservative

If F is conservative on an open connected domain

any 2 potential functions of F differ by a constant

line integral formula

integral from a to b of f(r(t)) magnitude(r’(t))dt

line integral linear path parametrization

r(t) = u + t(v-u)

vector line integral formula

the integral over C of F dot dr