The curl of vector field A and divergence of the curl.

Question

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Answer 1

Question

Chapter 3, Problem 41P

(a)

To determine

The curl of vector field $A$ and divergence of the curl.

(a)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $x2yax+y2zay−2xzaz$ .

Calculation:

Calculate the curl of vector field $(∇×A)$ using the relation.

$∇×A=(∂Az∂y−∂Ay∂z)ax+(∂Ax∂z−∂Az∂x)ay+(∂Ay∂x−∂Ax∂y)az$

$∇×A=(∂(−2xz)∂y−∂(y2z)∂z)ax+(∂(x2y)∂z−∂(−2xz)∂x)ay+(∂(y2z)∂x−∂(x2y)∂y)az=(0−y2)ax+(0+2z)ay+(0−x2)az=−y2ax+2zay−x2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=∂(∇×A)x∂x+∂(∇×A)y∂y+∂(∇×A)z∂z$

$∇⋅(∇×A)=∂(−y2)∂x+∂(2z)∂y+∂(−x2)∂z=0$

Thus, the curl of vector field $A$ is $−y2ax+2zay−x2az_$ and divergence of the curl is $0_$ .

(b)

To determine

The curl of vector field $A$ and divergence of the curl.

(b)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $ρ2zaρ+ρ3aϕ+3ρz2ϕaz$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=(1ρ∂Az∂ϕ−∂Aϕ∂z)aρ+(∂Aρ∂z−∂Az∂ρ)aϕ+1ρ(∂Aϕ∂ρ−∂Aρ∂ϕ)az$

$∇×A=[(1ρ∂(3ρz2)∂ϕ−∂(ρ3)∂z)aρ+(∂(ρ2z)∂z−∂(3ρz2)∂ρ)aϕ+(∂(ρ3)∂ρ−∂(ρ2z)∂ϕ)az]=[0−0]aρ+[ρ2−3z2]aϕ+[4ρ2−0]az=(ρ2−3z2)aϕ+4ρ2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=1ρ∂∂ρ(ρ(∇×A)ρ)+1ρ∂∂ϕ(∇×A)ϕ+∂∂z(∇×A)z$

$∇⋅(∇×A)=1ρ∂∂ρ(ρ×0)+1ρ∂∂ϕ(ρ2−3z2)+∂∂z(4ρ2)=0$

Thus, the curl of vector field $A$ is $(ρ2−3z2)aϕ+4ρ2az_$ and divergence of the curl is $0_$ .

(c)

To determine

The curl of vector field $A$ and divergence of the curl.

(c)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $sinϕr2ar−cosϕr2aθ$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=[1rsinθ(∂(Aϕsinθ)∂θ−∂Aθ∂ϕ)ar+1r(1sinθ∂Ar∂ϕ−∂(rAϕ)∂r)aθ+1r(∂(rAθ)∂r−∂Ar∂θ)aϕ]$

$∇×A=[1rsinθ(∂(0)∂θ−∂(−cosϕr2)∂ϕ)ar+1r(1sinθ∂(sinϕr2)∂ϕ−∂(0)∂r)aθ+1r(∂(r(−cosϕr2))∂r−∂(sinϕr2)∂θ)aϕ]=[1rsinθ(0−sinϕr2)ar+1r(1sinθcosϕr2)aθ+1r(cosϕr2−0)aϕ]=−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ$

$∇⋅(∇×A)=0$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=[1r2∂∂r(r2(∇×A)r)+1rsinθ∂∂θ((∇×A)θsinθ)+1rsinθ∂∂ϕ(∇×A)ϕ]$

$∇⋅(∇×A)=[1r2∂∂r(r2(−sinϕr3sinθ))+1rsinθ∂∂θ(cosϕr3sinθsinθ)+1rsinθ∂∂ϕ(cosϕr3)]=1r2(sinϕr2sinθ)+1rsinθ(0)−1rsinθ(cosϕr3)=sinϕr4sinθ−sinϕr4sinθ$

Thus, the curl of vector field $A$ is $(−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ)_$ and divergence of the curl is $0_$ .

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Answer 2

Question

Chapter 3, Problem 41P

(a)

To determine

The curl of vector field $A$ and divergence of the curl.

(a)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $x2yax+y2zay−2xzaz$ .

Calculation:

Calculate the curl of vector field $(∇×A)$ using the relation.

$∇×A=(∂Az∂y−∂Ay∂z)ax+(∂Ax∂z−∂Az∂x)ay+(∂Ay∂x−∂Ax∂y)az$

$∇×A=(∂(−2xz)∂y−∂(y2z)∂z)ax+(∂(x2y)∂z−∂(−2xz)∂x)ay+(∂(y2z)∂x−∂(x2y)∂y)az=(0−y2)ax+(0+2z)ay+(0−x2)az=−y2ax+2zay−x2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=∂(∇×A)x∂x+∂(∇×A)y∂y+∂(∇×A)z∂z$

$∇⋅(∇×A)=∂(−y2)∂x+∂(2z)∂y+∂(−x2)∂z=0$

Thus, the curl of vector field $A$ is $−y2ax+2zay−x2az_$ and divergence of the curl is $0_$ .

(b)

To determine

The curl of vector field $A$ and divergence of the curl.

(b)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $ρ2zaρ+ρ3aϕ+3ρz2ϕaz$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=(1ρ∂Az∂ϕ−∂Aϕ∂z)aρ+(∂Aρ∂z−∂Az∂ρ)aϕ+1ρ(∂Aϕ∂ρ−∂Aρ∂ϕ)az$

$∇×A=[(1ρ∂(3ρz2)∂ϕ−∂(ρ3)∂z)aρ+(∂(ρ2z)∂z−∂(3ρz2)∂ρ)aϕ+(∂(ρ3)∂ρ−∂(ρ2z)∂ϕ)az]=[0−0]aρ+[ρ2−3z2]aϕ+[4ρ2−0]az=(ρ2−3z2)aϕ+4ρ2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=1ρ∂∂ρ(ρ(∇×A)ρ)+1ρ∂∂ϕ(∇×A)ϕ+∂∂z(∇×A)z$

$∇⋅(∇×A)=1ρ∂∂ρ(ρ×0)+1ρ∂∂ϕ(ρ2−3z2)+∂∂z(4ρ2)=0$

Thus, the curl of vector field $A$ is $(ρ2−3z2)aϕ+4ρ2az_$ and divergence of the curl is $0_$ .

(c)

To determine

The curl of vector field $A$ and divergence of the curl.

(c)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $sinϕr2ar−cosϕr2aθ$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=[1rsinθ(∂(Aϕsinθ)∂θ−∂Aθ∂ϕ)ar+1r(1sinθ∂Ar∂ϕ−∂(rAϕ)∂r)aθ+1r(∂(rAθ)∂r−∂Ar∂θ)aϕ]$

$∇×A=[1rsinθ(∂(0)∂θ−∂(−cosϕr2)∂ϕ)ar+1r(1sinθ∂(sinϕr2)∂ϕ−∂(0)∂r)aθ+1r(∂(r(−cosϕr2))∂r−∂(sinϕr2)∂θ)aϕ]=[1rsinθ(0−sinϕr2)ar+1r(1sinθcosϕr2)aθ+1r(cosϕr2−0)aϕ]=−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ$

$∇⋅(∇×A)=0$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=[1r2∂∂r(r2(∇×A)r)+1rsinθ∂∂θ((∇×A)θsinθ)+1rsinθ∂∂ϕ(∇×A)ϕ]$

$∇⋅(∇×A)=[1r2∂∂r(r2(−sinϕr3sinθ))+1rsinθ∂∂θ(cosϕr3sinθsinθ)+1rsinθ∂∂ϕ(cosϕr3)]=1r2(sinϕr2sinθ)+1rsinθ(0)−1rsinθ(cosϕr3)=sinϕr4sinθ−sinϕr4sinθ$

Thus, the curl of vector field $A$ is $(−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ)_$ and divergence of the curl is $0_$ .

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Answer 3

Question

Chapter 3, Problem 41P

(a)

To determine

The curl of vector field $A$ and divergence of the curl.

(a)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $x2yax+y2zay−2xzaz$ .

Calculation:

Calculate the curl of vector field $(∇×A)$ using the relation.

$∇×A=(∂Az∂y−∂Ay∂z)ax+(∂Ax∂z−∂Az∂x)ay+(∂Ay∂x−∂Ax∂y)az$

$∇×A=(∂(−2xz)∂y−∂(y2z)∂z)ax+(∂(x2y)∂z−∂(−2xz)∂x)ay+(∂(y2z)∂x−∂(x2y)∂y)az=(0−y2)ax+(0+2z)ay+(0−x2)az=−y2ax+2zay−x2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=∂(∇×A)x∂x+∂(∇×A)y∂y+∂(∇×A)z∂z$

$∇⋅(∇×A)=∂(−y2)∂x+∂(2z)∂y+∂(−x2)∂z=0$

Thus, the curl of vector field $A$ is $−y2ax+2zay−x2az_$ and divergence of the curl is $0_$ .

(b)

To determine

The curl of vector field $A$ and divergence of the curl.

(b)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $ρ2zaρ+ρ3aϕ+3ρz2ϕaz$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=(1ρ∂Az∂ϕ−∂Aϕ∂z)aρ+(∂Aρ∂z−∂Az∂ρ)aϕ+1ρ(∂Aϕ∂ρ−∂Aρ∂ϕ)az$

$∇×A=[(1ρ∂(3ρz2)∂ϕ−∂(ρ3)∂z)aρ+(∂(ρ2z)∂z−∂(3ρz2)∂ρ)aϕ+(∂(ρ3)∂ρ−∂(ρ2z)∂ϕ)az]=[0−0]aρ+[ρ2−3z2]aϕ+[4ρ2−0]az=(ρ2−3z2)aϕ+4ρ2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=1ρ∂∂ρ(ρ(∇×A)ρ)+1ρ∂∂ϕ(∇×A)ϕ+∂∂z(∇×A)z$

$∇⋅(∇×A)=1ρ∂∂ρ(ρ×0)+1ρ∂∂ϕ(ρ2−3z2)+∂∂z(4ρ2)=0$

Thus, the curl of vector field $A$ is $(ρ2−3z2)aϕ+4ρ2az_$ and divergence of the curl is $0_$ .

(c)

To determine

The curl of vector field $A$ and divergence of the curl.

(c)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $sinϕr2ar−cosϕr2aθ$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=[1rsinθ(∂(Aϕsinθ)∂θ−∂Aθ∂ϕ)ar+1r(1sinθ∂Ar∂ϕ−∂(rAϕ)∂r)aθ+1r(∂(rAθ)∂r−∂Ar∂θ)aϕ]$

$∇×A=[1rsinθ(∂(0)∂θ−∂(−cosϕr2)∂ϕ)ar+1r(1sinθ∂(sinϕr2)∂ϕ−∂(0)∂r)aθ+1r(∂(r(−cosϕr2))∂r−∂(sinϕr2)∂θ)aϕ]=[1rsinθ(0−sinϕr2)ar+1r(1sinθcosϕr2)aθ+1r(cosϕr2−0)aϕ]=−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ$

$∇⋅(∇×A)=0$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=[1r2∂∂r(r2(∇×A)r)+1rsinθ∂∂θ((∇×A)θsinθ)+1rsinθ∂∂ϕ(∇×A)ϕ]$

$∇⋅(∇×A)=[1r2∂∂r(r2(−sinϕr3sinθ))+1rsinθ∂∂θ(cosϕr3sinθsinθ)+1rsinθ∂∂ϕ(cosϕr3)]=1r2(sinϕr2sinθ)+1rsinθ(0)−1rsinθ(cosϕr3)=sinϕr4sinθ−sinϕr4sinθ$

Thus, the curl of vector field $A$ is $(−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ)_$ and divergence of the curl is $0_$ .

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Answer 4

Question

Chapter 3, Problem 41P

(a)

To determine

The curl of vector field $A$ and divergence of the curl.

(a)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $x2yax+y2zay−2xzaz$ .

Calculation:

Calculate the curl of vector field $(∇×A)$ using the relation.

$∇×A=(∂Az∂y−∂Ay∂z)ax+(∂Ax∂z−∂Az∂x)ay+(∂Ay∂x−∂Ax∂y)az$

$∇×A=(∂(−2xz)∂y−∂(y2z)∂z)ax+(∂(x2y)∂z−∂(−2xz)∂x)ay+(∂(y2z)∂x−∂(x2y)∂y)az=(0−y2)ax+(0+2z)ay+(0−x2)az=−y2ax+2zay−x2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=∂(∇×A)x∂x+∂(∇×A)y∂y+∂(∇×A)z∂z$

$∇⋅(∇×A)=∂(−y2)∂x+∂(2z)∂y+∂(−x2)∂z=0$

Thus, the curl of vector field $A$ is $−y2ax+2zay−x2az_$ and divergence of the curl is $0_$ .

(b)

To determine

The curl of vector field $A$ and divergence of the curl.

(b)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $ρ2zaρ+ρ3aϕ+3ρz2ϕaz$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=(1ρ∂Az∂ϕ−∂Aϕ∂z)aρ+(∂Aρ∂z−∂Az∂ρ)aϕ+1ρ(∂Aϕ∂ρ−∂Aρ∂ϕ)az$

$∇×A=[(1ρ∂(3ρz2)∂ϕ−∂(ρ3)∂z)aρ+(∂(ρ2z)∂z−∂(3ρz2)∂ρ)aϕ+(∂(ρ3)∂ρ−∂(ρ2z)∂ϕ)az]=[0−0]aρ+[ρ2−3z2]aϕ+[4ρ2−0]az=(ρ2−3z2)aϕ+4ρ2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=1ρ∂∂ρ(ρ(∇×A)ρ)+1ρ∂∂ϕ(∇×A)ϕ+∂∂z(∇×A)z$

$∇⋅(∇×A)=1ρ∂∂ρ(ρ×0)+1ρ∂∂ϕ(ρ2−3z2)+∂∂z(4ρ2)=0$

Thus, the curl of vector field $A$ is $(ρ2−3z2)aϕ+4ρ2az_$ and divergence of the curl is $0_$ .

(c)

To determine

The curl of vector field $A$ and divergence of the curl.

(c)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $sinϕr2ar−cosϕr2aθ$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=[1rsinθ(∂(Aϕsinθ)∂θ−∂Aθ∂ϕ)ar+1r(1sinθ∂Ar∂ϕ−∂(rAϕ)∂r)aθ+1r(∂(rAθ)∂r−∂Ar∂θ)aϕ]$

$∇×A=[1rsinθ(∂(0)∂θ−∂(−cosϕr2)∂ϕ)ar+1r(1sinθ∂(sinϕr2)∂ϕ−∂(0)∂r)aθ+1r(∂(r(−cosϕr2))∂r−∂(sinϕr2)∂θ)aϕ]=[1rsinθ(0−sinϕr2)ar+1r(1sinθcosϕr2)aθ+1r(cosϕr2−0)aϕ]=−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ$

$∇⋅(∇×A)=0$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=[1r2∂∂r(r2(∇×A)r)+1rsinθ∂∂θ((∇×A)θsinθ)+1rsinθ∂∂ϕ(∇×A)ϕ]$

$∇⋅(∇×A)=[1r2∂∂r(r2(−sinϕr3sinθ))+1rsinθ∂∂θ(cosϕr3sinθsinθ)+1rsinθ∂∂ϕ(cosϕr3)]=1r2(sinϕr2sinθ)+1rsinθ(0)−1rsinθ(cosϕr3)=sinϕr4sinθ−sinϕr4sinθ$

Thus, the curl of vector field $A$ is $(−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ)_$ and divergence of the curl is $0_$ .

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Answer 5

Chapter 3, Problem 41P

(a)

To determine

The curl of vector field $A$ and divergence of the curl.

(a)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $x2yax+y2zay−2xzaz$ .

Calculation:

Calculate the curl of vector field $(∇×A)$ using the relation.

$∇×A=(∂Az∂y−∂Ay∂z)ax+(∂Ax∂z−∂Az∂x)ay+(∂Ay∂x−∂Ax∂y)az$

$∇×A=(∂(−2xz)∂y−∂(y2z)∂z)ax+(∂(x2y)∂z−∂(−2xz)∂x)ay+(∂(y2z)∂x−∂(x2y)∂y)az=(0−y2)ax+(0+2z)ay+(0−x2)az=−y2ax+2zay−x2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=∂(∇×A)x∂x+∂(∇×A)y∂y+∂(∇×A)z∂z$

$∇⋅(∇×A)=∂(−y2)∂x+∂(2z)∂y+∂(−x2)∂z=0$

Thus, the curl of vector field $A$ is $−y2ax+2zay−x2az_$ and divergence of the curl is $0_$ .

(b)

To determine

The curl of vector field $A$ and divergence of the curl.

(b)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $ρ2zaρ+ρ3aϕ+3ρz2ϕaz$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=(1ρ∂Az∂ϕ−∂Aϕ∂z)aρ+(∂Aρ∂z−∂Az∂ρ)aϕ+1ρ(∂Aϕ∂ρ−∂Aρ∂ϕ)az$

$∇×A=[(1ρ∂(3ρz2)∂ϕ−∂(ρ3)∂z)aρ+(∂(ρ2z)∂z−∂(3ρz2)∂ρ)aϕ+(∂(ρ3)∂ρ−∂(ρ2z)∂ϕ)az]=[0−0]aρ+[ρ2−3z2]aϕ+[4ρ2−0]az=(ρ2−3z2)aϕ+4ρ2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=1ρ∂∂ρ(ρ(∇×A)ρ)+1ρ∂∂ϕ(∇×A)ϕ+∂∂z(∇×A)z$

$∇⋅(∇×A)=1ρ∂∂ρ(ρ×0)+1ρ∂∂ϕ(ρ2−3z2)+∂∂z(4ρ2)=0$

Thus, the curl of vector field $A$ is $(ρ2−3z2)aϕ+4ρ2az_$ and divergence of the curl is $0_$ .

(c)

To determine

The curl of vector field $A$ and divergence of the curl.

(c)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $sinϕr2ar−cosϕr2aθ$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=[1rsinθ(∂(Aϕsinθ)∂θ−∂Aθ∂ϕ)ar+1r(1sinθ∂Ar∂ϕ−∂(rAϕ)∂r)aθ+1r(∂(rAθ)∂r−∂Ar∂θ)aϕ]$

$∇×A=[1rsinθ(∂(0)∂θ−∂(−cosϕr2)∂ϕ)ar+1r(1sinθ∂(sinϕr2)∂ϕ−∂(0)∂r)aθ+1r(∂(r(−cosϕr2))∂r−∂(sinϕr2)∂θ)aϕ]=[1rsinθ(0−sinϕr2)ar+1r(1sinθcosϕr2)aθ+1r(cosϕr2−0)aϕ]=−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ$

$∇⋅(∇×A)=0$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=[1r2∂∂r(r2(∇×A)r)+1rsinθ∂∂θ((∇×A)θsinθ)+1rsinθ∂∂ϕ(∇×A)ϕ]$

$∇⋅(∇×A)=[1r2∂∂r(r2(−sinϕr3sinθ))+1rsinθ∂∂θ(cosϕr3sinθsinθ)+1rsinθ∂∂ϕ(cosϕr3)]=1r2(sinϕr2sinθ)+1rsinθ(0)−1rsinθ(cosϕr3)=sinϕr4sinθ−sinϕr4sinθ$

Thus, the curl of vector field $A$ is $(−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ)_$ and divergence of the curl is $0_$ .

Answer 6

Chapter 3, Problem 41P

(a)

To determine

The curl of vector field $A$ and divergence of the curl.

(a)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $x2yax+y2zay−2xzaz$ .

Calculation:

Calculate the curl of vector field $(∇×A)$ using the relation.

$∇×A=(∂Az∂y−∂Ay∂z)ax+(∂Ax∂z−∂Az∂x)ay+(∂Ay∂x−∂Ax∂y)az$

$∇×A=(∂(−2xz)∂y−∂(y2z)∂z)ax+(∂(x2y)∂z−∂(−2xz)∂x)ay+(∂(y2z)∂x−∂(x2y)∂y)az=(0−y2)ax+(0+2z)ay+(0−x2)az=−y2ax+2zay−x2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=∂(∇×A)x∂x+∂(∇×A)y∂y+∂(∇×A)z∂z$

$∇⋅(∇×A)=∂(−y2)∂x+∂(2z)∂y+∂(−x2)∂z=0$

Thus, the curl of vector field $A$ is $−y2ax+2zay−x2az_$ and divergence of the curl is $0_$ .

Answer 7

Chapter 3, Problem 41P

(a)

To determine

The curl of vector field $A$ and divergence of the curl.

Answer 8

(a)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $x2yax+y2zay−2xzaz$ .

Calculation:

Calculate the curl of vector field $(∇×A)$ using the relation.

$∇×A=(∂Az∂y−∂Ay∂z)ax+(∂Ax∂z−∂Az∂x)ay+(∂Ay∂x−∂Ax∂y)az$

$∇×A=(∂(−2xz)∂y−∂(y2z)∂z)ax+(∂(x2y)∂z−∂(−2xz)∂x)ay+(∂(y2z)∂x−∂(x2y)∂y)az=(0−y2)ax+(0+2z)ay+(0−x2)az=−y2ax+2zay−x2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=∂(∇×A)x∂x+∂(∇×A)y∂y+∂(∇×A)z∂z$

$∇⋅(∇×A)=∂(−y2)∂x+∂(2z)∂y+∂(−x2)∂z=0$

Thus, the curl of vector field $A$ is $−y2ax+2zay−x2az_$ and divergence of the curl is $0_$ .

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

(b)

To determine

The curl of vector field $A$ and divergence of the curl.

(b)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $ρ2zaρ+ρ3aϕ+3ρz2ϕaz$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=(1ρ∂Az∂ϕ−∂Aϕ∂z)aρ+(∂Aρ∂z−∂Az∂ρ)aϕ+1ρ(∂Aϕ∂ρ−∂Aρ∂ϕ)az$

$∇×A=[(1ρ∂(3ρz2)∂ϕ−∂(ρ3)∂z)aρ+(∂(ρ2z)∂z−∂(3ρz2)∂ρ)aϕ+(∂(ρ3)∂ρ−∂(ρ2z)∂ϕ)az]=[0−0]aρ+[ρ2−3z2]aϕ+[4ρ2−0]az=(ρ2−3z2)aϕ+4ρ2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=1ρ∂∂ρ(ρ(∇×A)ρ)+1ρ∂∂ϕ(∇×A)ϕ+∂∂z(∇×A)z$

$∇⋅(∇×A)=1ρ∂∂ρ(ρ×0)+1ρ∂∂ϕ(ρ2−3z2)+∂∂z(4ρ2)=0$

Thus, the curl of vector field $A$ is $(ρ2−3z2)aϕ+4ρ2az_$ and divergence of the curl is $0_$ .

Answer 13

(b)

To determine

The curl of vector field $A$ and divergence of the curl.

Answer 14

(b)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $ρ2zaρ+ρ3aϕ+3ρz2ϕaz$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=(1ρ∂Az∂ϕ−∂Aϕ∂z)aρ+(∂Aρ∂z−∂Az∂ρ)aϕ+1ρ(∂Aϕ∂ρ−∂Aρ∂ϕ)az$

$∇×A=[(1ρ∂(3ρz2)∂ϕ−∂(ρ3)∂z)aρ+(∂(ρ2z)∂z−∂(3ρz2)∂ρ)aϕ+(∂(ρ3)∂ρ−∂(ρ2z)∂ϕ)az]=[0−0]aρ+[ρ2−3z2]aϕ+[4ρ2−0]az=(ρ2−3z2)aϕ+4ρ2az$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=1ρ∂∂ρ(ρ(∇×A)ρ)+1ρ∂∂ϕ(∇×A)ϕ+∂∂z(∇×A)z$

$∇⋅(∇×A)=1ρ∂∂ρ(ρ×0)+1ρ∂∂ϕ(ρ2−3z2)+∂∂z(4ρ2)=0$

Thus, the curl of vector field $A$ is $(ρ2−3z2)aϕ+4ρ2az_$ and divergence of the curl is $0_$ .

Answer 15

(b)

Expert Solution

Answer 16

(b)

Expert Solution

Answer 17

Expert Solution

Answer 18

(c)

To determine

The curl of vector field $A$ and divergence of the curl.

(c)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $sinϕr2ar−cosϕr2aθ$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=[1rsinθ(∂(Aϕsinθ)∂θ−∂Aθ∂ϕ)ar+1r(1sinθ∂Ar∂ϕ−∂(rAϕ)∂r)aθ+1r(∂(rAθ)∂r−∂Ar∂θ)aϕ]$

$∇×A=[1rsinθ(∂(0)∂θ−∂(−cosϕr2)∂ϕ)ar+1r(1sinθ∂(sinϕr2)∂ϕ−∂(0)∂r)aθ+1r(∂(r(−cosϕr2))∂r−∂(sinϕr2)∂θ)aϕ]=[1rsinθ(0−sinϕr2)ar+1r(1sinθcosϕr2)aθ+1r(cosϕr2−0)aϕ]=−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ$

$∇⋅(∇×A)=0$

Calculate the divergence of the curl $(∇⋅(∇×A))$ using the relation.

$∇⋅(∇×A)=[1r2∂∂r(r2(∇×A)r)+1rsinθ∂∂θ((∇×A)θsinθ)+1rsinθ∂∂ϕ(∇×A)ϕ]$

$∇⋅(∇×A)=[1r2∂∂r(r2(−sinϕr3sinθ))+1rsinθ∂∂θ(cosϕr3sinθsinθ)+1rsinθ∂∂ϕ(cosϕr3)]=1r2(sinϕr2sinθ)+1rsinθ(0)−1rsinθ(cosϕr3)=sinϕr4sinθ−sinϕr4sinθ$

Thus, the curl of vector field $A$ is $(−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ)_$ and divergence of the curl is $0_$ .

Answer 19

(c)

To determine

The curl of vector field $A$ and divergence of the curl.

Answer 20

(c)

Expert Solution

Explanation of Solution

Given:

The vector field $(A)$ is $sinϕr2ar−cosϕr2aθ$ .

Calculation:

Calculate the curl of the vector field $(∇×A)$ using the relation.

$∇×A=[1rsinθ(∂(Aϕsinθ)∂θ−∂Aθ∂ϕ)ar+1r(1sinθ∂Ar∂ϕ−∂(rAϕ)∂r)aθ+1r(∂(rAθ)∂r−∂Ar∂θ)aϕ]$

$∇×A=[1rsinθ(∂(0)∂θ−∂(−cosϕr2)∂ϕ)ar+1r(1sinθ∂(sinϕr2)∂ϕ−∂(0)∂r)aθ+1r(∂(r(−cosϕr2))∂r−∂(sinϕr2)∂θ)aϕ]=[1rsinθ(0−sinϕr2)ar+1r(1sinθcosϕr2)aθ+1r(cosϕr2−0)aϕ]=−sinϕr3sinθar+cosϕr3sinθaθ+cosϕr3aϕ$