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Dear Tianyu Zhang,

regarding the 3 first words:

1. "for some studies",

2. "in this text",

3. "we introduce",

the mentioned equality is well-defined and sufficient.

Generally we can't deduce with safety & abstractly the assertion about the equality of the material & directional derivative and besides this, the mentioned "Gâteaux Operator" is the (?):

• Gâteaux Differential, with linearity & continuity conditions or

• Gâteaux Variation a weaker version of the GD, without the demand of the linearity & continuity but with the homogeneity.

Therefore:

∴1 GO ∈ {GD,GV} (logically GO = GV),

∴2 GD ⇒ GV (generally the converse doesn't hold).

The solution for this misunderstanding is:

• let a t-independent (static) tensor field T = T(x), cause of the 1-dimensionality / "scaleness(*)" of the time,

• demonstrated through the 3 below parts

(Intro., 3-Calc., 3-Comp.).

(*): There are some interesting cases which the time notion is not a scalar but inherently-anisotropically a 2-adic quantity (2nd order tensor) as a characteristic time or weight τ0 with (i,j)-mixed-variant component 0τi j for some differential operators, such as the one Lt of the Lord-Shulman model:

Lt = I + τ0∂t= or tLi j = δi j + 0τi j ∂t.

• Introduction

A. Let the index of the:=

(1) tensor order τ ∈ ℕ ⊔ {0},.s

(2) dimension δ ∈ ℕ,

the hyperreal number ε ∈ ℝ, |ε| ≪ 1,

and the product P of the non-homogenous dimension (δi ≠ δj):

P = Π∀τ ∈ ℕ(δτ) = δ1 × δ2 × ... × δτ × ... .

B. Let the differential operators:

(10) gradient or x-partial derivative: ∇ ≡ ∂x ≡ ∂/∂x,

(11) right gradient: {○}⊗∇ ≡ ∇{○},

(12) left gradient: ∇⊗{○} ≡ (∇{○})T,

(2) rate or t-partial derivative: {○}• ≡ ∂t ≡ ∂/∂t,

(31) material-total t-derivative: {○}■m ≡ dt ≡ d/dt,

(32) conservative-total t-derivative: {○}■c ≡ Dt ≡ D/Dt

(4) directional derivative (along the vector v):

∇v{○(x,t)} = ∇{○(x,t)} • v ≡

≡ d{○(x + εv,t)}/dε|ε=0 ≡ dε{○(x + εv,t)}|ε=0

(5) convective derivative: ∇{○} • v

Γ. Let the physical/geometrical entities:

(1) xt-pair (x,t) ∈ Ω × I ⊂ ℝ3 × ℝ ≅ ℝ4,

(2) arbitrary-order ℝ-tensor field T:

T: Ω × I → ℝP | (x,t) ↦ T(x,t) = T.

• 3-Calculation

Α. The material-total t-derivative {○}■m

of the arbitrary-order ℝ-tensor field T = T(x,t) is:

dT(x,t) = (∂T/∂x) • dx + (∂T/∂t) dt

∴ T ■m = ∇T • v + T • (✓).

Β. The conservative-total t-derivative {○}■c

of the arbitrary-order ℝ-tensor field T = T(x,t) is:

T ■c = T ■m + T (∇ • v) = (T ⊗ v) • ∇ + T • (✓).

Γ. The directional derivative dε{○}|ε=0

of the arbitrary-order ℝ-tensor field T = T(x,t) is:

dT(x + εv,t) = [∂T/∂(x + εv)] • d(x + εv) + (∂T/∂t) dt

dεT(x + εv,t) = [∂T/∂(x + εv)] • v + 0

∴ dεT(x + εv,t)|ε=0 = ∇T • v = ∇vT (✓).

• 3-Comparison

(1) The material- & conservative- total t-derivative {○}■m & {○}■c are equal, iff the convective term is zero:

∇ • v = 0 ⇔ divergenless/solenoidal field v {*1}.

This is also called incompressibility condition, iff v is the velocity field (e.g. in Fluid Mechanics incompressible flows).

(2) The material-total t-derivative {○}■m & directional derivative dε{○}|ε=0 are equal, iff the arbitrary ℝ-tensor field is independent of t:

T = T(x) ⇔ static field {*2}.

∴ T ■c = T ■m = ∇vT, iff {*1,2} holds. ☐