Vector fields, line integral, green's theorem

Question 1

What is a 2D vector field?

Answer 1

A function that assigns (x, y) in subset D of R^2 to a vector F(x,y) in R^2.
There exists P, Q of two variables such that
F(x,y)= [P(x,y), Q(x, y)] FORALL (x,y) in D
F(x,y) = Pi + Qj

Question 2

What is a conservative vector field, how is it calculated, how is it interpreted, and what does the FToLI tell us about it?

Answer 2

Let F = vector field. VF is conservative If there exists a function f of two/three vars, with F=GV of f, then f is a potential function of F.

Any point on in the Vector field, F points in the direction of the greatest rate of increase of the function f.

By the FToLI, value of line integrals depends solely on endpoints, making it inherently path-independent.

Question 3

When is a curve given by a vector function smooth?

Answer 3

If r’ is continuous and r’(t) dne 0 forall t in [a, b].

Question 4

What is a piecewise-smooth curve?

Answer 4

If there exists set of curves with connected points, then the union of all C is a path and integral defined as sum of integrals of respective curves

Question 5

What is a closed path?

Answer 5

Closed if r(a) = r(b)

Question 6

What is a:
- Connected set
- Simply-connected set
- open, connected set

Answer 6

• Subset D connected if any two poitns in D can be connected by a path in D
• D simply connected if D connected and every simple closed curve in D encompasses only points in D
• Set that contains no boundary points and every point has small neighbourhood entirly within set.

Question 7

What is a simple curve/path?

Answer 7

Curve given by VF within an interval is simple if
r(l) != r(k) forall l, k in (a,b) : l != k

Question 8

What is the integral of a function f of two variables defined on a smooth curve given a vector function?

Answer 8

If continuous
Integral from a -> b f[x(t), y(t)] x sqrt[dx/dt^2 + dy/dt^2] dt

Question 9

What is the line integral of a function of two variables with respect to x, y respectively?

Answer 9

Wrt x: f * x’(t)
Wrt y: f * y’(t)

Question 10

What is the integral of -C, a curve with the same sample points but opposite orientation?

Answer 10

Integral ito components are negative, however integral of -C is equal to that of C.

Question 11

What is the FT of LI?

Answer 11

• C is smooth curve given by VF
• f differentiable function of 2/3 variables with continuous GV

Integral between bounds of C of gradient vector ⋅ dr = f(r(a))
OR
f(r(b)) - f(r(a))

Question 12

What is path independence?

Answer 12

For any two points A, B - the line integral from A to B is same for all possible paths between A, B
If F is a vector field, then the line integral is indepent of path if integral of C1 F ⋅ dr = integral of C2 F ⋅ dr forall C1,C2 in D with the same initial/terminal points

Question 13

When is a closed path path independent?

Answer 13

iff integral of C for VF F ⋅ dr = 0 forall closed paths C in D

Question 14

What is the integral of F defined on an open, connected set?

Answer 14

If Line integral is path indepedent, F is a conservative VF.

Question 15

What is Green’s theorem?

Answer 15

Suppose F continuous VF

Let C be simple closed curve with positive orientation in xy-plane

Let D = closed region bounded by C / C = partial D

If P, Q continuous first order PD on open region D, then the integral of F:
Integral of C for P(x, y)dx + Q(x, y)dy = Double integral of D for the difference between partial derivative of Q wrt x and P wrt y