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 2 V and λ 2 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 2 V and λ 2 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 2 V and λ 2 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 2 V and λ 2 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 2 V and λ 2 R, a covector is a map α : V ! R satisfying α(u + λw) = α(u) + λα(w) Definition Covectors form the dual space V ∗ 0 7 1 u 2 V can be represented as @11A 13 there is a corresponding basis ea of V ∗ α 2 V ∗ can be represented as 0 1 0 Naturally, α : V 7! R 0 7 1 0 1 0 @11A = (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 2 V ∗ can be represented as 0 1 0 Naturally, α : V 7! R 0 7 1 0 1 0 @11A = (0)7 + 1(11) + (0)13 = 11 13 Linear algebra 101 Given a basis ea of V , 0 7 1 u 2 V can be represented as @11A 13 α 2 V ∗ can be represented as 0 1 0 Naturally, α : V 7! R 0 7 1 0 1 0 @11A = (0)7 + 1(11) + (0)13 = 11 13 Linear algebra 101 Given a basis ea of V , 0 7 1 u 2 V can be represented as @11A 13 there is a corresponding basis ea of V ∗ Naturally, α : V 7! R 0 7 1 0 1 0 @11A = (0)7 + 1(11) + (0)13 = 11 13 Linear algebra 101 Given a basis ea of V , 0 7 1 u 2 V can be represented as @11A 13 there is a corresponding basis ea of V ∗ α 2 V ∗ can be represented as 0 1 0 0 7 1 0 1 0 @11A = (0)7 + 1(11) + (0)13 = 11 13 Linear algebra 101 Given a basis ea of V , 0 7 1 u 2 V can be represented as @11A 13 there is a corresponding basis ea of V ∗ α 2 V ∗ can be represented as 0 1 0 Naturally, α : V 7! R 0 7 1 @11A = (0)7 + 1(11) + (0)13 = 11 13 Linear algebra 101 Given a basis ea of V , 0 7 1 u 2 V can be represented as @11A 13 there is a corresponding basis ea of V ∗ α 2 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 , 0 7 1 u 2 V can be represented as @11A 13 there is a corresponding basis ea of V ∗ α 2 V ∗ can be represented as 0 1 0 Naturally, α : V 7! R 0 7 1 0 1 0 @11A 13 Linear algebra 101 Given a basis ea of V , 0 7 1 u 2 V can be represented as @11A 13 there is a corresponding basis ea of V ∗ α 2 V ∗ can be represented as 0 1 0 Naturally, α : V 7! R 0 7 1 0 1 0 @11A = (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 ) 0 1 v1 1 0 tensor @v2A 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 0 1 v1 @v2A 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 1 0 tensor 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 0 1 v1 1 0 tensor @v2A v3 α1 α2 α3 1 1 1 1 2 t1 t2 v1 1 tensor α α 2 2 t1 t2 v2 1 2 tensor ??? Representing tensors with components 0 1 v1 1 0 tensor @v2A v3 0 1 tensor 1 1 1 1 2 t1 t2 v1 1 tensor α α 2 2 t1 t2 v2 1 2 tensor ??? Representing tensors with components 0 1 v1 1 0 tensor @v2A v3 0 1 2 3 1 tensor α α α 1 1 1 2 t1 t2 v1 α α 2 2 t1 t2 v2 1 2 tensor ??? Representing tensors with components 0 1 v1 1 0 tensor @v2A v3 0 1 2 3 1 tensor α α α 1 1 tensor 1 2 tensor ??? Representing tensors with components 0 1 v1 1 0 tensor @v2A 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 ??? Representing tensors with components 0 1 v1 1 0 tensor @v2A 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 0 1 v1 1 0 tensor @v2A 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 ??? Definition p p A tensor field of type q assigns a tensor of type q to every point in a manifold (more general kind of mathematical space than Euclidean space)1 Tensor fields 1value of tensor field must vary smoothly from point to point Tensor fields Definition p p A tensor field of type q assigns a tensor of type q to every point in a manifold (more general kind of mathematical space than Euclidean space)1 1value of tensor field must vary smoothly from point to point Tensor fields Definition p p A tensor field of type q assigns a tensor of type q to every point in a manifold (more general kind of mathematical space than Euclidean space)1 1value of tensor field must vary smoothly from point to point Tensor fields Definition p p A tensor field of type q assigns a tensor of type q to every point in a manifold (more general kind of mathematical space than Euclidean space)1 1value of tensor field must vary smoothly from point to point 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