Final

Question 1

Set up for line integrals with multiple parts (Without Green’s theorem)

Answer 1

Paramaterize the curves using
(1-t)⟨x0,y0⟩ + t⟨x1,y1⟩
same for x1,y1 -> x2,y2
same for x2,y2 -> x0,y0
this will give you r(t)
∫f(r(t))|r’(t)|dt is the integral you use
The bounds for all the integrals now become t=0 to t=1
Then the whole integral for all the line segments, using the parameterization for each
Then you add them together

Question 2

∫∮⟨⟩√∇·

Answer 2

yeah

Question 3

If F(x, y) is a conservative vector field then ∮F · dr = 0 for
any simple closed curve C.

Answer 3

True

Question 4

If F is a vector field defined on R3 whose component functions have continuous second-order partial derivatives, then div(curl(F))= 0.

Answer 4

True

Question 5

If F is a vector field defined on R3 whose component functions have continuous partial derivatives and curl(F) =/= 0, then F is a conservative vector field.

Answer 5

False

Question 6

(u(t) x v(t))’ = u’(t) x v’(t)?

Answer 6

True

Question 7

Surface integral where you are given x, y, and z bounds and a function that can be solved into z = f(x,y)
(∫∫f(x,y,z)dS)

Answer 7

∫∫f(x,y,z)dS = ∫∫f(x,y,f(x,y))√[(dz/dx)² + (dz/dy)² + 1]dA

Question 8

Line integrals where C is given to you in ∫c f(x,y,z)dr

Answer 8

Parameterize the CURVE such that r(t) = ⟨x(t),y(t),z(t)⟩ 0<t<1 follows the curve. Multiply f(r(t)) that with |r’(t)| and integrate from t = 0 to t = 1

Question 9

Stoke’s theorem
∫F·dr = ?

Answer 9

∫F·dr = ∬curl(F)·n ds
thus
∬curl(F)·(rx x ry) ds

Question 10

Distance between two parallel planes

Answer 10

Find an x,y,z that satisfies the first equation
then find the coefficients of the other plane (a,b,c, and d) if you solve it for zero
use |ax + by + cz + d| / √[a² + b² + c²]

Question 11

Fundamental theorem of line integrals?
What are the conditions?

Answer 11

∫∇f·dr = f(b) - f(a)

dP/dy = dQ/dx

Question 12

Green’s theorem?
What are the conditions?

Answer 12

∫F·dr or ∮F·nds or ∮Pdx + Qdy =
∫∫(dQ/dx - dP/dy)dA
Simple closed curve (going counter-clockwise)

Question 13

Reverse Green’s theorem?
What are the substitutions?

Answer 13

P(x,y) = 1, -y, -1/2 y
Q(x,y) = 0, x, 1/2 x
∫∫1dA = 1/2∮xdy + ydx

Question 14

Surface area of a graph of a function?
(of the form x=x, y=y, z=f(x,y))

Answer 14

∫∫√[1 + (dz/dx)^2 + (dz/dy)^2]dA

Question 15

Surface integrals over parametric equations?

Answer 15

∫∫f(x,y,z)dS = ∫∫f(r(u,v))|ru x rv|dA

Question 16

Surface integrals over vector fields?
When z = g(x,y)?

Answer 16

∫∫FdS or ∫∫F·ndS =
∫∫F·(ru x rv)dA
∫∫[-P(dg/dx)-Q(dg/dy)+R]dA

Question 17

Stoke’s theorem?

Answer 17

∫F·dr = ∫∫curlF·ndS or ∫∫curlF·(ru x rv)dS

Question 18

Divergence Theorem?

Answer 18

∫∫F·dr = ∫∫∫divFdV