Chapter 5 - Line Integrals

Question 1

Scalar field

Answer 1

A function that assigns a real number to each point (x,y,z) in a region of R^3

Question 2

Vector field

Answer 2

A function that assigns a vector v (x,y,z) in a 3-dimensional space to each point (x,y,z) in a region of R^3

Question 3

Smooth curve

Answer 3

Dr/dt is continuous and |dr/dt| isn’t equal to 0

Question 4

Piecewise smooth curve

Answer 4

A continuous curve made up of 2 or more smooth curves such that the end point of one smooth curve coincides with the start point of the next smooth curve

Question 5

Method to calculate line integrals of scalar fields along a smooth curve C from A to B

Answer 5

1) Choose an appropriate parameterisation such that the position vector of a point on C is r(t), ta

Question 6

Simple closed curve

Answer 6

A curve that doesn’t cross itself and ends at the same point that it begins

Question 7

Line integral of a vector field v(x,y,z) along a smooth curve C

Answer 7

I = integral over C of [v•dr]
Since r=(x,y,z), dr=(dx,dy,dz)
I=integral over C of [v1dx + v2dy + v3dz]

Question 8

Evaluating integral if we specify C parametrically with position vector r (t) given as a function of a parameter t

Answer 8

Then we can write I = integral over c of [v• dr/dt] dt.
Then evaluate the scalar porduct as a function of t and evaluate thr definite integral with respect to t as for scalar fields

Question 9

Evaluating integral if we can specify c as an explicit functional relationship, such as y=f (x), z=g (x) where f (x) and g (x) are continuous differentiable functions of x

Answer 9

Then I = integral over C of [v1 + v2f’(x) + v3g’(x)] dx

And evaluate this as a definite integral with respect to x

Question 10

Length of C

Answer 10

Integral (from ta to tb) of [ |dr/dt| ]

Question 11

What happens to the sign of the line integral of a vector field?

Answer 11

Unlike the integrals of scalar fields, the value od the line integral of a vector field changes sign when the sense of description of the curve C is reversed

Question 12

Circulation of v around c

Answer 12

K = closed integral over c of [ v•dr ]

This is hoe the line integral of a vector field v aroubd a simple closed curve is denoted.

Question 13

Green’s theorem

Answer 13

Let R be a simply connected closed region of the plane whose boundary is a simple, piecewise smooth, closed curve C oriented anti-clockwise.

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

Closed integral over C of [ f(x,y)dx + g(x,y)dy]
= double integral over R of [(dg/dx - df/dy) dxdy]

Question 14

Area of the region R enclosed by the curve C

Answer 14

A = (1/2)* closed integral over c of [ -ydx + dy]
= closed integral over c of [xdy]
= closed integral over c of [-ydx]

Question 15

Path independent line integrals

Answer 15

The value of the integral depends only on the location of the end points of the curve, and not on the particular path taken between them.

I = integral (A to B) of [v•dr]
where A and B denote the start and end points of the curve over which integration is performed

Question 16

Theorem?

Answer 16

The line integral of a vector field v is path independent in a region R of space if and only if there exists a function phi (r) on R such that
v = (dphi/dx, dphi/dy, dphi/dz) = grad (phi)

Here, phi is called a potential function for the vector field v

Question 17

Conservative vector field

Answer 17

Dv1/dy = dv2/dx
Dv1/dz = dv3/dx
Dv2/dz = dv3/dy