Formulas and Equations

Question 1

Parametric equation of a line

Answer 1

x = x0 + at
y= y0 + bt
z= z0 + ct
(know how to get all of these equal by setting them equal to T)

Question 2

Vector equation of a line

Answer 2

r= r0 + at

Question 3

Equation of a plane

Answer 3

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

Question 4

property of Normal vectors of planes :

Answer 4

n1 x n2 = 0

Question 5

the angle between two planes

Answer 5

cos (x) = (n1 * n2) / |n1||n2|

Question 6

distance from a point to a plane

Answer 6

|(n * v)/ |n||

Question 7

What makes a function continuous (lim)?

Answer 7

lim r(t) = r(t0)

Question 8

velocity

Answer 8

r’(t)

Question 9

acceleration

Answer 9

v’(t) = r’‘(t)

Question 10

speed

Answer 10

|r’(t)|

Question 11

arc length

Answer 11

INTEG( a_> b) |r’(t)| dt

Question 12

Unit Tangent vector (T’(t))

Answer 12

r’(t)/ |r’(t)|

Question 13

Unit Normal Vector (N(t))

Answer 13

T’(t)/ |T’(t)|

Question 14

Binormal Unit vector (B(t) )

Answer 14

T(t) x N(t)

Question 15

parametric eqts of normal line

Answer 15

x = x0 + fx (t)
y= y0 + fy (t)
z= z0 + fz(t)

Question 16

Unit vector projection

Answer 16

GRADIENT ( f) * n

Question 17

Max unit vector projection

Answer 17

|GRADIENT f(x,y,z) |

Question 18

Min unit vector projection

Answer 18

• |GRADIENT f(x,y,z)|

Question 19

Second Deriv test

Answer 19

(fx2)(fy2) - (fxy)^2

Question 20

What if the second deriv is negative?

Answer 20

saddle point

Question 21

What constitutes a local maximum?

Answer 21

Second partial Deriv < 0

Question 22

What constitutes a local Min?

Answer 22

Second Partial Deriv > 0

Question 23

Lagrange

Answer 23

F= GRAD( f) - Lambda (GRAD g)

Question 24

surface area

Answer 24

DBL INTEG |fu x fv| dudv

Question 25

What are cylindrical coordinated similar to

Answer 25

polar coordinates

Question 26

Spherical coordinates

Answer 26

x= rcos (t) sin(p) y = rsin(t)sin(p) z= rcos (p) r^2 = x^2 + y^2 + z^2 R= row, t= theta, p= phi

Question 27

line integral

Answer 27

INTEG F * dr = F(r(t)) * r'(t) dt

Question 28

When is a line integral path independent?

Answer 28

INTEG F* dr = 0

Question 29

Fundamental theorem of line integrals

Answer 29

INTEG(a ->b) F(r(t)) * r'(t) dt = P(r(b) - r(a))

Question 30

Greens Thm

Answer 30

INTEG Mdx +Ndy = DBL INTEG( Nx - My) dxdy

Question 31

div F

Answer 31

GRADIENT * F

Question 32

curl F

Answer 32

GRADIENT x F

Question 33

When is a parameterized surface smooth?

Answer 33

Partial ders - > ru x rv = 0

Question 34

unit normal vector

Answer 34

(ru x rv) / |ru x rv|

Question 35

vector-valued differential of surface

Answer 35

dS= (ru x rv)dudv

Question 36

surface integral

Answer 36

DBL INTEG f(r(u,v)) |ru x rv|dudv

Question 37

Stokes Thm

Answer 37

DBL INTEG (GRADIENT x F) * dS = INTEG F * dr

Question 38

Divergence thm

Answer 38

TRP INTEG (GRAD * F(x,y,z) dV = DBL INTEG F*dS

Question 39

Assumptions of stokes thm

Answer 39

S must be a positively oriented piecewise smooth surface with a simple closed piecewise smooth boundary curve. (F has cont partial deriv)

Question 40

Assumptions of Divergence thm

Answer 40

Let E be a simple solid region, S be a boundary surface of E with an outward (positive) orientation. (F has cont partial deriv)