To derive : The integral theorem: ∫ volume τ ∇ ϕ d τ = ∫ surface enclosing τ ϕ n d τ .

Question

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Answer 1

Question

Chapter 6.11, Problem 17P

a.

To determine

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

b.

To determine

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

c.

To determine

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

d.

To determine

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

e.

To determine

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

f.

To determine

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

g.

To determine

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

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Answer 2

Question

Chapter 6.11, Problem 17P

a.

To determine

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

b.

To determine

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

c.

To determine

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

d.

To determine

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

e.

To determine

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

f.

To determine

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

g.

To determine

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

Expert Solution & Answer

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Answer 3

Question

Chapter 6.11, Problem 17P

a.

To determine

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

b.

To determine

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

c.

To determine

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

d.

To determine

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

e.

To determine

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

f.

To determine

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

g.

To determine

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 6.11, Problem 17P

a.

To determine

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

b.

To determine

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

c.

To determine

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

d.

To determine

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

e.

To determine

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

f.

To determine

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

g.

To determine

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 5

Chapter 6.11, Problem 17P

a.

To determine

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

b.

To determine

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

c.

To determine

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

d.

To determine

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

e.

To determine

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

f.

To determine

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

g.

To determine

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

Answer 6

Chapter 6.11, Problem 17P

a.

To determine

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

Answer 7

Chapter 6.11, Problem 17P

a.

To determine

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

Answer 8

b.

To determine

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

Answer 9

b.

To determine

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

Answer 10

c.

To determine

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

Answer 11

c.

To determine

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

Answer 12

d.

To determine

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

Answer 13

d.

To determine

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

Answer 14

e.

To determine

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

Answer 15

e.

To determine

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

Answer 16

f.

To determine

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

Answer 17

f.

To determine

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

Answer 18

g.

To determine

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

Answer 19

g.

To determine

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

Answer 20

Expert Solution & Answer

Answer 21

Expert Solution & Answer

Answer 22

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Answer 23

Ch. 6.3 - If A=2ijk,B=2i3j+k,C=j+k, find (AB)C,A(BC),(AB)C,...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - A force F=2i3j+k acts at the point (1,5,2). Find...Ch. 6.3 - Prob. 8P Ch. 6.3 - Prob. 9P Ch. 6.3 - In Figure 3.5, let r be another vector from O to...

To derive : The integral theorem: ∫ volume τ ∇ ϕ d τ = ∫ surface enclosing τ ϕ n d τ .

Solution Summary: The Gauss divergence theorem states that if fis a continuously differentiable vector point function and V is the volume bounded by the surface Sthen.

Concept explainers

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

Want to see the full answer?

Chapter 6 Solutions

To derive : The integral theorem: ∫ volume τ ∇ ϕ d τ = ∫ surface enclosing τ ϕ n d τ .

Solution Summary: The Gauss divergence theorem states that if fis a continuously differentiable vector point function and V is the volume bounded by the surface Sthen.

Concept explainers

To derive: The integral theorem: ∫volume τ∇ϕdτ=∫surface enclosing τϕ n dτ .

To derive: The integral theorem: ∫volume τ∇×Vdτ=∫surface enclosing τn× Vdτ

To derive: The integral theorem: ∫curve bounding σϕdr=∫surface σ(n× ∇ϕ)dσ

To derive: The integral theorem: ∮C= curve bounding σdr× V=∫surface σ(n×∇)× Vdσ

To derive: The integral theorem: ∫volume τϕ∇⋅Vdτ=∮surface enclosing τ ϕV⋅ndσ-∫volume τ V⋅∇ϕdτ

To derive: The integral theorem: ∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

To derive: The integral theorem: ∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr

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Chapter 6 Solutions

To derive: The integral theorem:

∫volume τV⋅(∇×U)dτ=∫volume τ U⋅(∇×V)dτ+∮surface enclosing τ (U×V)⋅ndσ

To derive: The integral theorem:

∫surface of σϕ(∇×V)⋅ndσ=∫surface of σ (∇×∇ϕ)⋅ndσ+∮curve bounding σ (ϕV)⋅dr