5)Line Integrals

Question 1

Scalar and vector fields and direction of line integral

Answer 1

Scalar field line integral is independent of direction of C
Vector field: if C is reversed line integral changes sign

Question 2

What is a smooth curve ( line Integrals)

Answer 2

Smooth if position vector r
dr/dot is continuous and magnitude never 0
|dr/dt| not equal to 0

No stationary points

Question 3

Helix centred on z-axis

Answer 3

x(t)= cost
y(t)= Sint
z(t)= kt

Question 4

Line integral of a scalar field

Answer 4

Int( f(x,y,z).ds)
c

   t_b = int( f( x(t), y(t), z(t) ) |dr/dt| .dt
   t_a

MAGNITUDE SQRT OF components squared dr/dt

Given Smooth curve C from A to B and scalar field f(x,y,z) defined in some region containing c

Question 5

Closed curve line Integral

Answer 5

Closed curve C line integral

IntO( f(vector (x)).ds
c

Value is independent of point from which arc length measured

Question 6

Line integral of a vector field

Def

Answer 6

Given a vector field BOLD v(x,y,z) defined over some region containing c.
^
Unit tangent vector T = d(r)/dt
-

I = int ( bold(v)(x,y,z).d(bold(r)))
      c
                          ^
= int( v(x,y,z) • T.ds)
     c                   -
                     ^
= int( v_1.dx + v_2.dy +v_3.dz)
     c                   

Unit tangent = derivative of position vector
Changes sign and is reversed if c reversed!

Question 7

Line integral of vector field evaluating

Answer 7

PARAMETRICALLY WRT t
DOT PRODUCT

I = int( vector(v) • ( d(vector(r))/dt) .dt )
c

FUNCTIONs of x y=f(x), z=g(x) WRT x

I= int( v_1 + v_2 f’(x) + v_3 g’(x)) .dx
c

y,z must be singly values over domain split-> invertible

Question 8

Circulation:

If in conservative field/ if in path independent field

Answer 8

Circulation k =

IntO( vector(v) •d (vector(r)))

Of vector(v) around c

Is the line integral of vector field vector(v) around a simple closed curve

For path independent variable circulation k=0
Closed curve line integral =0 for any simple closed curve In Conservative field

Question 9

Length of curve

Answer 9

Length of curve is line Integral

Int(.ds)
c

Question 10

Positional vector derivative

Answer 10

dr/dt vector

.ds = |dr/dt| .dt

Question 11

Greens theorem

Calculating the area enclosed by simple closed

Answer 11

IntO( f(x,y) .dx + g(x,y).dy)
c
= closed 
intO( (f(x,y), g(x,y) ) • (dx/dt, dy/dt) .dt )
  c

= double int(part g wrt x - part f wrt y) dx.dy
R

f(x,y) and g(x,y) continuous and have continuous first partial derivatives on some open set containing R.

Change of variable if in t

Use greens when we can’t parameterise and functions of both x and y.

Question 12

Subregion (line integral)

Answer 12

Subregions are simply connected regions split into subregions.
Must have boundary curves that are cut by any line parallel to either coordinate axes.

Opposites cancel.

Question 13

Area enclosed by smooth piece wise curve: by greens

Answer 13

A = 0.5 intO( -y.dx +x.dy)

For area of region enclosed by curve

As greens gives double of partials take away to give one, area

Alternatively:
g_x - f_y =1 or other value

E.g. f(x,y) =0 g(x,y) =x
Or f(x,y) =-y g(x,y) =0

Direction matters in this

Question 14

Path independence of line integral of c over vector field

Answer 14

I = int( vector(v) . d(vector(r)))
c

                                      ^ = int( vector(v) • vector(T) .dt)
c.                                   - Is path independent on R if the value of I is independent of path for a smooth curve from A to B

Circulation k=0 if Integral is path I dependent

Question 15

Conservative field

Answer 15

If a line integral is path independent on R

Then for any closed curve we also have 
IntO( vector(v) . dvector(r)) =0
For any simple closed curve in R
 Then the field Vector(v)
Is a CONSERVATIVE FIELD V

Question 16

Theorem line integral for a vector field v that is path independent

Answer 16

Theorem: the line integral of a vector field vector(v) is path independent in a region R of space if and only if there exists a function phi(vector(r) on R st

Vector (v) = partials of Phi
= ( φ_x, φ_y, φ_z)
= grad ( φ)

Question 17

If vector(v) is conservative in some region R

Answer 17

There exists a scalar field φ(x,y,z) on R such that
For curve c for t in [t_a, t_b]
t_b
Int(vector(v).dvector(r)) = int( (dφ/dt).dt)
c. t_a

                                      =φ(t_b) - φ(t_a)

As we have
∂φ dx ∂φ dy ∂φ dz dφ
— — + — — + — — = —
∂x dt ∂y dt ∂z dt dt

Question 18

A vector field vector(v) = (v_1,v_2,v_3)

Defined on a simply connected region is conservative if

Answer 18

A vector field vector(v) = (v_1,v_2,v_3)

Defined on a simply connected region is conservative if and only if

∂v_1 ∂v_2 ∂v_1 ∂v_3 ∂v_2 ∂v_3
—- = —- , —- = —- , —- = —-
∂y ∂x ∂z ∂x ∂z ∂y

Once finding phi(x,z,y) check

Constant =0

Remember on a closed curve equals circulation =0