Week 7 Surface Integrals, Flux and Divergence Theorem

Question 1

define the vector dS

Answer 1

dS = n^ * ds 
dS = the normal unit vector * scalar ds

Question 2

define n^

Answer 2

n^ = ∇f / |∇f|

Question 3

what happens to the unit vector and elemental surface vector when the elemental surface is inclined at an angle θ to the plane

Answer 3

cos θ = n^ * k^

so dxdy = dS * k^

Question 4

what is the differential element of solid angle in spherical polar coords

Answer 4

dΩ = sinθdθdΦ = ds / r^2

Question 5

what is the total flux equation

Answer 5

Flux = ∫ A * dS

Question 6

what are the 4 flux rules/observations

Answer 6

> The modulus of the total flux is maximal if the field is always perpendicular to the surface
The flux is maximally pos/neg when the field lines are outwards/inwards wrt surface
The flux will be zero if the field is everywhere parallel to the surface
The flux may be zero if there are equal contributions from the field entering and leaving the volume

Question 7

what does Divergence theorem do

Answer 7

Relates the total flux of a vector field through a close surface to the divergence of the vector field inside the volume bounded by the surface

Question 8

what is the divergence theorem equation

Answer 8

∫ A * dS = ∫ ∇ * A dV