Multi Flashcards
Magnitude
Unit vector
Dot product
Projection of u onto v
((uv)/(vv))v
Component of u onto v
magnitude of proj or (u*v)/||v||
cross product
<23-32, 31-13, 12-21>
parametric form
x = x0 + at …
Symmetric equations
remove t from parametric form
Tangent plane given unit vector n = <a, b, c>
a(x-x0) + b(y - y0) + c(z-z0) = 0
Velocity
Speed
Acceleration
ds/dt
Unit tangent vector
T = v / ||v||
unit normal vector
T’ (t) / ||T ‘ (t)||
a sub T (two expressions)
a * T = v * a / ||v|| = d^2/dt^2
a sub N (two expressions)
||v x a|| / ||v|| = k(ds/dt)^2
arc length
integral of speed
Curvature (K) (two expressions)
||T’(t)||/||v(t)||=||v(t) x a(t)||/||v(t)||^3
Tangent plane
dz = fx dx + fydy
Gradient
<fx, fy>
Directional derivative (Du f(x,y))
Gradient * unit vector
Direction of max increase
Gradient
Maximum value of directional derivative
||gradient||
Gradient is normal to what
Level curve
Q(x)
Critical points (d equation)
What conditions for rel min/max/saddle/inconclusive?
Lagrange
fx = lambda gx…
dV for rectangular coordinates
dV for cylindrical coordinates
dV for spherical coordinates
center of mass coordinates
(Myz/m, Mxz/m, Mxy/m)
Line integral of F on curve C
integral of F in terms of r dot product with dr
What does the line integral tell you?
Work done
How to check if F is conservative?
My = Nx
Fundamental Theorem of Line Integrals
If F is conservative, the line integral is independent of path so find the potential function and find f(x2, y2) - f(x1, y1)
What does Nx - My give you?
Circulation density
Green’s Theorem (2D)
line integral over closed loop of F * dr = double integral of Nx - My
Stoke’s Theorem (3D)
line integral over C of F * dr = double integral over surface S of curl F (aka del operator x F) dot product with N ds
N ds -> <-gx, -gy, 1> if oriented upwards and <gx, gy, -1> if oriented downwards
Surface area
double integral over surface of f = double integral over D of ||ru x rv|| du dv
Surface integral
double integral over surface of f = double integral of f in terms of r times ru x rv du dv
Divergence Theorem
double integral over S of F * N dS is equivalent to the triple integral of div F dV, where div F is Mx + Ny + Pz
