Lectures 13, 14 and 15

Question 1

How can we define a vector displacement from A to B?

Answer 1

r(AB) = r(B) - r(A)

Question 2

What kind of integral do we normally need to perform when a field is present?

Answer 2

A line integral.

Question 3

What are the three cases to consider for fields and what are their integrals?

Answer 3

A scalar field Ф (integral from A to B of Ф.dr), vector field a (integral from A to B of a.dr), and vector field a (integral from A to B of aXdr)

Question 4

How do you work out if the integral is path dependent?

Answer 4

Do the integral, and split it up into two separate paths. Calculate each part and if they are not equal then the integral is path dependent.

Question 5

Is the integral path independent or dependent for a conservative field?

Answer 5

Path independent.

Question 6

What are the 4 conditions for a vector field to be conservative?

Answer 6

• Integral a.dr independent of path taken
• Exists scalar function Ф(x,y,z) such that a=∇Ф
• Curl of a = 0
• a.dr is an exact differential

Question 7

What are two good example fields we can use?

Answer 7

G = x i and H = x j

Question 8

For G = x i, what are the paths OAB and OB equal to, where OAB is a right angled triangle?

Answer 8

OAB and OB = 1/2.

Question 9

For H = x j, what are the paths OAB and OB equal to, where OAB is a right angled triangle?

Answer 9

OAB = 1 and OB = 1/2.

Question 10

What do these paths OAB and OB mean for the two different fields?

Answer 10

• G=x i is conservative, has no curl, and integral is path dependent.
• H=x j is not conservative, has curl, and integral is path dependent.

Question 11

How do you find if a field is conservative or not?

Answer 11

Take the integral of the field.dr (split up if need to), and do this for multiple paths. If they are all equal then it is conservative and vice versa.

Question 12

What is Green’s theorem in the plane?

Answer 12

integral(closed) of F.dr = integral(closed) of (Pdx + Qdy) = double integral over R of (dQ/dx - dP/dy)dA

Question 13

What does Green’s theorem allow us to do?

Answer 13

Relate an integral over a closed loop to an integral over the region R inside the loop.

Question 14

What is the loop integral equal to for a conservative field and what does this make the integral over the region R?

Answer 14

Equals zero, therefore dQ/dx - dP/dy = 0

Question 15

How can we use Green’s theorem to calculate areas?

Answer 15

Set (dQ/dx - dP/dy) = 1, making the integral over R equal to 1.dA = A.

Question 16

What is the equation for the area using Green’s theorem when P = -y and Q = x?

Answer 16

A = 1/2 * integral(closed) of x dy - y dx

Question 17

What is the integral which will give the total flux out of region R across the boundary made by curve C?

Answer 17

integral(closed) over C of F.dn

Question 18

What is the dn vector equal to in terms of i and j?

Answer 18

i dy - j dx

Question 19

What is the divergence theorem in two dimensions?

Answer 19

integral (closed) of F.dn = double integral over R of ∇.F dA

Question 20

In what case is there a divergence-free field?

Answer 20

For example, a velocity field along x axis going through our closed loop C. The integrals cancel each other out at each side of the loop, hence the whole integral is equal to zero.