Multi

Question 1

Magnitude

Question 2

Unit vector

Question 3

Dot product

Question 4

Projection of u onto v

Answer 4

((uv)/(vv))v

Question 5

Component of u onto v

Answer 5

magnitude of proj or (u*v)/||v||

Question 6

cross product

Answer 6

<23-32, 31-13, 12-21>

Question 7

parametric form

Answer 7

x = x0 + at …

Question 8

Symmetric equations

Answer 8

remove t from parametric form

Question 9

Tangent plane given unit vector n = <a, b, c>

Answer 9

a(x-x0) + b(y - y0) + c(z-z0) = 0

Question 10

Velocity

Question 11

Speed

Question 12

Acceleration

Question 13

ds/dt

Question 14

Unit tangent vector

Answer 14

T = v / ||v||

Question 15

unit normal vector

Answer 15

T’ (t) / ||T ‘ (t)||

Question 16

a sub T (two expressions)

Answer 16

a * T = v * a / ||v|| = d^2/dt^2

Question 17

a sub N (two expressions)

Answer 17

||v x a|| / ||v|| = k(ds/dt)^2

Question 18

arc length

Answer 18

integral of speed

Question 19

Curvature (K) (two expressions)

Answer 19

||T’(t)||/||v(t)||=||v(t) x a(t)||/||v(t)||^3

Question 20

Tangent plane

Answer 20

dz = fx dx + fydy

Question 21

Gradient

Answer 21

<fx, fy>

Question 22

Directional derivative (Du f(x,y))

Answer 22

Gradient * unit vector

Question 23

Direction of max increase

Answer 23

Gradient

Question 24

Maximum value of directional derivative

Answer 24

||gradient||

Question 25

Gradient is normal to what

Answer 25

Level curve

Question 26

Q(x)

Question 27

Critical points (d equation)

Question 28

What conditions for rel min/max/saddle/inconclusive?

Question 29

Lagrange

Answer 29

fx = lambda gx…

Question 30

dV for rectangular coordinates

Question 31

dV for cylindrical coordinates

Question 32

dV for spherical coordinates

Question 33

center of mass coordinates

Answer 33

(Myz/m, Mxz/m, Mxy/m)

Question 34

Line integral of F on curve C

Answer 34

integral of F in terms of r dot product with dr

Question 35

What does the line integral tell you?

Answer 35

Work done

Question 36

How to check if F is conservative?

Answer 36

My = Nx

Question 37

Fundamental Theorem of Line Integrals

Answer 37

If F is conservative, the line integral is independent of path so find the potential function and find f(x2, y2) - f(x1, y1)

Question 38

What does Nx - My give you?

Answer 38

Circulation density

Question 39

Green’s Theorem (2D)

Answer 39

line integral over closed loop of F * dr = double integral of Nx - My

Question 40

Stoke’s Theorem (3D)

Answer 40

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

Question 41

Surface area

Answer 41

double integral over surface of f = double integral over D of ||ru x rv|| du dv

Question 42

Surface integral

Answer 42

double integral over surface of f = double integral of f in terms of r times ru x rv du dv

Question 43

Divergence Theorem

Answer 43

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