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
surface area
Parametrize shape in terms of two variables. Find the partial derivatives and the magnitude of their cross product. Integrate this magnitude with respect to the two parameters. (Can convert coordinates after finding magnitude)
Mass
Triple integral of density function
