Important Formulas Flashcards
Linear Approximation
f(p) + (q-p) * gradient f (p)
Direction of Maximum Rate of Change of f
gradient of f / magnitude of gradient of f
Maximum Rate of Change of f
magnitude of gradient of f
Directional Derivative
gradient f (p) * v / |v|
Chain Rule
gradient of f * d/dt(x(t), y(t))
normal vector for tangent hyperplane to level surface at point p
gradient f (p)
level set singularities
gradient f = 0
Cross product
Assign unit vectors to components, multiply, sum like terms, and convert out of unit form
cosine of the angle between a and b
dot product of a and b / (magnitude a * magnitude b)
critical points of f
gradient f = 0
saddle points of f
critical points where D < 0
local maxima of f
critical points where D > 0 and fxx < 0
local minima of f
critical points where D > 0 and fxx > 0
Hessian matrix
[ fxx fxy; fyx fyy ]
D
fxx*fyy - fxy^2
dA in polar
r dr dtheta
Lagrange multipliers
Given f bounded by g, first find the gradients of f and g. Check where gradient f is 0 (crit point). Then create a list of equations, starting with g (given in equality form) and followed by those of the form f1 = lambda g1, … Solve these as a system to find crtical points
Area in R2
Double integral of 1 dA
Volume
Triple integral of 1 dV
dV = ? d rho d phi d theta
rho^2 sin phi
dA = ? dr d theta
r
x in spherical
rho sin phi cos theta
y in spherical
rho sin phi sin theta
z in spherical
rho cos phi
