Chapter 6: Q25MP (page 338)

$\iint V - ndσ$ over the curved part of the hemisphere in Problem $24$ , if role="math" localid="1657355269158" $V = curl (yi - xj)$ .

Short Answer

Expert verified

The Solution to the problem is $\iint_{curved} V \times ndσ = - 18 π$

Step by step solution

Given Information.

$V = curl (yi - xj)$

Definition of Divergence Theorem.

The 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.

Find the solution.

Use Divergence theorem.

$∭_{T} \nabla \times VdT = \iint_{\partial T} V \times ndσ$

Where $\partial T$ is the surface area that encloses the volume $T$ .

$\nabla \times V = \nabla \times (\nabla \times (yi - xj))$

But the divergence of any curve is zero.

Use the triplet scale product.

$\iint V \times ndσ = \iint_{cunved} V \times ndσ + \iint_{plane} V \times ndσ$

$= 0$

By simplifying it is enough to calculate the integral over the plane part.

$\iint_{plane} V \times ndσ = \iint_{disk} 2 dxdy - 2 π (3)^{2}$

$= 18 π$

Thus, the integrated part in this question is

$\iint_{curved} V \times ndσ = - 18 π$

Hence, The Solution to the problem is $\iint_{curved} V \times ndσ = - 18 π$

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$\iint V - ndσ$ over the curved part of the hemisphere in Problem $24$ , if role="math" localid="1657355269158" $V = curl (yi - xj)$ .

Short Answer

Step by step solution

Given Information.

Definition of Divergence Theorem.

Find the solution.

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∬V−ndσover the curved part of the hemisphere in Problem 24, if role="math" localid="1657355269158" V=curl(yi−xj).

Short Answer

Step by step solution

Given Information.

Definition of Divergence Theorem.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Linear Momentum

Quantum Physics

Astrophysics

Electrostatics

Nuclear Physics

Study anywhere. Anytime. Across all devices.

$\iint V - ndσ$ over the curved part of the hemisphere in Problem $24$ , if role="math" localid="1657355269158" $V = curl (yi - xj)$ .