Week 1

Question 1

Important rules of Einstein notation

Answer 1

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

Question 2

Answer 2

This is just a convention, there isn’t a specific intuition behind it

Question 3

Einstein for cross prod

Answer 3

Introduce LC tensor with indices below
put indices above for vectors being curled
One free index

Question 4

Changing index of vector w Kronecker delta

Answer 4

Crucially if indices match but are opposite, they change to the other letter on that Kronecker delta see line 4 to line 5

Question 5

Each extra letter in index notation refers to

Answer 5

An extra dimension eg

Question 6

Kronecker delta in index notation

Answer 6

Plays role of identity matrix eg:

Question 7

Euclidean metric tensor

Answer 7

All entries are zero except below and same but for 3

Question 8

Answer 8

each kronecker delta only appys to one factor (only converts one letter each)

Question 9

Put this in Einstein notation

Answer 9

Question 10

Using Euclidean metric tensor (end inverse)

Answer 10

Dot prod is equal for transpose

Question 11

Use anti symmetric property of Levi cevitica

Answer 11

Question 12

Electrostatics

Answer 12

The forces due to a static array of electric charges

Question 13

Magnetostatics

Answer 13

The forces due to a static magnetic field

Question 14

Answer 14

Where θ is the angle between gradF and dl

Question 15

div of function of vector
div of cross prod of 2 vectors

Answer 15

Question 16

Curl product rules

Answer 16

Question 17

Answer 17

Question 18

Laplacian of a vector

Answer 18

Question 19

What is a line integral?

Answer 19

Integral of a field, vector or scalar along a path.

Aka path integral

Question 20

Line integral of scalar function

Answer 20

Question 21

Line integral of vector field

Answer 21

Question 22

dl = ?

Answer 22

Infinitesimal line element, eg (dx,dy,dz)

Question 23

Surface integral?

Answer 23

Question 24

Flux of v through S

Answer 24

Or equivalent for a closed surface

Question 25

Summing flux of sub surfaces within a surface?

Answer 25

Shared sub surfaces cancel each other out, therefore we can sum over infinitesimal cubes to find larger surface’s flux

Question 26

when to change order of indices of LC

Answer 26

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

Question 27

Volume integral of scalar function

Answer 27

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

Question 28

Volume integral of vector field

Answer 28

Question 29

Curl of a vector field across a closed surface

Answer 29

Question 30

Curl in Einstein SC

Answer 30

Question 31

Relate LC to Kronecker delta

Answer 31