Conservative vector fields

Question 1

Fundamental theorem of calculus

Answer 1

If g is a smooth function defined on the interval [a,b], then
∫(over a to b) g’(t) .dt = g(b) - g(a)

Question 2

Fundamental theorem of calculus for higher dimensions

Answer 2

Let Ω C |R^n and f: Ω -> |R be a smooth scalar field and assume that r: [a,b] -> |R^n is a piecewise smooth parametrisation of a path whose image is included in Ω.

Then ∫ (over C) (∇f) .dr = f(r(b)) - f(r(a)).

Question 3

Smooth function depends on the end points

Answer 3

If F = ∇f and f is smooth, then the line integral ∫(over C) F.dr does not depend on the path C but just on its end points.

Question 4

Conservative vector field

Answer 4

A vector field F: Ω -> |R^n is called a conservative vector field if its line integral does not depend on the path. More precisely, for any C1 and C2 piecewise smooth paths in Ω with respective parameterisations r, s : [a,b] -> Ω so that r(a) = s(a) and r(b) = s(b) we have
∫ (over C1) F.dr = ∫ (over C2) F . ds

Question 5

Gradient of vector fields theorem

Answer 5

Gradient of vector fields are conservative

Question 6

Connected path

Answer 6

A set Ω C |R^n is called a connected path if for any x, y e Ω , there exists a path r [a,b] -> Ω so that r(a) = x and r(b) =y.

Question 7

Smooth scalar field theorem

Answer 7

Suppose F is a smooth conservative vector field defined on a connected set Ω . Fix a e Ω and define the functionf: Ω ->||R given by
f(x) = ∫(over C) F.dr
where r is the parametrisation of a piecewise smooth path C connected a and x. Then f is a smooth scalar field and ∇f= F.

Question 8

Equivalent definitions of F

Answer 8

Let F be a vector field that is continuous on a connected set Ω C |R^n. The following are equivalent

1) There exists a scalar field f so that F = ∇f
2) F is conservative
3) The integral of F along a closed path is zero.

Question 9

The sum and scalar multiplication of F and G are conservative

Answer 9

If F and G are conservative vector fields with a , b e |R, then aF + bG is a conservative vector field.