Calculus on Manifolds Tensors, guaranteed and verified 100% Pass

1


Tensors

Let 𝑉 be a vector space over ℝ, and let 𝑉 𝑚 = 𝑉 × 𝑉 × … × 𝑉 be the
𝑚-fold product. A function, 𝑇: 𝑉 𝑚 → ℝ, is called multilinear if for each
𝑖, 1 ≤ 𝑖 ≤ 𝑚, and 𝑐 ∈ ℝ we have:

1) 𝑇(𝑣1 , … , 𝑣𝑖 + 𝑣𝑖′ , … , 𝑣𝑚 ) = 𝑇(𝑣1 , … , 𝑣𝑖 , … , 𝑣𝑛 ) + 𝑇(𝑣1 , … , 𝑣𝑖′ , … 𝑣𝑚 )


2) 𝑇(𝑣1 , … , 𝑐𝑣𝑖 , … , 𝑣𝑚 ) = 𝑐𝑇 (𝑣1 , … , 𝑣𝑖 , … , 𝑣𝑚 )


Ex. The dot product of vectors is a bilinear function on ℝ𝑛 × ℝ𝑛 :

𝑇: ℝ𝑛 × ℝ𝑛 → ℝ

(𝑣1 , 𝑣2 ) → 𝑣1 ⋅ 𝑣2

𝑇(𝑣1 + 𝑣1′ , 𝑣2 ) = (𝑣1 + 𝑣1′ ) ⋅ 𝑣2 = 𝑣1 ⋅ 𝑣2 + 𝑣1′ ⋅ 𝑣2

= 𝑇(𝑣1 , 𝑣2 ) + 𝑇(𝑣1′ , 𝑣2 )


𝑇(𝑣1 , 𝑣2 + 𝑣2′ ) = (𝑣1 ) ⋅ (𝑣2 + 𝑣2′ ) = 𝑣1 ⋅ 𝑣2 + 𝑣1 ⋅ 𝑣2′

= 𝑇 (𝑣1 , 𝑣2 ) + 𝑇(𝑣1 , 𝑣2′ )


𝑇(𝑎𝑣1 , 𝑣2 ) = (𝑎𝑣1 ) ⋅ 𝑣2 = 𝑎 (𝑣1 ⋅ 𝑣2 ) = 𝑎 𝑇(𝑣1 , 𝑣2 )

𝑇(𝑣1 , 𝑎 𝑣2 ) = 𝑣1 ⋅ (𝑎𝑣2 ) = 𝑎 (𝑣1 ⋅ 𝑣2 ) = 𝑎 𝑇(𝑣1 , 𝑣2 ).

, 2


Ex. The determinant is an 𝑛-linear function. 𝑇: 𝑉 𝑛 → ℝ by:

𝑣1
𝑣2
𝑇(𝑣1 , … , 𝑣𝑛 ) = det ( ⋮ )
𝑣𝑛
Since:
𝑣1 𝑣1 𝑣1
𝑣2 𝑣2 𝑣2
⋮ ⋮ ⋮
det 𝑣 + 𝑣 ′ = det 𝑣 + det 𝑣 ′
𝑖 𝑖 𝑖 𝑖
⋮ ⋮ ⋮
( 𝑣𝑛 ) (𝑣𝑛 ) (𝑣𝑛 )
𝑣1 𝑣1
𝑣2 𝑣2
⋮ ⋮
det 𝑎 𝑣 = 𝑎 det 𝑣 .
𝑖 𝑖
⋮ ⋮
( 𝑣𝑛 ) (𝑣𝑛 )


Def. A multilinear function, 𝑇: 𝑉 𝑚 → ℝ, is called a 𝒎-tensor on 𝑉.
The set of all 𝑚-tensors, denoted 𝒯 𝑚 (𝑉), is a vector space over ℝ.

(𝑆 + 𝑇)(𝑣1 , … , 𝑣𝑚 ) = 𝑆(𝑣1 , … , 𝑣𝑚 ) + 𝑇(𝑣1 , … , 𝑣𝑚 )
(𝑎 𝑆)(𝑣1 , … , 𝑣𝑚 ) = 𝑎 (𝑆(𝑣1 , … , 𝑣𝑚 )).



If 𝑆 ∈ 𝒯 𝑚 (𝑉) and 𝑇 ∈ 𝒯 𝑙 (𝑉), then we define the tensor product,
𝑆 ⨂ 𝑇 ∈ 𝒯 𝑚+𝑙 (𝑉), by:

𝑆 ⨂ 𝑇 (𝑣1 , … , 𝑣𝑚 , 𝑣𝑚+1 , . . , 𝑣𝑚+𝑙 ) = (𝑆(𝑣1 , … , 𝑣𝑚 ))(𝑇(𝑣𝑚+1 , … , 𝑣𝑚+𝑙 )).