Chapter 2 Formulas

Question 1

Line Integral of Scalar Functions

Answer 1

integral from a to b of (f(r(t)) times |v(t)| dt)

Question 2

ds (given r(t))

Answer 2

|v(t)| dt

Question 3

Mass

Answer 3

int(c) of density ds

Question 4

First moment M(yz)

Answer 4

int(c) of x times density ds

Question 5

First Moment M(xz)

Answer 5

int(c) y times density ds

Question 6

First Moment M(xy)

Answer 6

int(c) z times density ds

Question 7

Center of mass

Answer 7

1/M times (M(yz), M(xz), M(xy))

Question 8

Line Integrals of Vector Fields

Answer 8

int(c) F dot T ds = int(c) F dot dr = int(c) F(r(t)) dot r’(t)

Question 9

Work

Answer 9

int(c) F dot T ds = F dot dr

Question 10

Circulation of F around C (fluid which aligns with C)

Answer 10

int(c) F dot dr

Question 11

Flux (fluid exiting through C)

Answer 11

int(c) F dot N ds = int(c) Pdy - Qdx

Question 12

Fundamental Thm of Line Integrals

Answer 12

int(c) grad(f) dot dr = f(r(b)) - f(r(a))

Question 13

When Curl(F) = 0

Answer 13

F = grad(f) (f = potential function)

Question 14

Divergence of F (div(F))

Answer 14

d(F1)dx + d(F2)du + d(F3)dz = grad() dot F

Question 15

Green’s Thm. Circulation Form

Answer 15

double int(curl(F) dA)

Question 16

Area (using Green’s Thm.)

Answer 16

pos Int(c) x dy - pos Int(c) y dx = 1/2 pos Int(c) xdy - ydx.

Question 17

x, y, and z in cylindrical coordinates

Answer 17

x = rcos(θ), y = rsin(θ), z = z

Question 18

x, y, and z in spherical coordinates (ρ, φ, θ)

Answer 18

x = ρsinφcosθ, y = ρsinφsin, z = ρcosφ

Question 19

Normal Vector of a surface S parametrized by r(t)

Answer 19

dr(t)/du cross dr(t)/dv

Question 20

dσ

Answer 20

|dr(t)/du cross dr(t)/dv|

Question 21

Surface Area

Answer 21

double int(D) |dr(t)/du cross dr(t)/dv| dA = double int(D) dσ