Integral Theorems

Question 1

How long is a piece of string?

Answer 1

L = (b ∫ a) dS

dS = line element

Question 2

Divergence theorem

Answer 2

( ∮ S) a . dS = ( ∫ V) ∇ .a dV

Question 3

Stoke’s Theorem

Answer 3

( ∮ c) a . dr = ( ∫ s) ∇ .a n(hat) dS

Question 4

a quarter circle - coords and limits

Answer 4

cylindrical coords

0 < Φ < π

Question 5

Volume element

Answer 5

dV = dxdydz

or

dV = dz dΦ ρdρ

Question 6

ellipsoid coordinate system

to find limits

Answer 6

cylindrical

0 < Φ < 2π

and ρ = 0 gives z and solve for function = 0 for ρ

Question 7

divergence of the curl

Answer 7

=0

Question 8

along the negative x-axis

Answer 8

along the straight path y = 0

Question 9

along a semi-circular path

Answer 9

parameterize

r = (cosΦ, sinΦ, z)

0 < Φ < π

Question 10

line integral

Answer 10

ds = |dr/dt| dt = √(dx/dt)^2+(dy/dt)^2+(dz/dt)^2

parameterise by substituting r into f(x,y)

Question 11

diagonal line path

Answer 11

y = x and dy = dx

Question 12

divergence theorem definition

Answer 12

S is the closed surface of the volume V and n(hat) the normal vector to the surface

The flux through a closed surface equals the sum of all sources minus all sinks contained in this volume

Question 13

surface of length L = π centred around the origin

Answer 13

limits -π/2 < x < π/2

Question 14

Area integral

Answer 14

= weight or mass

∫∫ dx dy

Question 15

how to find volume limits
z = 1 - y - x^2

Answer 15

z = 1 - y - x^2
x = when z = 0 => x = √1-y
y = when x and z = 0 => y = 1