Important Formulas

Question 1

Linear Approximation

Answer 1

f(p) + (q-p) * gradient f (p)

Question 1

Direction of Maximum Rate of Change of f

Answer 1

gradient of f / magnitude of gradient of f

Question 2

Maximum Rate of Change of f

Answer 2

magnitude of gradient of f

Question 3

Directional Derivative

Answer 3

gradient f (p) * v / |v|

Question 4

Chain Rule

Answer 4

gradient of f * d/dt(x(t), y(t))

Question 5

normal vector for tangent hyperplane to level surface at point p

Answer 5

gradient f (p)

Question 6

level set singularities

Answer 6

gradient f = 0

Question 7

Cross product

Answer 7

Assign unit vectors to components, multiply, sum like terms, and convert out of unit form

Question 8

cosine of the angle between a and b

Answer 8

dot product of a and b / (magnitude a * magnitude b)

Question 9

critical points of f

Answer 9

gradient f = 0

Question 10

saddle points of f

Answer 10

critical points where D < 0

Question 11

local maxima of f

Answer 11

critical points where D > 0 and fxx < 0

Question 12

local minima of f

Answer 12

critical points where D > 0 and fxx > 0

Question 13

Hessian matrix

Answer 13

[ fxx fxy; fyx fyy ]

Question 14

D

Answer 14

fxx*fyy - fxy^2

Question 15

dA in polar

Answer 15

r dr dtheta

Question 16

Lagrange multipliers

Answer 16

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

Question 17

Area in R2

Answer 17

Double integral of 1 dA

Question 18

Volume

Answer 18

Triple integral of 1 dV

Question 19

dV = ? d rho d phi d theta

Answer 19

rho^2 sin phi

Question 20

dA = ? dr d theta

Answer 20

r

Question 21

x in spherical

Answer 21

rho sin phi cos theta

Question 22

y in spherical

Answer 22

rho sin phi sin theta

Question 23

z in spherical

Answer 23

rho cos phi

Question 24

surface area

Answer 24

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)

Question 25

Mass

Answer 25

Triple integral of density function