Chapter 6: Q8P (page 323)

Evaluate each of the integrals in Problems $3$ to $8$ as either a volume integral or a surface integral, whichever is easier.
$∭ \nabla \times Vdτ$ over the volumerole="math" localid="1657334446941" $x^{2} + y^{2} \leq 4, 0 \leq z \leq 5, V = (\sqrt{x^{2} + y^{2}}) (ix + jy)$

Short Answer

Expert verified

The solution of the integrals is $∭ \nabla \times V d τ = 80 π$ .

Step by step solution

Given Information.

The given integral is $∭ \nabla \times V d τ .$

Definition of Divergence’s Theorem.

Divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed. According to this theorem, the surface integral of a vector field over a closed surface, also known as the flux through the surface, equals the volume integral of the divergence over the region inside the surface.

Apply Gauss’ Theorem.

Apply Gauss' theorem and use the fact mentioned below.

$V \times \hat{ρ} = ρ^{2},$

$ρ = 2$

In cylindrical coordinates. $d σ = ρ d θ d z$

Solve further as shown below.

$\iint V \times \hat{ρ} = \int_{0}^{5} \int_{0}^{2 π} 2^{3} d θ d z$

$= 8 (2 π) (5)$

$= 80 π$

Hence, the solution of the integrals is $∭ \nabla \times V d τ = 80 π$ .

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Evaluate each of the integrals in Problems $3$ to $8$ as either a volume integral or a surface integral, whichever is easier.
$∭ \nabla \times Vdτ$ over the volumerole="math" localid="1657334446941" $x^{2} + y^{2} \leq 4, 0 \leq z \leq 5, V = (\sqrt{x^{2} + y^{2}}) (ix + jy)$

Short Answer

Step by step solution

Given Information.

Definition of Divergence’s Theorem.

Apply Gauss’ Theorem.

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Evaluate each of the integrals in Problems3to 8as either a volume integral or a surface integral, whichever is easier.∭∇×Vdτover the volumerole="math" localid="1657334446941" x2+y2≤4,0≤z≤5,V=(x2+y2)(ix+jy)

Short Answer

Step by step solution

Given Information.

Definition of Divergence’s Theorem.

Apply Gauss’ Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Fields in Physics

Solid State Physics

Conservation of Energy and Momentum

Turning Points in Physics

Linear Momentum

Study anywhere. Anytime. Across all devices.

Evaluate each of the integrals in Problems $3$ to $8$ as either a volume integral or a surface integral, whichever is easier.
$∭ \nabla \times Vdτ$ over the volumerole="math" localid="1657334446941" $x^{2} + y^{2} \leq 4, 0 \leq z \leq 5, V = (\sqrt{x^{2} + y^{2}}) (ix + jy)$