Week 1 Flashcards

1
Q

Important rules of Einstein notation

A

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

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2
Q
A

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

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3
Q

Einstein for cross prod

A

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

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4
Q

Changing index of vector w Kronecker delta

A

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

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5
Q

Each extra letter in index notation refers to

A

An extra dimension eg

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6
Q

Kronecker delta in index notation

A

Plays role of identity matrix eg:

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7
Q

Euclidean metric tensor

A

All entries are zero except below and same but for 3

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8
Q
A

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

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9
Q

Put this in Einstein notation

A
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10
Q

Using Euclidean metric tensor (end inverse)

A

Dot prod is equal for transpose

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11
Q

Use anti symmetric property of Levi cevitica

A
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12
Q

Electrostatics

A

The forces due to a static array of electric charges

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13
Q

Magnetostatics

A

The forces due to a static magnetic field

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14
Q
A

Where θ is the angle between gradF and dl

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15
Q

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

A
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16
Q

Curl product rules

A
17
Q
A
18
Q

Laplacian of a vector

A
19
Q

What is a line integral?

A

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

Aka path integral

20
Q

Line integral of scalar function

A
21
Q

Line integral of vector field

A
22
Q

dl = ?

A

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

23
Q

Surface integral?

A
24
Q

Flux of v through S

A

Or equivalent for a closed surface

25
Q

Summing flux of sub surfaces within a surface?

A

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

26
Q

when to change order of indices of LC

A

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

27
Q

Volume integral of scalar function

A

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

28
Q

Volume integral of vector field

A
29
Q

Curl of a vector field across a closed surface

A
30
Q

Curl in Einstein SC

A
31
Q

Relate LC to Kronecker delta

A