Tensorderivat (kontinuummekanik) - Tensor derivative (continuum mechanics)

De derivater af skalarer , vektorer og anden ordens tensorer med hensyn til anden ordens tensorer er af betydelig anvendelse i kontinuum mekanik . Disse derivater bruges i teorierne om ikke-lineær elasticitet og plasticitet , især i designet af algoritmer til numeriske simuleringer .

Den retningsbestemte derivat giver en systematisk måde at finde disse derivater.

Derivater med hensyn til vektorer og andenordens tensorer

Definitionerne af retningsderivater til forskellige situationer er angivet nedenfor. Det antages, at funktionerne er tilstrækkeligt glatte til, at derivater kan tages.

Derivater af skalarværdige funktioner for vektorer

Lad f ( v ) være en reel værdi af vektoren v . Derefter den afledte af f ( v ) i forhold til v (eller i v ) er den vektor defineres gennem dets dot produkt med enhver vektor u er

${\ displaystyle {\ frac {\ partial f} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} = Df (\ mathbf {v}) [\ mathbf {u}] = \ left [{\ frac {\ rm {d}} {{\ rm {d}} \ alpha}} ~ f (\ mathbf {v} + \ alpha ~ \ mathbf {u}) \ højre] _ {\ alpha = 0}}$

for alle vektorer u . Ovenstående punktprodukt giver en skalar, og hvis u er en enhedsvektor giver retningsderivatet af f ved v , i u- retningen.

Ejendomme:

1. Hvis så${\ displaystyle f (\ mathbf {v}) = f_ {1} (\ mathbf {v}) + f_ {2} (\ mathbf {v})}$${\ displaystyle {\ frac {\ partial f} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} = \ left ({\ frac {\ partial f_ {1}} {\ partial \ mathbf {v }}} + {\ frac {\ partial f_ {2}} {\ partial \ mathbf {v}}} \ right) \ cdot \ mathbf {u}}$
2. Hvis så${\ displaystyle f (\ mathbf {v}) = f_ {1} (\ mathbf {v}) ~ f_ {2} (\ mathbf {v})}$${\ displaystyle {\ frac {\ partial f} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} = \ left ({\ frac {\ partial f_ {1}} {\ partial \ mathbf {v }}} \ cdot \ mathbf {u} \ right) ~ f_ {2} (\ mathbf {v}) + f_ {1} (\ mathbf {v}) ~ \ left ({\ frac {\ partial f_ {2 }} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} \ right)}$
3. Hvis så${\ displaystyle f (\ mathbf {v}) = f_ {1} (f_ {2} (\ mathbf {v}))}$${\ displaystyle {\ frac {\ partial f} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} = {\ frac {\ partial f_ {1}} {\ partial f_ {2}}} ~ {\ frac {\ partial f_ {2}} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u}}$

Derivater af vektorværdiansatte funktioner af vektorer

Lad f ( v ) være en vektor-værdsat funktion af vektoren v . Derefter den afledte af f ( v ) i forhold til v (eller i v ) er den anden ordens tensor defineres gennem dets dot produkt med enhver vektor u er

${\ displaystyle {\ frac {\ partial \ mathbf {f}} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} = D \ mathbf {f} (\ mathbf {v}) [\ mathbf { u}] = \ venstre [{\ frac {\ rm {d}} {{\ rm {d}} \ alpha}} ~ \ mathbf {f} (\ mathbf {v} + \ alpha ~ \ mathbf {u} ) \ højre] _ {\ alpha = 0}}$

for alle vektorer u . Ovenstående punktprodukt giver en vektor, og hvis u er en enhedsvektor, giver retningsderivatet af f ved v , i retning u .

Ejendomme:

1. Hvis så${\ displaystyle \ mathbf {f} (\ mathbf {v}) = \ mathbf {f} _ {1} (\ mathbf {v}) + \ mathbf {f} _ {2} (\ mathbf {v})}$${\ displaystyle {\ frac {\ partial \ mathbf {f}} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} = \ left ({\ frac {\ partial \ mathbf {f} _ {1 }} {\ partial \ mathbf {v}}} + {\ frac {\ partial \ mathbf {f} _ {2}} {\ partial \ mathbf {v}}} \ right) \ cdot \ mathbf {u}}$
2. Hvis så${\ displaystyle \ mathbf {f} (\ mathbf {v}) = \ mathbf {f} _ {1} (\ mathbf {v}) \ times \ mathbf {f} _ {2} (\ mathbf {v}) }$${\ displaystyle {\ frac {\ partial \ mathbf {f}} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} = \ left ({\ frac {\ partial \ mathbf {f} _ {1 }} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} \ right) \ times \ mathbf {f} _ {2} (\ mathbf {v}) + \ mathbf {f} _ {1} (\ mathbf {v}) \ times \ left ({\ frac {\ partial \ mathbf {f} _ {2}} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} \ right)}$
3. Hvis så${\ displaystyle \ mathbf {f} (\ mathbf {v}) = \ mathbf {f} _ {1} (\ mathbf {f} _ {2} (\ mathbf {v}))}$${\ displaystyle {\ frac {\ partial \ mathbf {f}} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u} = {\ frac {\ partial \ mathbf {f} _ {1}} { \ partial \ mathbf {f} _ {2}}} \ cdot \ left ({\ frac {\ partial \ mathbf {f} _ {2}} {\ partial \ mathbf {v}}} \ cdot \ mathbf {u } \ret)}$

Afledte af skalarvurderede funktioner af andenordens tensorer

Lad være en reel værdsat funktion af anden ordens tensor . Derefter differentialkvotienten af med hensyn til (eller i ) i retningen er den anden ordens tensor defineret som ${\ displaystyle f ({\ boldsymbol {S}})}$${\ displaystyle {\ boldsymbol {S}}}$${\ displaystyle f ({\ boldsymbol {S}})}$${\ displaystyle {\ boldsymbol {S}}}$${\ displaystyle {\ boldsymbol {S}}}$${\ displaystyle {\ boldsymbol {T}}}$

${\ displaystyle {\ frac {\ partial f} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = Df ({\ boldsymbol {S}}) [{\ boldsymbol {T}} ] = \ venstre [{\ frac {\ rm {d}} {{\ rm {d}} \ alpha}} ~ f ({\ boldsymbol {S}} + \ alpha ~ {\ boldsymbol {T}}) \ højre] _ {\ alpha = 0}}$

for alle andenordens tensorer . ${\ displaystyle {\ boldsymbol {T}}}$

Ejendomme:

1. Hvis så${\ displaystyle f ({\ boldsymbol {S}}) = f_ {1} ({\ boldsymbol {S}}) + f_ {2} ({\ boldsymbol {S}})}$${\ displaystyle {\ frac {\ partial f} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = \ left ({\ frac {\ partial f_ {1}} {\ partial { \ boldsymbol {S}}}} + {\ frac {\ partial f_ {2}} {\ partial {\ boldsymbol {S}}}} \ right): {\ boldsymbol {T}}}$
2. Hvis så${\ displaystyle f ({\ boldsymbol {S}}) = f_ {1} ({\ boldsymbol {S}}) ~ f_ {2} ({\ boldsymbol {S}})}$${\ displaystyle {\ frac {\ partial f} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = \ left ({\ frac {\ partial f_ {1}} {\ partial { \ boldsymbol {S}}}}: {\ boldsymbol {T}} \ højre) ~ f_ {2} ({\ boldsymbol {S}}) + f_ {1} ({\ boldsymbol {S}}) ~ \ venstre ({\ frac {\ partial f_ {2}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} \ right)}$
3. Hvis så${\ displaystyle f ({\ boldsymbol {S}}) = f_ {1} (f_ {2} ({\ boldsymbol {S}}))}$${\ displaystyle {\ frac {\ partial f} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = {\ frac {\ partial f_ {1}} {\ partial f_ {2} }} ~ \ left ({\ frac {\ partial f_ {2}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} \ right)}$

Afledte af tensorværdige funktioner af andenordens tensorer

Lad være en anden ordens tensorværdieret funktion af den anden ordens tensor . Derefter differentialkvotienten af med hensyn til (eller i ) i retningen er den fjerde orden tensor defineret som ${\ displaystyle {\ boldsymbol {F}} ({\ boldsymbol {S}})}$${\ displaystyle {\ boldsymbol {S}}}$${\ displaystyle {\ boldsymbol {F}} ({\ boldsymbol {S}})}$${\ displaystyle {\ boldsymbol {S}}}$${\ displaystyle {\ boldsymbol {S}}}$${\ displaystyle {\ boldsymbol {T}}}$

${\ displaystyle {\ frac {\ partial {\ boldsymbol {F}}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = D {\ boldsymbol {F}} ({\ boldsymbol {S}}) [{\ boldsymbol {T}}] = \ venstre [{\ frac {\ rm {d}} {{\ rm {d}} \ alpha}} ~ {\ boldsymbol {F}} ({ \ boldsymbol {S}} + \ alpha ~ {\ boldsymbol {T}}) \ højre] _ {\ alpha = 0}}$

for alle andenordens tensorer . ${\ displaystyle {\ boldsymbol {T}}}$

Ejendomme:

1. Hvis så${\ displaystyle {\ boldsymbol {F}} ({\ boldsymbol {S}}) = {\ boldsymbol {F}} _ {1} ({\ boldsymbol {S}}) + {\ boldsymbol {F}} _ { 2} ({\ boldsymbol {S}})}$${\ displaystyle {\ frac {\ partial {\ boldsymbol {F}}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = \ left ({\ frac {\ partial {\ boldsymbol {F}} _ {1}} {\ partial {\ boldsymbol {S}}}} + {\ frac {\ partial {\ boldsymbol {F}} _ {2}} {\ partial {\ boldsymbol {S}} }} \ højre): {\ boldsymbol {T}}}$
2. Hvis så${\ displaystyle {\ boldsymbol {F}} ({\ boldsymbol {S}}) = {\ boldsymbol {F}} _ {1} ({\ boldsymbol {S}}) \ cdot {\ boldsymbol {F}} _ {2} ({\ boldsymbol {S}})}$${\ displaystyle {\ frac {\ partial {\ boldsymbol {F}}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = \ left ({\ frac {\ partial {\ boldsymbol {F}} _ {1}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} \ right) \ cdot {\ boldsymbol {F}} _ {2} ({\ boldsymbol { S}}) + {\ boldsymbol {F}} _ {1} ({\ boldsymbol {S}}) \ cdot \ left ({\ frac {\ partial {\ boldsymbol {F}} _ {2}} {\ delvis {\ boldsymbol {S}}}}: {\ boldsymbol {T}} \ højre)}$
3. Hvis så${\ displaystyle {\ boldsymbol {F}} ({\ boldsymbol {S}}) = {\ boldsymbol {F}} _ {1} ({\ boldsymbol {F}} _ {2} ({\ boldsymbol {S} }))}$${\ displaystyle {\ frac {\ partial {\ boldsymbol {F}}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = {\ frac {\ partial {\ boldsymbol {F} } _ {1}} {\ partial {\ boldsymbol {F}} _ {2}}}: \ left ({\ frac {\ partial {\ boldsymbol {F}} _ {2}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} \ højre)}$
4. Hvis så${\ displaystyle f ({\ boldsymbol {S}}) = f_ {1} ({\ boldsymbol {F}} _ {2} ({\ boldsymbol {S}}))}$${\ displaystyle {\ frac {\ partial f} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} = {\ frac {\ partial f_ {1}} {\ partial {\ boldsymbol { F}} _ {2}}}: \ left ({\ frac {\ partial {\ boldsymbol {F}} _ {2}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T} }\ret)}$

Gradient af et tensorfelt

Den gradient , på en tensor felt i retning af en arbitrær konstant vektor c defineres som: ${\ displaystyle {\ boldsymbol {\ nabla}} {\ boldsymbol {T}}}$${\ displaystyle {\ boldsymbol {T}} (\ mathbf {x})}$

${\ displaystyle {\ boldsymbol {\ nabla}} {\ boldsymbol {T}} \ cdot \ mathbf {c} = \ lim _ {\ alpha \ rightarrow 0} \ quad {\ cfrac {d} {d \ alpha}} ~ {\ boldsymbol {T}} (\ mathbf {x} + \ alpha \ mathbf {c})}$

Gradienten af ​​et tensorfelt af orden n er et tensorfelt af orden n +1.

Kartesiske koordinater

Bemærk: Einstein-summeringskonventionen om opsummering på gentagne indekser bruges nedenfor.

Hvis er basisvektorerne i et kartesisk koordinatsystem med koordinater for punkter angivet med ( ), så gives gradienten af ​​tensorfeltet ved ${\ displaystyle \ mathbf {e} _ {1}, \ mathbf {e} _ {2}, \ mathbf {e} _ {3}}$${\ displaystyle x_ {1}, x_ {2}, x_ {3}}$${\ displaystyle {\ boldsymbol {T}}}$

${\ displaystyle {\ boldsymbol {\ nabla}} {\ boldsymbol {T}} = {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial x_ {i}}} \ otimes \ mathbf {e} _ {jeg}}$

Bevis

Vektorerne x og c kan skrives som og . Lad y  : = x + α c . I så fald er gradienten givet ved ${\ displaystyle \ mathbf {x} = x_ {i} ~ \ mathbf {e} _ {i}}$${\ displaystyle \ mathbf {c} = c_ {i} ~ \ mathbf {e} _ {i}}$ ${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} {\ boldsymbol {T}} \ cdot \ mathbf {c} & = \ left. {\ cfrac {\ rm {d}} {{\ rm {d}} \ alpha}} ~ {\ boldsymbol {T}} (x_ {1} + \ alpha c_ {1}, x_ {2} + \ alpha c_ {2}, x_ {3} + \ alpha c_ { 3}) \ højre | _ {\ alpha = 0} \ equiv \ venstre. {\ Cfrac {\ rm {d}} {{\ rm {d}} \ alpha}} ~ {\ boldsymbol {T}} (y_ {1}, y_ {2}, y_ {3}) \ right | _ {\ alpha = 0} \\ & = \ left [{\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial y_ { 1}}} ~ {\ cfrac {\ partial y_ {1}} {\ partial \ alpha}} + {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial y_ {2}}} ~ {\ cfrac {\ partial y_ {2}} {\ partial \ alpha}} + {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial y_ {3}}} ~ {\ cfrac {\ partial y_ {3 }} {\ partial \ alpha}} \ right] _ {\ alpha = 0} = \ left [{\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial y_ {1}}} ~ c_ {1 } + {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial y_ {2}}} ~ c_ {2} + {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial y_ { 3}}} ~ c_ {3} \ right] _ {\ alpha = 0} \\ & = {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial x_ {1}}} ~ c_ {1 } + {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial x_ {2}}} ~ c_ {2} + {\ cfrac {\ partia l {\ boldsymbol {T}}} {\ partial x_ {3}}} ~ c_ {3} \ equiv {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial x_ {i}}} ~ c_ {i} = {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial x_ {i}}} ~ (\ mathbf {e} _ {i} \ cdot \ mathbf {c}) = \ left [ {\ cfrac {\ partial {\ boldsymbol {T}}} {\ partial x_ {i}}} \ otimes \ mathbf {e} _ {i} \ right] \ cdot \ mathbf {c} \ qquad \ square \ end {align}}}$

Da basisvektorerne ikke varierer i et kartesisk koordinatsystem, har vi følgende relationer for gradienterne i et skalarfelt , et vektorfelt v og et andet ordens tensorfelt . ${\ displaystyle \ phi}$${\ displaystyle {\ boldsymbol {S}}}$

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ phi & = {\ cfrac {\ partial \ phi} {\ partial x_ {i}}} ~ \ mathbf {e} _ {i} = \ phi _ {, i} ~ \ mathbf {e} _ {i} \\ {\ boldsymbol {\ nabla}} \ mathbf {v} & = {\ cfrac {\ partial (v_ {j} \ mathbf {e} _ {j})} {\ partial x_ {i}}} \ otimes \ mathbf {e} _ {i} = {\ cfrac {\ partial v_ {j}} {\ partial x_ {i}}} ~ \ mathbf {e} _ {j} \ otimes \ mathbf {e} _ {i} = v_ {j, i} ~ \ mathbf {e} _ {j} \ otimes \ mathbf {e} _ {i} \\ {\ fed symbol {\ nabla}} {\ boldsymbol {S}} & = {\ cfrac {\ partial (S_ {jk} \ mathbf {e} _ {j} \ otimes \ mathbf {e} _ {k})} {\ delvis x_ {i}}} \ otimes \ mathbf {e} _ {i} = {\ cfrac {\ partial S_ {jk}} {\ partial x_ {i}}} ~ \ mathbf {e} _ {j} \ otimes \ mathbf {e} _ {k} \ otimes \ mathbf {e} _ {i} = S_ {jk, i} ~ \ mathbf {e} _ {j} \ otimes \ mathbf {e} _ {k} \ otimes \ mathbf {e} _ {i} \ end {justeret}}}$

Kurvlinære koordinater

Bemærk: Einstein-summeringskonventionen om opsummering på gentagne indekser bruges nedenfor.

Hvis er de kontravariant basisvektorer i et krumlinjært koordinatsystem med koordinater af punkter angivet med ( ), så er gradienten af ​​tensorfeltet givet ved (se et bevis.) ${\ displaystyle \ mathbf {g} ^ {1}, \ mathbf {g} ^ {2}, \ mathbf {g} ^ {3}}$ ${\ displaystyle \ xi ^ {1}, \ xi ^ {2}, \ xi ^ {3}}$${\ displaystyle {\ boldsymbol {T}}}$

${\ displaystyle {\ boldsymbol {\ nabla}} {\ boldsymbol {T}} = {\ frac {\ partial {\ boldsymbol {T}}} {\ partial \ xi ^ {i}}} \ otimes \ mathbf {g } ^ {i}}$

Fra denne definition har vi følgende relationer til gradienterne i et skalarfelt , et vektorfelt v og et andet ordens tensorfelt . ${\ displaystyle \ phi}$${\ displaystyle {\ boldsymbol {S}}}$

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ phi & = {\ frac {\ partial \ phi} {\ partial \ xi ^ {i}}} ~ \ mathbf {g} ^ {i } \\ {\ boldsymbol {\ nabla}} \ mathbf {v} & = {\ frac {\ partial \ left (v ^ {j} \ mathbf {g} _ {j} \ right)} {\ partial \ xi ^ {i}}} \ otimes \ mathbf {g} ^ {i} = \ left ({\ frac {\ partial v ^ {j}} {\ partial \ xi ^ {i}}} + v ^ {k} ~ \ Gamma _ {ik} ^ {j} \ højre) ~ \ mathbf {g} _ {j} \ otimes \ mathbf {g} ^ {i} = \ left ({\ frac {\ partial v_ {j}} {\ partial \ xi ^ {i}}} - v_ {k} ~ \ Gamma _ {ij} ^ {k} \ right) ~ \ mathbf {g} ^ {j} \ otimes \ mathbf {g} ^ {i } \\ {\ boldsymbol {\ nabla}} {\ boldsymbol {S}} & = {\ frac {\ partial \ left (S_ {jk} ~ \ mathbf {g} ^ {j} \ otimes \ mathbf {g} ^ {k} \ højre)} {\ partial \ xi ^ {i}}} \ otimes \ mathbf {g} ^ {i} = \ left ({\ frac {\ partial S_ {jk}} {\ partial \ xi _ {i}}} - S_ {lk} ~ \ Gamma _ {ij} ^ {l} -S_ {jl} ~ \ Gamma _ {ik} ^ {l} \ højre) ~ \ mathbf {g} ^ {j } \ otimes \ mathbf {g} ^ {k} \ otimes \ mathbf {g} ^ {i} \ end {justeret}}}$

hvor Christoffel-symbolet er defineret ved hjælp af ${\ displaystyle \ Gamma _ {ij} ^ {k}}$

${\ displaystyle \ Gamma _ {ij} ^ {k} ~ \ mathbf {g} _ {k} = {\ frac {\ partial \ mathbf {g} _ {i}} {\ partial \ xi ^ {j}} } \ quad \ antyder \ quad \ Gamma _ {ij} ^ {k} = {\ frac {\ partial \ mathbf {g} _ {i}} {\ partial \ xi ^ {j}}} \ cdot \ mathbf { g} ^ {k} = - \ mathbf {g} _ {i} \ cdot {\ frac {\ partial \ mathbf {g} ^ {k}} {\ partial \ xi ^ {j}}}}$

Cylindriske polære koordinater

I cylindriske koordinater er gradienten givet ved

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ phi = {} \ quad & {\ frac {\ partial \ phi} {\ partial r}} ~ \ mathbf {e} _ {r} + {\ frac {1} {r}} ~ {\ frac {\ partial \ phi} {\ partial \ theta}} ~ \ mathbf {e} _ {\ theta} + {\ frac {\ partial \ phi} { \ partial z}} ~ \ mathbf {e} _ {z} \\ {\ boldsymbol {\ nabla}} \ mathbf {v} = {} \ quad & {\ frac {\ partial v_ {r}} {\ partial r}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial v _ {\ theta}} {\ partial r}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {\ theta} + {\ frac {\ partial v_ {z}} {\ partial r}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {z} \\ {} + {} og {\ frac {1} {r}} \ venstre ({\ frac {\ partial v_ {r}} {\ partial \ theta}} - v _ {\ theta} \ til højre) ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {r} + {\ frac {1} {r}} \ left ({\ frac {\ partial v _ {\ theta}} {\ partial \ theta}} + v_ {r} \ right) ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {\ theta} + {\ frac {1} {r}} { \ frac {\ partial v_ {z}} {\ partial \ theta}} ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {z} \\ {} + {} & {\ frac {\ partial v_ {r}} {\ partial z}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial v _ {\ theta}} {\ partial z}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {\ theta} + { \ frac {\ partial v_ {z}} {\ partial z}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {z} \\ {\ boldsymbol {\ nabla}} {\ boldsymbol {S}} = {} \ quad & {\ frac {\ partial S_ {rr}} {\ partial r}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial S_ {rr}} {\ partial z}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {z} + {\ frac {1} {r}} \ venstre [{\ frac {\ partial S_ {rr}} {\ partial \ theta}} - (S _ {\ theta r} + S_ {r \ theta}) \ højre] ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {\ theta} \\ {} + {} & {\ frac {\ partial S_ {r \ theta}} {\ partial r}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial S_ {r \ theta}} {\ partial z}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf { e} _ {z} + {\ frac {1} {r}} \ venstre [{\ frac {\ partial S_ {r \ theta}} {\ partial \ theta}} + (S_ {rr} -S _ {\ theta \ theta}) \ right] ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {\ theta} \\ {} + {} & {\ frac {\ partial S_ {rz}} {\ partial r}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf { e} _ {z} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial S_ {rz}} {\ partial z}} ~ \ mathbf {e} _ {r} \ otimes \ mathbf { e} _ {z} \ otimes \ mathbf {e} _ {z} + {\ frac {1} {r}} \ left [{\ frac {\ partial S_ {rz}} {\ partial \ theta}} - S _ {\ theta z} \ right] ~ \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {\ theta} \\ {} + {} & {\ frac {\ partial S _ {\ theta r}} {\ partial r}} ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ { r} + {\ frac {\ partial S _ {\ theta r}} {\ partial z}} ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {r} \ otimes \ mathbf {e } _ {z} + {\ frac {1} {r}} \ venstre [{\ frac {\ partial S _ {\ theta r}} {\ partial \ theta}} + (S_ {rr} -S _ {\ theta \ theta}) \ højre] ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {r} \ otimes \ mathbf {e} _ {\ theta} \\ {} + {} & { \ frac {\ partial S _ {\ theta \ theta}} {\ partial r}} ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial S _ {\ theta \ theta}} {\ partial z}} ~ \ matematik bf {e} _ {\ theta} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {z} + {\ frac {1} {r}} \ left [{\ frac { \ partial S _ {\ theta \ theta}} {\ partial \ theta}} + (S_ {r \ theta} + S _ {\ theta r}) \ right] ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {\ theta} \\ {} + {} & {\ frac {\ partial S _ {\ theta z}} {\ partial r}} ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial S _ {\ theta z}} {\ partial z} } ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {z} + {\ frac {1} {r}} \ left [{\ frac {\ partial S _ {\ theta z}} {\ partial \ theta}} + S_ {rz} \ right] ~ \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {\ theta} \\ {} + {} & {\ frac {\ partial S_ {zr}} {\ partial r}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf { e} _ {r} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial S_ {zr}} {\ partial z}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf { e} _ {r} \ otimes \ mathbf {e} _ {z} + {\ frac {1} {r}} \ left [{\ frac {\ partial S_ {zr}} {\ partial \ theta}} - S_ {z \ theta} \ right] ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {r } \ otimes \ mathbf {e} _ {\ theta} \\ {} + {} & {\ frac {\ partial S_ {z \ theta}} {\ partial r}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial S_ {z \ theta}} {\ partial z}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {z} + {\ frac {1} {r}} \ left [{\ frac {\ partial S_ {z \ theta}} {\ partial \ theta}} + S_ {zr} \ right] ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {\ theta} \ otimes \ mathbf {e} _ {\ theta} \\ {} + {} & {\ frac {\ partial S_ {zz}} {\ partial r}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {r} + {\ frac {\ partial S_ {zz}} {\ partial z}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {z} + {\ frac {1} {r}} ~ {\ frac {\ partial S_ {zz}} {\ partial \ theta}} ~ \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {z} \ otimes \ mathbf {e} _ {\ theta} \ end {aligned}}}$

Divergens af et tensorfelt

Den divergens af en tensor felt , er defineret på rekursive relation ${\ displaystyle {\ boldsymbol {T}} (\ mathbf {x})}$

${\ displaystyle ({\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {T}}) \ cdot \ mathbf {c} = {\ boldsymbol {\ nabla}} \ cdot \ left (\ mathbf {c} \ cdot {\ boldsymbol {T}} ^ {\ textsf {T}} \ right) ~; \ qquad {\ boldsymbol {\ nabla}} \ cdot \ mathbf {v} = {\ text {tr}} ({\ boldsymbol { \ nabla}} \ mathbf {v})}$

hvor c er en vilkårlig konstant vektor, og v er et vektorfelt. Hvis er et tensorfelt af orden n > 1, så er divergensen af ​​feltet en tensor af orden n - 1. ${\ displaystyle {\ boldsymbol {T}}}$

Kartesiske koordinater

Bemærk: Einstein-summeringskonventionen om opsummering på gentagne indekser bruges nedenfor.

I et kartesisk koordinatsystem har vi følgende relationer for et vektorfelt v og et andet ordens tensorfelt . ${\ displaystyle {\ boldsymbol {S}}}$

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ cdot \ mathbf {v} & = {\ frac {\ partial v_ {i}} {\ partial x_ {i}}} = v_ {i , i} \\ {\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {S}} & = {\ frac {\ partial S_ {ik}} {\ partial x_ {i}}} \ cdot ~ \ mathbf { e} _ {k} = S_ {ik, i} \ cdot ~ \ mathbf {e} _ {k} \ end {aligned}}}$

hvor tensorindeksnotation for partielle derivater bruges i de yderste udtryk. Noter det

${\ displaystyle {\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {S}} \ neq {\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {S}} ^ {\ textsf {T}}.}$

For en symmetrisk andenordens tensor er divergensen ofte skrevet som

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {S}} & = {\ cfrac {\ partial S_ {ki}} {\ partial x_ {i}}} \ cdot ~ \ mathbf {e} _ {k} = S_ {ki, i} \ cdot ~ \ mathbf {e} _ {k} \ end {aligned}}}$

Ovenstående udtryk bruges undertiden som definitionen i kartesisk komponentform (ofte også skrevet som ). Bemærk, at en sådan definition ikke er i overensstemmelse med resten af ​​denne artikel (se afsnittet om krøllede koordinater). ${\ displaystyle {\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {S}}}$${\ displaystyle \ operatorname {div} {\ boldsymbol {S}}}$

Forskellen stammer fra, om differentieringen udføres med hensyn til rækkerne eller kolonnerne i og er konventionel. Dette demonstreres med et eksempel. I et kartesisk koordinatsystem er den anden ordens tensor (matrix) gradienten af ​​en vektorfunktion . ${\ displaystyle {\ boldsymbol {S}}}$${\ displaystyle \ mathbf {S}}$${\ displaystyle \ mathbf {v}}$

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ cdot \ left ({\ boldsymbol {\ nabla}} \ mathbf {v} \ right) & = {\ boldsymbol {\ nabla}} \ cdot \ left (v_ {i, j} ~ \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} \ right) = v_ {i, ji} ~ \ mathbf {e} _ {i} \ cdot \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} = \ left ({\ boldsymbol {\ nabla}} \ cdot \ mathbf {v} \ right) _ {, j} ~ \ mathbf {e} _ {j} = {\ boldsymbol {\ nabla}} \ venstre ({\ boldsymbol {\ nabla}} \ cdot \ mathbf {v} \ højre) \\ {\ boldsymbol {\ nabla}} \ cdot \ left [\ left ({\ boldsymbol {\ nabla}} \ mathbf {v} \ right) ^ {\ textsf {T}} \ right] & = {\ boldsymbol {\ nabla}} \ cdot \ left (v_ {j, i} ~ \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} \ right) = v_ {j, ii} ~ \ mathbf {e} _ {i} \ cdot \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} = {\ boldsymbol {\ nabla}} ^ {2} v_ {j} ~ \ mathbf {e} _ {j} = {\ boldsymbol { \ nabla}} ^ {2} \ mathbf {v} \ end {justeret}}}$

Den sidste ligning svarer til den alternative definition / fortolkning

${\ displaystyle {\ begin {align} \ left ({\ boldsymbol {\ nabla}} \ cdot \ right) _ {\ text {alt}} \ left ({\ boldsymbol {\ nabla}} \ mathbf {v} \ højre) = \ venstre ({\ boldsymbol {\ nabla}} \ cdot \ højre) _ {\ tekst {alt}} \ venstre (v_ {i, j} ~ \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} \ right) = v_ {i, jj} ~ \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} \ cdot \ mathbf {e} _ {j} = {\ boldsymbol {\ nabla}} ^ {2} v_ {i} ~ \ mathbf {e} _ {i} = {\ boldsymbol {\ nabla}} ^ {2} \ mathbf {v} \ end {aligned}} }$

Kurvlinære koordinater

Bemærk: Einstein-summeringskonventionen om opsummering på gentagne indekser bruges nedenfor.

I buelinjede koordinater, de divergerende et vektorfelt v og en anden ordens tensor felt er ${\ displaystyle {\ boldsymbol {S}}}$

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ cdot \ mathbf {v} & = \ left ({\ cfrac {\ partial v ^ {i}} {\ partial \ xi ^ {i} }} + v ^ {k} ~ \ Gamma _ {ik} ^ {i} \ right) \\ {\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {S}} & = \ left ({\ cfrac { \ partial S_ {ik}} {\ partial \ xi _ {i}}} - S_ {lk} ~ \ Gamma _ {ii} ^ {l} -S_ {il} ~ \ Gamma _ {ik} ^ {l} \ højre) ~ \ mathbf {g} ^ {k} \ end {justeret}}}$

Mere generelt,

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {S}} & = \ left [{\ cfrac {\ partial S_ {ij}} {\ partial q ^ {k} }} - \ Gamma _ {ki} ^ {l} ~ S_ {lj} - \ Gamma _ {kj} ^ {l} ~ S_ {il} \ right] ~ g ^ {ik} ~ \ mathbf {b} ^ {j} \\ [8pt] & = \ left [{\ cfrac {\ partial S ^ {ij}} {\ partial q ^ {i}}} + \ Gamma _ {il} ^ {i} ~ S ^ { lj} + \ Gamma _ {il} ^ {j} ~ S ^ {il} \ højre] ~ \ mathbf {b} _ {j} \\ [8pt] & = \ venstre [{\ cfrac {\ delvis S_ { ~ j} ^ {i}} {\ partial q ^ {i}}} + \ Gamma _ {il} ^ {i} ~ S_ {~ j} ^ {l} - \ Gamma _ {ij} ^ {l} ~ S_ {~ l} ^ {i} \ højre] ~ \ mathbf {b} ^ {j} \\ [8pt] & = \ venstre [{\ cfrac {\ delvis S_ {i} ^ {~ j}} { \ delvis q ^ {k}}} - \ Gamma _ {ik} ^ {l} ~ S_ {l} ^ {~ j} + \ Gamma _ {kl} ^ {j} ~ S_ {i} ^ {~ l } \ højre] ~ g ^ {ik} ~ \ mathbf {b} _ {j} \ end {justeret}}}$

Cylindriske polære koordinater

I cylindriske polære koordinater

${\ displaystyle {\ begin {align} {\ boldsymbol {\ nabla}} \ cdot \ mathbf {v} = \ quad & {\ frac {\ partial v_ {r}} {\ partial r}} + {\ frac { 1} {r}} \ left ({\ frac {\ partial v _ {\ theta}} {\ partial \ theta}} + v_ {r} \ right) + {\ frac {\ partial v_ {z}} {\ delvis z}} \\ {\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {S}} = \ quad & {\ frac {\ partial S_ {rr}} {\ partial r}} ~ \ mathbf {e} _ {r} + {\ frac {\ partial S_ {r \ theta}} {\ partial r}} ~ \ mathbf {e} _ {\ theta} + {\ frac {\ partial S_ {rz}} {\ partial r}} ~ \ mathbf {e} _ {z} \\ {} + {} & {\ frac {1} {r}} \ left [{\ frac {\ partial S _ {\ theta r}} {\ partial \ theta}} + (S_ {rr} -S _ {\ theta \ theta}) \ højre] ~ \ mathbf {e} _ {r} + {\ frac {1} {r}} \ venstre [{\ frac { \ delvis S _ {\ theta \ theta}} {\ partial \ theta}} + (S_ {r \ theta} + S _ {\ theta r}) \ højre] ~ \ mathbf {e} _ {\ theta} + {\ frac {1} {r}} \ left [{\ frac {\ partial S _ {\ theta z}} {\ partial \ theta}} + S_ {rz} \ right] ~ \ mathbf {e} _ {z} \ \ {} + {} & {\ frac {\ partial S_ {zr}} {\ partial z}} ~ \ mathbf {e} _ {r} + {\ frac {\ partial S_ {z \ theta}} {\ delvis z}} ~ \ mathbf {e} _ {\ theta} + {\ frac {\ partial S_ {zz}} {\ partial z}} ~ \ mathbf {e} _ {z} \ end {align}}}$

Krølle af et tensorfelt

Den krølle af en ordre- n > 1 tensor felt er også defineret ved hjælp af rekursive relation ${\ displaystyle {\ boldsymbol {T}} (\ mathbf {x})}$

${\ displaystyle ({\ boldsymbol {\ nabla}} \ times {\ boldsymbol {T}}) \ cdot \ mathbf {c} = {\ boldsymbol {\ nabla}} \ times (\ mathbf {c} \ cdot {\ fed symbol {T}}) ~; \ qquad ({\ boldsymbol {\ nabla}} \ times \ mathbf {v}) \ cdot \ mathbf {c} = {\ boldsymbol {\ nabla}} \ cdot (\ mathbf {v } \ times \ mathbf {c})}$

hvor c er en vilkårlig konstant vektor, og v er et vektorfelt.

Krølle af et første ordens tensor (vektor) felt

Overvej et vektorfelt v og en vilkårlig konstant vektor c . I indeksnotation er krydsproduktet givet af

${\ displaystyle \ mathbf {v} \ times \ mathbf {c} = \ varepsilon _ {ijk} ~ v_ {j} ~ c_ {k} ~ \ mathbf {e} _ {i}}$

hvor er permutationssymbolet , ellers kendt som Levi-Civita-symbolet. Derefter, ${\ displaystyle \ varepsilon _ {ijk}}$

${\ displaystyle {\ boldsymbol {\ nabla}} \ cdot (\ mathbf {v} \ times \ mathbf {c}) = \ varepsilon _ {ijk} ~ v_ {j, i} ~ c_ {k} = (\ varepsilon _ {ijk} ~ v_ {j, i} ~ \ mathbf {e} _ {k}) \ cdot \ mathbf {c} = ({\ boldsymbol {\ nabla}} \ times \ mathbf {v}) \ cdot \ mathbf {c}}$

Derfor,

${\ displaystyle {\ boldsymbol {\ nabla}} \ times \ mathbf {v} = \ varepsilon _ {ijk} ~ v_ {j, i} ~ \ mathbf {e} _ {k}}$

Krølle af et andet ordens tensorfelt

For en anden ordens tensor ${\ displaystyle {\ boldsymbol {S}}}$

${\ displaystyle \ mathbf {c} \ cdot {\ boldsymbol {S}} = c_ {m} ~ S_ {mj} ~ \ mathbf {e} _ {j}}$

Derfor bruger definitionen af ​​krøllen af ​​et første ordens tensorfelt,

${\ displaystyle {\ boldsymbol {\ nabla}} \ times (\ mathbf {c} \ cdot {\ boldsymbol {S}}) = \ varepsilon _ {ijk} ~ c_ {m} ~ S_ {mj, i} ~ \ mathbf {e} _ {k} = (\ varepsilon _ {ijk} ~ S_ {mj, i} ~ \ mathbf {e} _ {k} \ otimes \ mathbf {e} _ {m}) \ cdot \ mathbf { c} = ({\ boldsymbol {\ nabla}} \ times {\ boldsymbol {S}}) \ cdot \ mathbf {c}}$

Derfor har vi det

${\ displaystyle {\ boldsymbol {\ nabla}} \ times {\ boldsymbol {S}} = \ varepsilon _ {ijk} ~ S_ {mj, i} ~ \ mathbf {e} _ {k} \ otimes \ mathbf {e } _ {m}}$

Identiteter, der involverer krølningen af ​​et tensorfelt

Den mest almindeligt anvendte identitet, der involverer krumning af et tensorfelt , er ${\ displaystyle {\ boldsymbol {T}}}$

${\ displaystyle {\ boldsymbol {\ nabla}} \ times ({\ boldsymbol {\ nabla}} {\ boldsymbol {T}}) = {\ boldsymbol {0}}}$

Denne identitet gælder for tensorfelter af alle ordrer. I det vigtige tilfælde af en anden ordens tensor indebærer denne identitet det ${\ displaystyle {\ boldsymbol {S}}}$

${\ displaystyle {\ boldsymbol {\ nabla}} \ times ({\ boldsymbol {\ nabla}} {\ boldsymbol {S}}) = {\ boldsymbol {0}} \ quad \ antyder \ quad S_ {mi, j} -S_ {mj, i} = 0}$

Afledt af determinanten for en anden ordens tensor

Derivatet af determinanten af ​​en anden ordens tensor er givet ved ${\ displaystyle {\ boldsymbol {A}}}$

${\ displaystyle {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} \ det ({\ boldsymbol {A}}) = \ det ({\ boldsymbol {A}}) ~ \ venstre [{ \ boldsymbol {A}} ^ {- 1} \ højre] ^ {\ tekster {T}} ~.}$

I en ortonormalbasis, komponenterne i kan skrives som en matrix A . I så fald svarer højre side til matrixens medfaktorer. ${\ displaystyle {\ boldsymbol {A}}}$

Bevis

Lad være en anden ordens tensor og lad . Derefter har vi fra definitionen af ​​derivatet af en skalærværdifunktion af en tensor ${\ displaystyle {\ boldsymbol {A}}}$${\ displaystyle f ({\ boldsymbol {A}}) = \ det ({\ boldsymbol {A}})}$ ${\ displaystyle {\ begin {align} {\ frac {\ partial f} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} & = \ left. {\ cfrac {\ rm {d }} {{\ rm {d}} \ alpha}} \ det ({\ boldsymbol {A}} + \ alpha ~ {\ boldsymbol {T}}) \ højre | _ {\ alpha = 0} \\ & = \ venstre. {\ cfrac {\ rm {d}} {{\ rm {d}} \ alpha}} \ det \ left [\ alpha ~ {\ boldsymbol {A}} \ left ({\ cfrac {1} { \ alpha}} ~ {\ boldsymbol {\ mathit {I}}} + {\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ højre) \ højre] \ højre | _ {\ alpha = 0} \\ & = \ left. {\ cfrac {\ rm {d}} {{\ rm {d}} \ alpha}} \ left [\ alpha ^ {3} ~ \ det ({\ boldsymbol { A}}) ~ \ det \ left ({\ cfrac {1} {\ alpha}} ~ {\ boldsymbol {\ mathit {I}}} + {\ boldsymbol {A}} ^ {- 1} \ cdot {\ fed symbol {T}} \ højre) \ højre] \ højre | _ {\ alpha = 0}. \ slut {justeret}}}$ Determinanten af en tensor kan udtrykkes i form af en karakteristisk ligning med hensyn til invariants anvendelse ${\ displaystyle I_ {1}, I_ {2}, I_ {3}}$ ${\ displaystyle \ det (\ lambda ~ {\ boldsymbol {\ mathit {I}}} + {\ boldsymbol {A}}) = \ lambda ^ {3} + I_ {1} ({\ boldsymbol {A}}) ~ \ lambda ^ {2} + I_ {2} ({\ boldsymbol {A}}) ~ \ lambda + I_ {3} ({\ boldsymbol {A}}).}$ Ved hjælp af denne udvidelse kan vi skrive ${\ displaystyle {\ begin {align} {\ frac {\ partial f} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} & = \ left. {\ cfrac {\ rm {d }} {{\ rm {d}} \ alpha}} \ left [\ alpha ^ {3} ~ \ det ({\ boldsymbol {A}}) ~ \ left ({\ cfrac {1} {\ alpha ^ { 3}}} + I_ {1} \ left ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ right) ~ {\ cfrac {1} {\ alpha ^ {2} }} + I_ {2} \ left ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ right) ~ {\ cfrac {1} {\ alpha}} + I_ {3 } \ venstre ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ højre) \ højre) \ højre] \ højre | _ {\ alpha = 0} \\ & = \ venstre . \ det ({\ boldsymbol {A}}) ~ {\ cfrac {\ rm {d}} {{\ rm {d}} \ alpha}} \ venstre [1 + I_ {1} \ venstre ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ højre) ~ \ alpha + I_ {2} \ left ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol { T}} \ højre) ~ \ alpha ^ {2} + I_ {3} \ venstre ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ højre) ~ \ alpha ^ { 3} \ højre] \ højre | _ {\ alpha = 0} \\ & = \ venstre. \ Det ({\ boldsymbol {A}}) ~ \ venstre [I_ {1} ({\ boldsymbol {A}} ^ {-1} \ cdot {\ boldsymbol {T}}) + 2 ~ I_ {2} \ left ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ right) ~ \ alpha + 3 ~ I_ {3} \ venstre ({\ fed symbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ højre) ~ \ alpha ^ {2} \ højre] \ højre | _ {\ alpha = 0} \\ & = \ det ({ \ boldsymbol {A}}) ~ I_ {1} \ left ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ right) ~. \ end {align}}}$ Husk at invarianten er givet af ${\ displaystyle I_ {1}}$ ${\ displaystyle I_ {1} ({\ boldsymbol {A}}) = {\ text {tr}} {\ boldsymbol {A}}.}$ Derfor, ${\ displaystyle {\ frac {\ partial f} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} = \ det ({\ boldsymbol {A}}) ~ {\ text {tr} } \ left ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ right) = \ det ({\ boldsymbol {A}}) ~ \ left [{\ boldsymbol {A} } ^ {- 1} \ right] ^ {\ textsf {T}}: {\ boldsymbol {T}}.}$ Påberåber den vilkårlighed, som vi så har ${\ displaystyle {\ boldsymbol {T}}}$ ${\ displaystyle {\ frac {\ partial f} {\ partial {\ boldsymbol {A}}}} = \ det ({\ boldsymbol {A}}) ~ \ left [{\ boldsymbol {A}} ^ {- 1 } \ right] ^ {\ textsf {T}} ~.}$

Afledte af invarianter af en anden ordens tensor

De vigtigste invarianter af en anden ordens tensor er

${\ displaystyle {\ begin {align} I_ {1} ({\ boldsymbol {A}}) & = {\ text {tr}} {\ boldsymbol {A}} \\ I_ {2} ({\ boldsymbol {A }}) & = {\ frac {1} {2}} \ left [({\ text {tr}} {\ boldsymbol {A}}) ^ {2} - {\ text {tr}} {{\ boldsymbol {A}} ^ {2}} \ højre] \\ I_ {3} ({\ boldsymbol {A}}) & = \ det ({\ boldsymbol {A}}) \ slut {justeret}}}$

Derivaterne af disse tre invarianter med hensyn til er ${\ displaystyle {\ boldsymbol {A}}}$

${\ displaystyle {\ begin {align} {\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} & = {\ boldsymbol {\ mathit {1}}} \\ [3pt] {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} & = I_ {1} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} \\ [3pt] {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} & = \ det ({\ boldsymbol {A}}) ~ \ venstre [{\ boldsymbol {A}} ^ {- 1} \ højre] ^ {\ textsf {T}} = I_ {2} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} ~ \ left (I_ {1} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} \ right) = \ left ({\ boldsymbol {A}} ^ {2} -I_ {1} ~ {\ boldsymbol {A}} + I_ {2} ~ {\ boldsymbol {\ mathit {1}}} \ højre) ^ {\ tekster { T}} \ end {justeret}}}$

Bevis

Fra afledningen af ​​determinanten ved vi det ${\ displaystyle {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} = \ det ({\ boldsymbol {A}}) ~ \ left [{\ boldsymbol {A}} ^ {-1} \ right] ^ {\ textsf {T}} ~.}$ For afledningerne af de to andre invarianter, lad os gå tilbage til den karakteristiske ligning ${\ displaystyle \ det (\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A}}) = \ lambda ^ {3} + I_ {1} ({\ boldsymbol {A}}) ~ \ lambda ^ {2} + I_ {2} ({\ boldsymbol {A}}) ~ \ lambda + I_ {3} ({\ boldsymbol {A}}) ~.}$ Ved at bruge den samme tilgang som for determinanten af ​​en tensor, kan vi vise det ${\ displaystyle {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} \ det (\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A}}) = \ det (\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A}}) ~ \ left [(\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A }}) ^ {- 1} \ right] ^ {\ textsf {T}} ~.}$ Nu kan venstre side udvides som ${\ displaystyle {\ begin {align} {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} \ det (\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol { A}}) & = {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} \ left [\ lambda ^ {3} + I_ {1} ({\ boldsymbol {A}}) ~ \ lambda ^ {2} + I_ {2} ({\ boldsymbol {A}}) ~ \ lambda + I_ {3} ({\ boldsymbol {A}}) \ højre] \\ & = {\ frac {\ delvis I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {2} + {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda + {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} ~. \ end {justeret}}}$ Derfor ${\ displaystyle {\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {2} + {\ frac {\ partial I_ {2}} {\ partial {\ fed symbol {A}}}} ~ \ lambda + {\ frac {\ delvis I_ {3}} {\ delvis {\ boldsymbol {A}}}} = \ det (\ lambda ~ {\ boldsymbol {\ mathit {1} }} + {\ boldsymbol {A}}) ~ \ venstre [(\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A}}) ^ {- 1} \ højre] ^ {\ tekster fra {T}}}$ eller, ${\ displaystyle (\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A}}) ^ {\ textsf {T}} \ cdot \ left [{\ frac {\ partial I_ {1} } {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {2} + {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda + {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} \ right] = \ det (\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A}} ) ~ {\ boldsymbol {\ mathit {1}}} ~.}$ Udvidelse af højre side og adskillelse af termer på venstre side giver ${\ displaystyle \ left (\ lambda ~ {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A}} ^ {\ textsf {T}} \ right) \ cdot \ left [{\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {2} + {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda + {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} \ right] = \ left [\ lambda ^ {3} + I_ {1} ~ \ lambda ^ {2} + I_ {2} ~ \ lambda + I_ {3} \ right] {\ boldsymbol {\ mathit {1}}}}$ eller, ${\ displaystyle {\ begin {align} \ left [{\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {3} \ right. & \ left. + {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {2} + {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A }}}} ~ \ lambda \ right] {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {2} + {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {2}} {\ delvis {\ boldsymbol {A}}}} ~ \ lambda + {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A }}}} \\ & = \ left [\ lambda ^ {3} + I_ {1} ~ \ lambda ^ {2} + I_ {2} ~ \ lambda + I_ {3} \ right] {\ boldsymbol {\ matematik {1}}} ~. \ end {justeret}}}$ Hvis vi definerer og , kan vi skrive ovenstående som ${\ displaystyle I_ {0}: = 1}$${\ displaystyle I_ {4}: = 0}$ ${\ displaystyle {\ begin {align} \ left [{\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {3} \ right. & \ left. + {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {2} + {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A }}}} ~ \ lambda + {\ frac {\ partial I_ {4}} {\ partial {\ boldsymbol {A}}}} \ right] {\ boldsymbol {\ mathit {1}}} + {\ boldsymbol { A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {0}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {3} + {\ boldsymbol {A} } ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda ^ {2} + {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} ~ \ lambda + {\ boldsymbol {A}} ^ {\ textsf {T} } \ cdot {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} \\ & = \ left [I_ {0} ~ \ lambda ^ {3} + I_ {1} ~ \ lambda ^ {2} + I_ {2} ~ \ lambda + I_ {3} \ højre] {\ boldsymbol {\ mathit {1}}} ~. \ end {justeret}}}$ Samler termer, der indeholder forskellige beføjelser i λ, får vi ${\ displaystyle {\ begin {align} \ lambda ^ {3} & \ left (I_ {0} ~ {\ boldsymbol {\ mathit {1}}} - {\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {0}} {\ partial {\ boldsymbol {A}}}} til højre) + \ lambda ^ {2} \ left (I_ {1} ~ {\ boldsymbol {\ mathit {1}}} - {\ frac {\ partial I_ { 2}} {\ partial {\ boldsymbol {A}}}} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ delvis I_ {1}} {\ partial {\ boldsymbol {A}}}} til højre) + \\ & \ qquad \ qquad \ lambda \ left (I_ {2} ~ {\ boldsymbol {\ mathit {1}}} - {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T }} \ cdot {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} til højre) + \ left (I_ {3} ~ {\ boldsymbol {\ mathit {1}}} - {\ frac {\ partial I_ {4}} {\ partial {\ boldsymbol {A}}}} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T }} \ cdot {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} \ right) = 0 ~. \ end {aligned}}}$ Derefter har vi påkaldt λ's vilkårlighed ${\ displaystyle {\ begin {align} I_ {0} ~ {\ boldsymbol {\ mathit {1}}} - {\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {0}} {\ partial {\ boldsymbol {A}} }} & = 0 \\ I_ {1} ~ {\ boldsymbol {\ mathit {1}}} - {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} ~ {\ fed symbol {\ mathit {1}}} - I_ {2} ~ {\ boldsymbol {\ mathit {1}}} - {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A} }}} & = 0 \\ I_ {3} ~ {\ boldsymbol {\ mathit {1}}} - {\ frac {\ partial I_ {4}} {\ partial {\ boldsymbol {A}}}} ~ { \ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} \ cdot {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}} } & = 0 ~. \ End {justeret}}}$ Dette indebærer, at ${\ displaystyle {\ begin {align} {\ frac {\ partial I_ {1}} {\ partial {\ boldsymbol {A}}}} & = {\ boldsymbol {\ mathit {1}}} \\ {\ frac {\ partial I_ {2}} {\ partial {\ boldsymbol {A}}}} & = I_ {1} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ textsf {T}} \\ {\ frac {\ partial I_ {3}} {\ partial {\ boldsymbol {A}}}} & = I_ {2} ~ {\ boldsymbol {\ mathit {1}}} - {\ fed symbol {A}} ^ {\ textsf {T}} ~ \ venstre (I_ {1} ~ {\ boldsymbol {\ mathit {1}}} - {\ boldsymbol {A}} ^ {\ tekster {T}} \ højre) = \ venstre ({\ boldsymbol {A}} ^ {2} -I_ {1} ~ {\ boldsymbol {A}} + I_ {2} ~ {\ boldsymbol {\ mathit {1}}} \ højre) ^ {\ textsf {T}} \ end {justeret}}}$

Afledt af andenordens identitetstensor

Lad være andenordens identitetstensor. Derefter gives afledningen af ​​denne tensor med hensyn til en anden ordens tensor af ${\ displaystyle {\ boldsymbol {\ mathit {1}}}}$${\ displaystyle {\ boldsymbol {A}}}$

${\ displaystyle {\ frac {\ partial {\ boldsymbol {\ mathit {1}}}} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} = {\ boldsymbol {\ mathsf {0 }}}: {\ boldsymbol {T}} = {\ boldsymbol {\ mathit {0}}}}$

Dette er fordi er uafhængig af . ${\ displaystyle {\ boldsymbol {\ mathit {1}}}}$${\ displaystyle {\ boldsymbol {A}}}$

Afledt af en anden ordens tensor med hensyn til sig selv

Lad være en anden ordens tensor. Derefter ${\ displaystyle {\ boldsymbol {A}}}$

${\ displaystyle {\ frac {\ partial {\ boldsymbol {A}}} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} = \ left [{\ frac {\ partial} {\ delvis \ alpha}} ({\ boldsymbol {A}} + \ alpha ~ {\ boldsymbol {T}}) \ højre] _ {\ alpha = 0} = {\ boldsymbol {T}} = {\ boldsymbol {\ mathsf {I}}}: {\ boldsymbol {T}}}$

Derfor,

${\ displaystyle {\ frac {\ partial {\ boldsymbol {A}}} {\ partial {\ boldsymbol {A}}}} = {\ boldsymbol {\ mathsf {I}}}}$

Her er fjerde ordens identitetstensor. I indeksnotation med hensyn til et ortonormalt grundlag ${\ displaystyle {\ boldsymbol {\ mathsf {I}}}}$

${\ displaystyle {\ boldsymbol {\ mathsf {I}}} = \ delta _ {ik} ~ \ delta _ {jl} ~ \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} \ otimes \ mathbf {e} _ {k} \ otimes \ mathbf {e} _ {l}}$

Dette resultat indebærer det

${\ displaystyle {\ frac {\ partial {\ boldsymbol {A}} ^ {\ textsf {T}}} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} = {\ boldsymbol { \ mathsf {I}}} ^ {\ tekster {T}}: {\ boldsymbol {T}} = {\ boldsymbol {T}} ^ {\ tekster {T}}}$

hvor

${\ displaystyle {\ boldsymbol {\ mathsf {I}}} ^ {\ textsf {T}} = \ delta _ {jk} ~ \ delta _ {il} ~ \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} \ otimes \ mathbf {e} _ {k} \ otimes \ mathbf {e} _ {l}}$

Derfor, hvis tensoren er symmetrisk, så er afledningen også symmetrisk, og vi får ${\ displaystyle {\ boldsymbol {A}}}$

${\ displaystyle {\ frac {\ partial {\ boldsymbol {A}}} {\ partial {\ boldsymbol {A}}}} = {\ boldsymbol {\ mathsf {I}}} ^ {(s)} = {\ frac {1} {2}} ~ \ left ({\ boldsymbol {\ mathsf {I}}} + {\ boldsymbol {\ mathsf {I}}} ^ {\ textsf {T}} \ right)}$

hvor den symmetriske fjerde ordens identitetstensor er

${\ displaystyle {\ boldsymbol {\ mathsf {I}}} ^ {(s)} = {\ frac {1} {2}} ~ (\ delta _ {ik} ~ \ delta _ {jl} + \ delta _ {il} ~ \ delta _ {jk}) ~ \ mathbf {e} _ {i} \ otimes \ mathbf {e} _ {j} \ otimes \ mathbf {e} _ {k} \ otimes \ mathbf {e} _ {l}}$

Afledt af det omvendte af en anden ordens tensor

Lad og være to andenordens tensorer, så ${\ displaystyle {\ boldsymbol {A}}}$${\ displaystyle {\ boldsymbol {T}}}$

${\ displaystyle {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} \ left ({\ boldsymbol {A}} ^ {- 1} \ right): {\ boldsymbol {T}} = - {\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ cdot {\ boldsymbol {A}} ^ {- 1}}$

I indeksnotation med hensyn til et ortonormalt grundlag

${\ displaystyle {\ frac {\ partial A_ {ij} ^ {- 1}} {\ partial A_ {kl}}} ~ T_ {kl} = - A_ {ik} ^ {- 1} ~ T_ {kl} ~ A_ {lj} ^ {- 1} \ antyder {\ frac {\ partial A_ {ij} ^ {- 1}} {\ partial A_ {kl}}} = - A_ {ik} ^ {- 1} ~ A_ { lj} ^ {- 1}}$

Vi har også

${\ displaystyle {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} \ left ({\ boldsymbol {A}} ^ {- {\ textsf {T}}} \ right): {\ boldsymbol {T}} = - {\ boldsymbol {A}} ^ {- {\ textsf {T}}} \ cdot {\ boldsymbol {T}} ^ {\ textsf {T}} \ cdot {\ boldsymbol {A}} ^ {- {\ textsf {T}}}}$

I indeksnotation

${\ displaystyle {\ frac {\ partial A_ {ji} ^ {- 1}} {\ partial A_ {kl}}} ~ T_ {kl} = - A_ {jk} ^ {- 1} ~ T_ {lk} ~ A_ {li} ^ {- 1} \ antyder {\ frac {\ partial A_ {ji} ^ {- 1}} {\ partial A_ {kl}}} = - A_ {li} ^ {- 1} ~ A_ { jk} ^ {- 1}}$

Hvis tensoren er symmetrisk, så ${\ displaystyle {\ boldsymbol {A}}}$

${\ displaystyle {\ frac {\ partial A_ {ij} ^ {- 1}} {\ partial A_ {kl}}} = - {\ cfrac {1} {2}} \ left (A_ {ik} ^ {- 1} ~ A_ {jl} ^ {- 1} + A_ {il} ^ {- 1} ~ A_ {jk} ^ {- 1} \ højre)}$

Bevis

Husk det ${\ displaystyle {\ frac {\ partial {\ boldsymbol {\ mathit {1}}}} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} = {\ boldsymbol {\ mathit {0 }}}}$ Siden kan vi skrive ${\ displaystyle {\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {A}} = {\ boldsymbol {\ mathit {1}}}}$ ${\ displaystyle {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} \ left ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {A}} \ right): {\ boldsymbol {T}} = {\ boldsymbol {\ mathit {0}}}}$ Brug af produktreglen til andenordens tensorer ${\ displaystyle {\ frac {\ partial} {\ partial {\ boldsymbol {S}}}} [{\ boldsymbol {F}} _ {1} ({\ boldsymbol {S}}) \ cdot {\ boldsymbol {F }} _ {2} ({\ boldsymbol {S}})]: {\ boldsymbol {T}} = \ left ({\ frac {\ partial {\ boldsymbol {F}} _ {1}} {\ partial { \ boldsymbol {S}}}}: {\ boldsymbol {T}} \ højre) \ cdot {\ boldsymbol {F}} _ {2} + {\ boldsymbol {F}} _ {1} \ cdot \ left ({ \ frac {\ partial {\ boldsymbol {F}} _ {2}} {\ partial {\ boldsymbol {S}}}}: {\ boldsymbol {T}} \ right)}$ vi får ${\ displaystyle {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} ({\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {A}}): {\ boldsymbol { T}} = \ left ({\ frac {\ partial {\ boldsymbol {A}} ^ {- 1}} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} \ right) \ cdot {\ boldsymbol {A}} + {\ boldsymbol {A}} ^ {- 1} \ cdot \ left ({\ frac {\ partial {\ boldsymbol {A}}} {\ partial {\ boldsymbol {A}} }}: {\ boldsymbol {T}} \ right) = {\ boldsymbol {\ mathit {0}}}}$ eller, ${\ displaystyle \ left ({\ frac {\ partial {\ boldsymbol {A}} ^ {- 1}} {\ partial {\ boldsymbol {A}}}}: {\ boldsymbol {T}} \ right) \ cdot {\ boldsymbol {A}} = - {\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}}}$ Derfor, ${\ displaystyle {\ frac {\ partial} {\ partial {\ boldsymbol {A}}}} \ left ({\ boldsymbol {A}} ^ {- 1} \ right): {\ boldsymbol {T}} = - {\ boldsymbol {A}} ^ {- 1} \ cdot {\ boldsymbol {T}} \ cdot {\ boldsymbol {A}} ^ {- 1}}$

Integration af dele

En anden vigtig operation relateret til tensorderivater i kontinuummekanik er integration af dele. Formlen for integration af dele kan skrives som

${\ displaystyle \ int _ {\ Omega} {\ boldsymbol {F}} \ otimes {\ boldsymbol {\ nabla}} {\ boldsymbol {G}} \, {\ rm {d}} \ Omega = \ int _ { \ Gamma} \ mathbf {n} \ otimes ({\ boldsymbol {F}} \ otimes {\ boldsymbol {G}}) \, {\ rm {d}} \ Gamma - \ int _ {\ Omega} {\ boldsymbol {G}} \ otimes {\ boldsymbol {\ nabla}} {\ boldsymbol {F}} \, {\ rm {d}} \ Omega}$

hvor og er differentierbare tensorfelter af vilkårlig orden, er enheden udad normal i forhold til det domæne, som tensorfelterne er defineret over, repræsenterer en generaliseret tensorproduktoperatør og er en generaliseret gradientoperatør. Når er lig med identitetstensoren, får vi divergenssætningen${\ displaystyle {\ boldsymbol {F}}}$${\ displaystyle {\ boldsymbol {G}}}$${\ displaystyle \ mathbf {n}}$${\ displaystyle \ otimes}$${\ displaystyle {\ boldsymbol {\ nabla}}}$${\ displaystyle {\ boldsymbol {F}}}$

${\ displaystyle \ int _ {\ Omega} {\ boldsymbol {\ nabla}} {\ boldsymbol {G}} \, {\ rm {d}} \ Omega = \ int _ {\ Gamma} \ mathbf {n} \ tidspunkter {\ boldsymbol {G}} \, {\ rm {d}} \ Gamma \ ,.}$

Vi kan udtrykke formlen for integration af dele i kartesisk indeksnotation som

${\ displaystyle \ int _ {\ Omega} F_ {ijk ....} \, G_ {lmn ..., p} \, {\ rm {d}} \ Omega = \ int _ {\ Gamma} n_ { p} \, F_ {ijk ...} \, G_ {lmn ...} \, {\ rm {d}} \ Gamma - \ int _ {\ Omega} G_ {lmn ...} \, F_ { ijk ..., p} \, {\ rm {d}} \ Omega \ ,.}$

I det specielle tilfælde, hvor tensorproduktoperationen er en sammentrækning af et indeks, og gradientoperationen er en divergens, og begge og er andenordens tensorer, har vi ${\ displaystyle {\ boldsymbol {F}}}$${\ displaystyle {\ boldsymbol {G}}}$

${\ displaystyle \ int _ {\ Omega} {\ boldsymbol {F}} \ cdot ({\ boldsymbol {\ nabla}} \ cdot {\ boldsymbol {G}}) \, {\ rm {d}} \ Omega = \ int _ {\ Gamma} \ mathbf {n} \ cdot \ left ({\ boldsymbol {G}} \ cdot {\ boldsymbol {F}} ^ {\ textsf {T}} \ right) \, {\ rm { d}} \ Gamma - \ int _ {\ Omega} ({\ boldsymbol {\ nabla}} {\ boldsymbol {F}}): {\ boldsymbol {G}} ^ {\ textsf {T}} \, {\ rm {d}} \ Omega \ ,.}$

I indeksnotation,

${\ displaystyle \ int _ {\ Omega} F_ {ij} \, G_ {pj, p} \, {\ rm {d}} \ Omega = \ int _ {\ Gamma} n_ {p} \, F_ {ij } \, G_ {pj} \, {\ rm {d}} \ Gamma - \ int _ {\ Omega} G_ {pj} \, F_ {ij, p} \, {\ rm {d}} \ Omega \ ,.}$

Se også

• Kovariant derivat
• Ricci-beregning

Referencer