Tensors in 10 Minutes Or Less

Tensors In 10 Minutes Or Less

Sydney Timmerman Under the esteemed mentorship of Apurva Nakade

December 5, 2018 JHU Math Directed Reading Program Theorem If there is a relationship between two tensor ﬁelds in one coordinate system, that relationship holds in certain other coordinate systems

This means the laws of physics can be expressed as relationships between tensors!

Why tensors are cool, kids This means the laws of physics can be expressed as relationships between tensors!

Why tensors are cool, kids

Theorem If there is a relationship between two tensor ﬁelds in one coordinate system, that relationship holds in certain other coordinate systems Why tensors are cool, kids

Theorem If there is a relationship between two tensor ﬁelds in one coordinate system, that relationship holds in certain other coordinate systems

This means the laws of physics can be expressed as relationships between tensors! Example Einstein’s ﬁeld equations govern general relativity, 8πG G = T µν c4 µν 0|{z} 0|{z} (2) tensor (2) tensor

What tensors look like What tensors look like

Example Einstein’s ﬁeld equations govern general relativity, 8πG G = T µν c4 µν 0|{z} 0|{z} (2) tensor (2) tensor Let V be an n-dimensional vector space over R Deﬁnition Given u, w ∈ V and λ ∈ R, a covector is a map

α : V → R

satisfying α(u + λw) = α(u) + λα(w)

Deﬁnition Covectors form the dual space V ∗

The Dual Space Deﬁnition Given u, w ∈ V and λ ∈ R, a covector is a map

α : V → R

satisfying α(u + λw) = α(u) + λα(w)

Deﬁnition Covectors form the dual space V ∗

The Dual Space

Let V be an n-dimensional vector space over R satisfying α(u + λw) = α(u) + λα(w)

Deﬁnition Covectors form the dual space V ∗

The Dual Space

Let V be an n-dimensional vector space over R Deﬁnition Given u, w ∈ V and λ ∈ R, a covector is a map

α : V → R Deﬁnition Covectors form the dual space V ∗

The Dual Space

Let V be an n-dimensional vector space over R Deﬁnition Given u, w ∈ V and λ ∈ R, a covector is a map

α : V → R

satisfying α(u + λw) = α(u) + λα(w) The Dual Space

Let V be an n-dimensional vector space over R Deﬁnition Given u, w ∈ V and λ ∈ R, a covector is a map

α : V → R

satisfying α(u + λw) = α(u) + λα(w)

Deﬁnition Covectors form the dual space V ∗  7  u ∈ V can be represented as 11 13

there is a corresponding basis ea of V ∗

α ∈ V ∗ can be represented as 0 1 0

Naturally, α : V 7→ R

 7  0 1 0 11 = (0)7 + 1(11) + (0)13 = 11 13

Linear algebra 101

Given a basis ea of V , there is a corresponding basis ea of V ∗

α ∈ V ∗ can be represented as 0 1 0

Naturally, α : V 7→ R

 7  0 1 0 11 = (0)7 + 1(11) + (0)13 = 11 13

Linear algebra 101

Given a basis ea of V ,

 7  u ∈ V can be represented as 11 13 α ∈ V ∗ can be represented as 0 1 0

Naturally, α : V 7→ R

 7  0 1 0 11 = (0)7 + 1(11) + (0)13 = 11 13

Linear algebra 101

Given a basis ea of V ,

 7  u ∈ V can be represented as 11 13

there is a corresponding basis ea of V ∗ Naturally, α : V 7→ R

 7  0 1 0 11 = (0)7 + 1(11) + (0)13 = 11 13

Linear algebra 101

Given a basis ea of V ,

 7  u ∈ V can be represented as 11 13

there is a corresponding basis ea of V ∗

α ∈ V ∗ can be represented as 0 1 0  7  0 1 0 11 = (0)7 + 1(11) + (0)13 = 11 13

Linear algebra 101

Given a basis ea of V ,

 7  u ∈ V can be represented as 11 13

there is a corresponding basis ea of V ∗

α ∈ V ∗ can be represented as 0 1 0

Naturally, α : V 7→ R  7  11 = (0)7 + 1(11) + (0)13 = 11 13

Linear algebra 101

Given a basis ea of V ,

 7  u ∈ V can be represented as 11 13

there is a corresponding basis ea of V ∗

α ∈ V ∗ can be represented as 0 1 0

Naturally, α : V 7→ R

0 1 0 = (0)7 + 1(11) + (0)13 = 11

Linear algebra 101

Given a basis ea of V ,

 7  u ∈ V can be represented as 11 13

there is a corresponding basis ea of V ∗

α ∈ V ∗ can be represented as 0 1 0

Naturally, α : V 7→ R

 7  0 1 0 11 13 Linear algebra 101

Given a basis ea of V ,

 7  u ∈ V can be represented as 11 13

there is a corresponding basis ea of V ∗

α ∈ V ∗ can be represented as 0 1 0

Naturally, α : V 7→ R

 7  0 1 0 11 = (0)7 + 1(11) + (0)13 = 11 13 Deﬁnition A multilinear map is linear in each argument, i.e.

t(..., u + λw, ...) = t(..., u, ...) + λt(..., u, ...)

We’ve already seen multilinear maps:

Covectors, α : V 7→ R u → hα, ui ∗ Vectors, v : V 7→ R α → hα, ui (Think of scalars like a : ∅ 7→ R → a)

Multilinear maps We’ve already seen multilinear maps:

Covectors, α : V 7→ R u → hα, ui ∗ Vectors, v : V 7→ R α → hα, ui (Think of scalars like a : ∅ 7→ R → a)

Multilinear maps

Deﬁnition A multilinear map is linear in each argument, i.e.

t(..., u + λw, ...) = t(..., u, ...) + λt(..., u, ...) Covectors, α : V 7→ R u → hα, ui ∗ Vectors, v : V 7→ R α → hα, ui (Think of scalars like a : ∅ 7→ R → a)

Multilinear maps

Deﬁnition A multilinear map is linear in each argument, i.e.

t(..., u + λw, ...) = t(..., u, ...) + λt(..., u, ...)

We’ve already seen multilinear maps: Multilinear maps

Deﬁnition A multilinear map is linear in each argument, i.e.

t(..., u + λw, ...) = t(..., u, ...) + λt(..., u, ...)

We’ve already seen multilinear maps:

Covectors, α : V 7→ R u → hα, ui ∗ Vectors, v : V 7→ R α → hα, ui (Think of scalars like a : ∅ 7→ R → a) Deﬁnition p A tensor of type q is a multilinear map

∗ ∗ t : V × ... × V × V × ... × V → R | {zq } | {zp }

Example

1 ∗ a 0 tensor t : V → R is a vector 0 a 1 tensor t : V → R is a covector 0 a 0 tensor t :→ R is a scalar

Examples of tensors Example

1 ∗ a 0 tensor t : V → R is a vector 0 a 1 tensor t : V → R is a covector 0 a 0 tensor t :→ R is a scalar

Examples of tensors

Deﬁnition p A tensor of type q is a multilinear map

∗ ∗ t : V × ... × V × V × ... × V → R | {zq } | {zp } 0 a 1 tensor t : V → R is a covector 0 a 0 tensor t :→ R is a scalar

Examples of tensors

Deﬁnition p A tensor of type q is a multilinear map

∗ ∗ t : V × ... × V × V × ... × V → R | {zq } | {zp }

Example

1 ∗ a 0 tensor t : V → R is a vector 0 a 0 tensor t :→ R is a scalar

Examples of tensors

Deﬁnition p A tensor of type q is a multilinear map

∗ ∗ t : V × ... × V × V × ... × V → R | {zq } | {zp }

Example

1 ∗ a 0 tensor t : V → R is a vector 0 a 1 tensor t : V → R is a covector Examples of tensors

Deﬁnition p A tensor of type q is a multilinear map

∗ ∗ t : V × ... × V × V × ... × V → R | {zq } | {zp }

Example

1 ∗ a 0 tensor t : V → R is a vector 0 a 1 tensor t : V → R is a covector 0 a 0 tensor t :→ R is a scalar A tensor is completely determined by its values on all combinations of the basis vectors Deﬁnition These values are called components

c,...,d c d ta,...,b := t(ea, ..., eb; e , ..., e )

Components of a Tensor Deﬁnition These values are called components

c,...,d c d ta,...,b := t(ea, ..., eb; e , ..., e )

Components of a Tensor

A tensor is completely determined by its values on all combinations of the basis vectors Components of a Tensor

A tensor is completely determined by its values on all combinations of the basis vectors Deﬁnition These values are called components

c,...,d c d ta,...,b := t(ea, ..., eb; e , ..., e )   v1 1 0 tensor v2 v3

0 1 2 3 1 tensor α α α

1 1 1 1 2 t1 t2 v1 1 tensor α α 2 2 t1 t2 v2

1 2 tensor ???

Representing tensors with components   v1 v2 v3

0 1 2 3 1 tensor α α α

1 1 1 1 2 t1 t2 v1 1 tensor α α 2 2 t1 t2 v2

1 2 tensor ???