1 4. Differential Operations with Vectors, Tensors Del Operator

Home , Del, Differential operator, Einstein notation, Gradient, Vector operator

Mathematics Review Polymer Rheology

4. Differential Operations with Vectors, Tensors Scalars, vectors, and tensors are differentiated to determine rates of change (with respect to time, position) •To carryout the differentiation with respect to a single variable, differentiate each coefficient individually. •There is no change in order (vectors remain vectors, scalars remain scalars, etc.

≈ ∂w ’ ≈ ∂B ∂B ∂B ’ ∆ 1 ÷ ∆ 11 21 31 ÷ ∆ ∂t ÷ ∆ ∂t ∂t ∂t ÷ ∂w ∂w ∂B ∂B ∂B ∂B ∂α = ∆ 2 ÷ = ∆ 21 22 23 ÷ ∂t ∂t ∆ ∂t ÷ ∂t ∆ ∂t ∂t ∂t ÷ ∆ ∂w ÷ ∆ ∂B ∂B ∂B ÷ ∆ 3 ÷ ∆ 31 32 33 ÷ « ∂t ◊ « ∂t ∂t ∂t ◊ 123 123

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4. Differential Operations with Vectors, Tensors (continued)

•To carryout the differentiation with respect to 3D spatial variation, use the del (nabla) Del Operator operator. •This is a vector operator •Del may be applied in three different ways ≈ ∂ ’ •Del may operate on scalars, vectors, or tensors ∆ ÷ ∆ ∂x1 ÷ ∂ ∂ ∂ ∆ ∂ ÷ This is written in ∇ ≡ eˆ1 + eˆ2 + eˆ3 = ∆ ÷ Cartesian ∂x1 ∂x2 ∂x3 ∆ ∂x2 ÷ coordinates ∆ ∂ ÷ ∆ ÷ « ∂x3 ◊ 123 3 ∂ ∂ = ƒeˆp = eˆp p=1 ∂xp ∂xp

Einstein notation for del

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4. Differential Operations with Vectors, Tensors (continued)

A. Scalars - gradient ≈ ’ This is written in ∆ ∂β ÷ Cartesian Gibbs ∆ ∂x1 ÷ coordinates notation ∂ ∂ ∂ ∆ ∂β ÷ ∇β ≡ eˆ1 β + eˆ2 β + eˆ3 β = ∆ ÷ ∂x1 ∂x2 ∂x3 ∆ ∂x2 ÷ ∆ ∂β ÷ ∆ ÷ « ∂x3 ◊ 123 ∂β Gradient of a = eˆp scalar field ∂xp The gradient of a scalar field is a The gradient operation vector captures the total spatial variation of a scalar, •gradient operation increases the order of the vector, or tensor field. entity operated upon

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4. Differential Operations with Vectors, Tensors (continued) B. Vectors - gradient ∂ ∂ ∂ ∇w ≡ eˆ1 w + eˆ2 w + eˆ3 w This is all written ∂x1 ∂x2 ∂x3 in Cartesian coordinates (basis The basis vectors ∂ = eˆ1 (w1eˆ1 + w2eˆ2 + w3eˆ3 ) vectors are can move out of ∂x 1 constant) the derivatives ∂ because they are + eˆ2 (w1eˆ1 + w2eˆ2 + w3eˆ3 ) constant (do not ∂x2 change with ∂ position) + eˆ3 (w1eˆ1 + w2eˆ2 + w3eˆ3 ) ∂x3

∂w1 ∂w2 ∂w3 ∂w1 = eˆ1eˆ1 + eˆ1eˆ2 + eˆ1eˆ3 + eˆ2eˆ1 + ∂x1 ∂x1 ∂x1 ∂x2

∂w2 ∂w3 ∂w1 ∂w2 ∂w3 eˆ2eˆ2 + eˆ2eˆ3 + eˆ3eˆ1 + eˆ3eˆ2 + eˆ3eˆ3 ∂x2 ∂x2 ∂x3 ∂x3 ∂x3

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4. Differential Operations with Vectors, Tensors (continued)

B. Vectors - gradient (continued) constants may appear on either side of the Gradient of a differential operator vector field 3 3 ∂wk ∂wk ∂wk ∇w ≡ ƒƒeˆjeˆk = eˆjeˆk = eˆjeˆk j=1k=1 ∂xj ∂xj ∂xj

Einstein notation The gradient of for gradient of a a vector field is vector a tensor

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4. Differential Operations with Vectors, Tensors (continued)

C. Vectors - divergence

≈ ∂ ∂ ∂ ’ Divergence of a ∆ ÷ vector field ∇ ⋅ w ≡ eˆ1 + eˆ2 + eˆ3 ⋅ w1eˆ1 + w2eˆ2 + w3eˆ3 « ∂x1 ∂x2 ∂x3◊ ∂w ∂w ∂w = 1 + 2 + 3 ∂x1 ∂x2 ∂x3 The Divergence 3 ∂wi ∂wi of a vector field = ƒ = is a scalar i=1 ∂xi ∂xi

Einstein notation for gradient of a vector

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4. Differential Operations with Vectors, Tensors (continued)

C. Vectors - divergence (continued) This is all written in Cartesian constants may appear coordinates (basis on either side of the vectors are differential operator constant) Using Einstein notation

∂ ∂wj ∂wj ∇ ⋅ w ≡ eˆm ⋅ wjeˆj = eˆm ⋅eˆj = δmj ∂xm ∂xm ∂xm ∂w = j ∂xj

•divergence operation decreases the order of the entity operated upon

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4. Differential Operations with Vectors, Tensors (continued) D. Vectors - Laplacian

Using ∂ ∂ ∂ ∂ ∇ ⋅∇w ≡ eˆ ⋅eˆ w eˆ = w eˆ ⋅eˆ eˆ Einstein m p j j j ( m p) j ∂xm ∂xp ∂xm ∂xp notation: ∂ ∂ = wj (δmp)eˆj ∂xm ∂xp ∂ ∂ The Laplacian = wj eˆj of a vector field ∂xp ∂xp is a vector ≈ ∂2w ∂2w ∂2w ’ ∆ 1 + 1 + 1 ÷ ∆ ∂x1 ∂x2 ∂x3 ÷ ∆∂2w ∂2w ∂2w ÷ •Laplacian operation does = ∆ 2 + 2 + 2 ÷ not change the order of the ∆ ∂x1 ∂x2 ∂x3 ÷ entity operated upon ∆ ∂2w ∂2w ∂2w ÷ ∆ 3 + 3 + 3 ÷ « ∂x1 ∂x2 ∂x3 ◊ 123 © Faith A. Morrison, Michigan Tech U.

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4. Differential Operations with Vectors, Tensors (continued)

(impossible; cannot E. Scalar - divergence ∇ ⋅α decrease order of a scalar) F. Scalar - Laplacian ∇ ⋅∇α

G. Tensor - gradient ∇A

H. Tensor - divergence ∇ ⋅ A

I. Tensor - Laplacian ∇⋅∇A