Week 1 Flashcards
Important rules of Einstein notation
Repeated indices are implicitly summed over (therefore dissapear)
No index can appear more than twice in any term
Each term in an equation must have the same free indices
This is just a convention, there isn’t a specific intuition behind it
Einstein for cross prod
Introduce LC tensor with indices below
put indices above for vectors being curled
One free index
Changing index of vector w Kronecker delta
Crucially if indices match but are opposite, they change to the other letter on that Kronecker delta see line 4 to line 5
Each extra letter in index notation refers to
An extra dimension eg
Kronecker delta in index notation
Plays role of identity matrix eg:
Euclidean metric tensor
All entries are zero except below and same but for 3
each kronecker delta only appys to one factor (only converts one letter each)
Put this in Einstein notation
Using Euclidean metric tensor (end inverse)
Dot prod is equal for transpose
Use anti symmetric property of Levi cevitica
Electrostatics
The forces due to a static array of electric charges
Magnetostatics
The forces due to a static magnetic field
Where θ is the angle between gradF and dl
div of function of vector
div of cross prod of 2 vectors
Curl product rules
Laplacian of a vector
What is a line integral?
Integral of a field, vector or scalar along a path.
Aka path integral
Line integral of scalar function
Line integral of vector field
dl = ?
Infinitesimal line element, eg (dx,dy,dz)
Surface integral?
Flux of v through S
Or equivalent for a closed surface
Summing flux of sub surfaces within a surface?
Shared sub surfaces cancel each other out, therefore we can sum over infinitesimal cubes to find larger surface’s flux
when to change order of indices of LC
when RELATIVE order of factors to which LC refers change
eg: you can move them around without changing LC but if you change the order that they are in (other than cycling) you must change LC accordingly
Volume integral of scalar function
In Cartesian coordinates dv=dxdydz
If f=1, the integral gives volume
Else
If f is density (amount of Q per volume), integral gives total amount of Q in V
Volume integral of vector field
Curl of a vector field across a closed surface
Curl in Einstein SC
Relate LC to Kronecker delta
